This advice changed my view of the world. Not only did I realize that being a teenager with a Y-chromosome can't be easy either, it also explained why my male classmates were suddenly developing interests in things like Special Relativity or Scanning Tunnel Microscopes (Nobel Prize '86). It made also sense they were usually very irritated if a girl attempted to join them: all that was just suppressed hormones, the poor guys*. It further revealed a deep connection between General Relativity and potatoes that hadn't previously occurred to me. Most disturbingly however, it labeled General Relativity as unsexy, a fact that has bothered me ever since.

Over the course of years I moreover had to notice that General Relativity is a subject of great mystery to many, it's a word that has entered the colloquial language as the incomprehensible and ununderstandably complicated result of a genius' brain. My physics teacher notably told me when getting tired of my questions that there are maybe three people in the world who understand General Relativity, thereby repeating (as I found out later) a rumor that was more than half a century old (see Wikipedia on the History of General Relativity).

Special and General Relativity is also the topic I receive the most questions about. The twin paradox for example still seems to confuse many people, and only a couple of days ago I was again confronted with a misunderstanding that I've encountered repeatedly, though its origin is unclear to me. The twin paradox is not a paradox, so the explanation seems to go, because it doesn't take into account General Relativity. That's plain wrong. The twin paradox is not a paradox because it doesn't take into account acceleration (unless your spacetime allows closed timelike loops you will have to accelerate one of the twins to get them to meet again which breaks the symmetry).

The problem is that for reasons I don't know many people seem to believe Special Relativity is about constant velocities only, possibly a consequence of bad introductionary textbook. That is not the case. Heck, you can describe acceleration even in Newtonian mechanics! To make that very clear:

- The difference between Special and General Relativity is that the former is in flat space, whereas the latter is in a 'general', curved space.
- Flat space does not mean the metric tensor is diagonal with the entries (-1,1,1,1), this is just the case in a very specific coordinate system. Flat space means the curvature tensor identically vanishes (which is independent of the coordinate system).
- Of course one can describe accelerated observers in Special Relativity.

That leads me now directly to the Equivalence Principle, the cornerstone of General Relativity. Googling 'Equivalence Principle' it is somehow depressing. Wikipedia isn't wrong, but too specific (the Equivalence Principle doesn't have anything to do with standing on the surface of the Earth). The second hit is a NASA website which I find mostly confusing (saying all objects react equally to gravity doesn't tell you anything about the relation of gravitational to inertial mass). The third and fourth hits get it right, the fifth is wrong (the locality is a crucial ingredient).

So here it is:

**The Equivalence Principle:**Locally, the effects of gravitation (motion in a curved space) are the same as that of an accelerated observer in flat space.

That is what Einstein explains in his thought experiment with the elevator. If you are standing in the elevator (that is just a local patch, theoretically infinitesimally small) you can't tell whether you are pulled down because there is a planet underneath your feet, or because there is a flying pig pulling up the elevator. This website has two very nice mini-movies depicting the situation.

If you could make your elevator larger you could however eventually distinguish between flat and curved space because you could measure geodesic deviation, i.e. the curvature.

If you think of particles, the Equivalence Principle means that the inertial mass is equal to the gravitational mass, which has been measured with impressive precision. But the above formulation makes the mathematical consequences much clearer. To formulate your theory, you will have to introduce a tangential bundle on your curved manifold where you can deal with the 'local' quantities, and you will have to figure out how the cuts in this bundle (tensors) will transform under change of coordinates. If you want your theory to be independent of that choice of coordinates it will have to be formulated in tensor equations. Next thing to ask is then how to transport tensors from one point to the other, which leads you to a 'covariant' derivative.

The Equivalence Principle is thus a very central ingredient of General Relativity and despite its simplicity the base of a large mathematical apparatus, it's the kind of insight every theoretical physicist dreams of. It gives you a notion of a 'straightest line' in curved space (a geodesic) on which a testparticle moves. This curve most notably is independent of the mass of that particle: heavy and light things fall alike even in General Relativity (well, we already knew this to be the case in the Newtonian limit). For a very nice demonstration see the video on the NASA website. Please note that this holds for pointlike testparticles only, it is no loger true for extended or spinning objects, or for objects that significantly disturb the background.

The Equivalence Principle however is not sufficient to give you Einstein's field equations that describe how space is curved by its matter content. But that's a different story. It remains to be said all this is standard textbook knowledge and General Relativity is today not usually considered a large mystery. There are definitely more than 3 people who understand it. We have moved on quite a bit since 1905.

**Bottomline**

General Relativity is sexy.

Though I doubt there's more than three people in the world who really understand potatoes.

Footnote:

* In the more advanced stages of confusion they start referring to physical theories as women.

Josh, this one's for you.

## 104 comments:

Thanks for this great post for the weekend!

I really wonder why this story that "only three people understand GR" can be so persistent.

Perhaps it's an easy excuse for not even trying to understand it on the one hand, and can give an elitist feeling to some of those who have mastered it, on the other hand? Anyway, it's a great example of really bad framing!

Curved spacetime is a very cool concept, albeit a bit hard to visualize ;-)

Cheers, Stefan

Thanks a lot, Bee.

It's interesting to note that Lewis Carroll (of

Alice in Wonderlandfame!) actually proposed a form of the equivalence principle in a novel in 1889 - more than two decades before Einstein. Here's the extract:Excerpt from

Sylvie and BrunoWhen I think of general relativity I get a hard on. Does that make me weird??

No, it proves my case that GR is sexier than potatoes. Maybe you should tell Cosmopolitan, it could do wonders for the issue of women in theoretical physics.

Thanks for mentioning Carroll, I didn't know that, that's quite interesting!

Dear Bee,

How far does the Equivalence Principle survive Quantum Field Theory?

-Arun

BTW, typo - "That leads me now directly to the Equivalence Principle, the

fundingstone of General Relativity."I'm afraid I have to say that I really hate the Equivalence Principle. It has caused more unnecessary confusion than almost anything in the subject. For example, you say very truly that

"If you could make your elevator larger you could however eventually distinguish between flat and curved space because you could measure geodesic deviation, i.e. the curvature."

In other words, if you could perform more precise experiments, you *could* distinguish the effects of acceleration and gravity. And this statement is true no matter how small the elevator is. Thus, the "equivalence" of gravity and acceleration is an illusion born of the imprecision of our experimental apparatus. That being the case, is the EP really "fundamental"? Hardly.It just makes some people [not you!] think that they can transform away the curvature tensor. They may not formulate it in this obviously incorrect way, but that is the physical picture they have in mind. And it's wrong.

Better to put things this way. Suppose you are a genius and you have the idea that gravitation is not a "force", merely the result of spacetime curvature. Your idea is that when objects fall, this is merely a manifestation of Newton's *first* law: the path in spacetime is a geodesic. Then some mathematician comes along and shows you an extremely powerful theorem which states that geodesics are determined entirely by their initial position and tangent vector, *and nothing else*. You realise with a shock that this means that your theory can *only* work if the trajectories of particles in gravitational fields are *independent of their composition*. Then Baron L von E comes along and assures you that this is, in fact, true! Amazing! Right, so spacetime is indeed curved. Now to search for the field equations......

The point: you can do all this without ever talking about the EP, without ever getting hopelessly confused about "local flatness", without ever talking about coordinates or "local Lorentz invariance" or any of that junk.

There is a real problem here. I have seen trained physicists who think that "special relativity is not generally covariant" because in SR "you only use partial derivatives, not covariant derivatives"; one argued to me that you can *only* use covariant derivatives in GR because only in GR do you have the EP ! [Apparently these people have never seen Maxwell's equations written in polar coordinates...]

In short, the EP was historically important, but now we can forget it. More than this: the EP is very often used by people who want to pretend that GR is really just something defined on a Minkowskian "background". So we should not just forget about the EP: let's try to stamp it out. :-)

The point: you can do all this without ever talking about the EP...Hindsight is great, fact is nobody did.

Actually, (MTW, Chapter 12) one has to say that a freely falling observer is in a locally Lorentzian frame of reference. If one says the laws of mechanics are locally Newtonian, one ends up with a very different theory.

Dear Arun,

Thanks, I've fixed the typo. You are of course right, what I mean with accelerated observer in flat space is an accelerated observer as described by Special Relativity. Best,

B.

Hi Bee,

`

A much appreciated post as it has been sometime since I’ve seen a discussion of the equivalence principle as it relates to GR. As you indicate it was the key insight that Einstein had and held fast to which greatly aided him in his discovery. That is along with what SR had already mandated, considering time as a true dimension and the discarding of the fifth postulate. Richard Feynman once said that no matter how hard he tried he could never fathom the process Einstein followed that lead to this revelation. However, as you seem to indicate it was the trust of truth in this principle that eventually forced the greater conclusion.

In contrast today the whole idea of large guiding principles or characters of nature is more or less frowned upon. I however find such things revealing and at the same time wondrous. I’m also awestruck that one person introduced two such principles in a single lifetime. The first being of course that the laws of nature are the same or rather indistinguishable for all observers in uniform motion, the consequence of which literally forces one to SR and the one you explained which if insisted upon makes GR almost equally inescapable. For me there has always been a truth born of elegance to such principles that leaves one to consider if perhaps there is no other way to form what we call reality. Its seems however there are many like Dr. Who which insist this is all but an illusion and yet as Einstein I still wonder why it is so stubbornly persistent.

Best,

Phil

Hi Bee/Stefan, this is completely off-topic but I thought fellow bloggers and readers might be interested in a simple puzzle concerning Hubble's law that I've posted on my blog at

http://coraifeartaigh.wordpress.com

The prize is a guest post for the correct answer in simple language!

Hi Bee,

I don't understand the explanation you give for the 'twin paradox'.

The period of acceleration can be made arbitrarily short -- and the effect arbitrarily small, right? An integral over the gamma factor from t0 to t0+delta can be made arbitrarily small, despite the large acceleration.

Some people argue that the fact that the twin has to accelerate at all breaks the symmetry between the two twins and 'solves' the paradox. But I don't buy that at all -- it requires that the universe 'know' whether the twin is going to break the symmetry beforehand.

Cheers,

Orin

Hi Bee,

In regards to the equivalence principle and practical application in relation to its utility as to problem solving I always liked the one involving the helium filled balloon and its action in a moving car. Despite what Dr. Who may say the insight provided by Einstein allows visualization/conceptualization methodology most useful in the solving of problems that previously were either very complicated in their explanation/calculation or not approachable at all.

Best,

Phil

Wikipedia explains it quite well:

"....Special relativity does not claim that all observers are equivalent, only that all observers at rest in inertial reference frames are equivalent. But the space ship jumps frames (accelerates) when it performs a U-turn. In contrast, the twin who stays home remains in the same inertial frame for the whole duration of his brother's flight. No accelerating or decelerating forces apply to the homebound twin.

There are indeed not two but three relevant inertial frames: the one in which the stay-at-home twin remains at rest, the one in which the traveling twin is at rest on his outward trip, and the one in which he is at rest on his way home. It is during the acceleration at the U-turn that the traveling twin switches frames. That is when he must adjust his calculated age of the twin at rest."

How is the Einstein Equivalence Principle different from Newton's equivalence principle? In both, gravitational mass is equal to inerial mass, right?

Hi Arun,

“It is during the acceleration at the U-turn that the traveling twin switches frames. That is when he must adjust his calculated age of the twin at rest.”

Let’s say that while the twin that was traveling away he transmitted the time on his spaceship's clock back to his twin on earth. Do you mean to suggest that only during/after the U turn that (counting in the signals speed and distance) there would be an observable difference? As far as I understand it acceleration is acceleration, whether it be manifest within a straight path or resultant of a continual positional change experienced in turning.

Best,

Phil

Hi Orin,

You don't have to 'buy' it, you can just compute it. It doesn't matter whether you cram the acceleration into a long or short time interval, point is if you accelerate you are not an inertial frame. The twin paradox comes along by saying, well either twin thinks the other is younger, so what happens when they meet again? The point is they can't meet again without either one of them being accelerated which removes the apparent paradox. To then compute the difference in age, you'll have to integrate over the curve of each twin to get the eigentime passed. Exactly how long it is depends on the curve (i.e. it might differ if you accelerate differently). This of course does not require the universe to 'know' in advance what the twins are going to do. Best,

B.

Booker,

The Equivalence Principle says locally the laws are that of Special Relativity. Newton didn't know anything about that. Best,

B.

Einstein's elevator postulates isotropic vacuum in the massed sector. ALL isotropic constructions can be dissected into chiral halves. (Styrofoam balls from Christmas ornaments are better than apples.) Abhay Ashtekar did it for GR by 2002.

Put on two left shoes, close your eyes, try to walk a straight line.

The Equivalence Principle will failfor chemically identical opposite parity mass distributions - without violating any prior observation in any venue at any scale.Sub-nanometer emergent parity divergence requires crystallography not physics. Single crystal alpha-quartz solid spheres (no direction bias) of enantiomorphic space group P3(1)21 (right-handed) opposing P3(2)21 (left-handed) in a parity Eötvös experiment can violate the Equivalence Principle. Somebody should look.

Hi Bee:

A couple of questions. How exactly is the EP stated in the GR equation? Would you say that the EP is a postulate of GR or a prediction? And that the speed of light is a constant is a postulate or a prediction of SR?

If EP is not valid GR is not valid correct?

Cecil,

The equivalence principle amounts to treating inertial mass and gravitational mass as exactly the same thing in general relativity. So all accelerations are curvatures due to the gravitational field, i.e. curved spacetime.

It's difficult to find a good mathematical treatment from a physical perspective, and when you do find the mathematical and physical facts, they aren't always satisfying. Here are some useful online links for the technical details:

Riemann tensor for curvature (acceleration)

Stress-energy tensor (source of gravitation

This page is good for explaining exactly why Einstein's law differs from Newton's: e.g., when Einstein wrote Newton's law in terms of spacetime curvature, he had to add a term containing the trace of the Ricci tensor to satisfy conservation of mass-energy. The effective divergence of the stress-energy tensor must be zero (i.e. the vector sum of gravitational field lines from any mass/energy must be zero). Since it isn't zero, an extra term had to be included in the field equation to make sure that there is zero curvature wherever the divergence of the stress-energy tensor is not zero. This term is the reason why Einstein's field equation predicts twice the deflection of starlight by gravity that you get predicted by Newtonian gravity.

Physically, if a non-relativistic particle or bullet passes by the sun, half of the acquired gravitational potential energy from the approach is used for speeding up the bullet, and half is used for changing the direction (deflecting) the bullet. For a photon moving at c, none of the energy gained from approaching the sun can be used to speed up the photon, so twice as much deflection occurs than would occur for a non-relativistic bullet (100% instead of 50% of the acquired gravitational potential energy gets used to change the direction of the photon).

This is the kind of physics that general relativity delivers: it's a kind of accountancy. Feynman gives a nice explanation of curvature in general relativity in his lectures, pointing out that the Earth's gravitational field makes the Earth's radius contract by 1.5 millimetres or MG/(3c^2).

It's pretty interesting that you can get this contraction from the Lorentz transformation (1 - V^2 /c^2)^{1/2} factor for lengths of moving bodies.

Clearly, in quantum gravity you have exchanges of gravitons between masses. Therefore a moving body will experience front-side graviton interactions which may cause the contraction (possibly like the net air pressure on the nose of a moving aircraft, or the water pressure on the front of a moving ship).

If so, whatever graviton effect causes the contraction of length of moving bodies, will also cause the radial contraction of static masses.

Because of the equivalence principle between inertial and gravitational masses, there should be an equivalence between the contraction you get when moving at relativistic velocity and that you get in a strong gravitational field. One way to relate these is by the fall of a small particle from a long distance in a gravitational field. The velocity gained when a small particle is dropped from an immense distance and falls to the earth's surface (ignoring air drag) is be equal to the escape velocity from the earth's surface. The relativistic contraction of that small particle due to freefall from a very large distance should be identical to the amount of contraction of static mass you get due to gravity at the earth's surface, if inertial and gravitational mass effects are indistinguishable.

If you put the escape velocity law (V^2 = 2GM/r) into the Lorentz contraction, and then expand the result by the binomial expansion, as a first approximation this predicts that gravity contracts length by the amount GM/c^2. However in the case of a moving body only one dimension gets contracted (that in the direction of motion), whereas three dimensions are contracted by gravity effects on static masses, so the average contraction amount per dimension will be one-third of GM/c^2, which gives the result Feynman gives from general relativity in his lecture on curvature in 'Lectures on Physics'.

So it's easy to understand the physics behind the mathematical laws in general relativity. All contractions in relativity are real effects from graviton exchanges occuring between masses.

Curvature is not real at the fundamental (quantum field) level because quantum particles (gravitons) will accelerate masses in a large number of small steps from individual discrete interactions, not as a continuous smooth spacetime curvature. For large masses and large distances, the graviton interactions produce effects that average out to look like a smooth curvature. The argument for 4-dimensional spacetime in general relativity is, as Feynman pointed out, based on the fact that in a gravitational field the radial field lines are contracted (e.g. the earth's radius is contracted by 1.5 mm), but the transverse lines (like the earth's circumference) aren't contracted. So this contraction of radius but not circumference would produce an increase in Pi if Euclidean 3-dimensional space were true. Having 4-dimensional spacetime is justified because it means that you can keep Pi fixed and account for the distortion by having an effective extra dimension appear! One thing I don't like about general relativity is that the source for the gravitational field (the stress-energy tensor) is a continuously variable differential equation system, and we know that mass comes in discrete particles. So all the solutions of general relativity which have ever been done are fiddles, using smoothed distributions of mass-energy. There is no way to get general relativity to work by having discrete particles produce the field: it only works for statistically averaged smooth distributions of matter and energy. You have to assume that the source of a gravitational field is a perfect fluid with no particulate qualities, so it varies smoothly and works with the differential equations being used.

So despite the fact that general relativity has made many accurate predictions, it's mathematical framework is that of classical physics (differential equations for fields), instead of being inherently compatible with quantum fields. It's certainly accurate as an approximation where large numbers of gravitons are involved, but even Einstein himself had very serious reservations on whether continuous field structures were right:

'I consider it quite possible that physics cannot be based on the [smooth geometric] field principle, i.e., on continuous structures. In that case, nothing remains of my entire castle in the air ...'

- Albert Einstein in a letter to friend Michel Besso, 1954.

(Apostrophe error near end of my comment above: "it's mathematical framework" should read "its mathematical framework". Sorry!)

Hi Nige,

A most interesting synopsis, yet I have often wondered why it is insisted by most that since matter and energy as being discrete it should therefore follow that space-time must be the same. Just like the naturals are realized as embedded within the continuum of the reals, as in set theory, why is it so hard to imagine there may be a similar relation between matter/energy and space-time? This of course has strayed off topic as it has little to do with the equivalence principle; that is other then its connection with GR’s space-time. It is also interesting however to note that while Einstein’s theories do set a limit on (inertial) speed at the same time it imposes no such limit on acceleration in the most general sense; that is at least as I understand it.

Best,

Phil

I've always thought a useful way of getting at the equivalence principle is to pose a simple question -

Once SR established the equivalence of frames at rest or in motion at constant velocity, the next question is whether an accelerated frame is equivalent to anything?

Answer: yes;the effects of gravitation are the same as that of an accelerated observer in flat space

Hi Nigel:

I don't think you answered my questions but nice info anyway. If the EP is an absolute valid principle of our universe then string theory (ST) is inavlid from the get go. So back to the original question: EP is an assumption built into GR somehow, but how? If we accept that if EP is valid even for a quantum theory of gravity then ideas like ST are invalid, correct?

Hi Cecil,

A couple of questions. How exactly is the EP stated in the GR equation? Would you say that the EP is a postulate of GR or a prediction? And that the speed of light is a constant is a postulate or a prediction of SR?

If EP is not valid GR is not valid correct?

Which 'GR equation'? I told you in the post above that the EP alone doesn't give you Einstein's field equations in case that's what you're asking. Besides this, please keep in mind that there are always different ways to axiomatize a theory, i.e. different postulates that are amount to the same.

As far as I am concerned the EP in GR is an hypothesis that is experimentally testable. Same with the constancy of the speed of light in SR. Whether or not that's a pre- or postdiction involves a lot of history.

I don't know what you mean with 'not valid'. If the EP would turn out to be violated it would mean we'd have to improve our current theories. GR works incredibly well so far, it doesn't suddenly become non-valid. Newtonian mechanics hasn't become non-valid either with the discovery of SR, it just only holds within certain limiting cases.

Best,

B.

Hi Bee,

It doesn't matter whether you cram the acceleration into a long or short time interval, point is if you accelerate you are not an inertial frame.Do you agree that both twins are in inertial frames before the trip's halfway point? If so, then how does the first leg of the trip figure into the twin's difference in age upon meeting? Same question for the second leg of the trip.

The period during which the twin is in a non-inertial frame can be made arbitrarily short, and I compute it to be zero for arbitrarily fast turnarounds (do you disagree?).

To then compute the difference in age, you'll have to integrate over the curve of each twin to get the eigentime passed.Yes I remember doing that in undergraduate GR class -- in fact this geometrical notion of different paths through space-time (one path has a kink) was the resolution of the twin paradox that we were taught. But I don't buy it -- not within SR alone at least. SR does not provide a prescription for declaring that just because one path has a kink that this means that you can throw out the validity of SR to the first and last legs of the trip, where all frames are inertial. Isn't SR supposed to be valid in inertial frames? Well, both twins are in inertial frames for the first and last legs of the trip, and therefore according to SR for those legs of the trip the aging should be symmetric. And according to SR the time dilation during the non-inertial turnaround can be made arbitrarily small.

I have no problem with SR, and in fact I use it in my work in HEP routinely -- but I also must be honest and say that I think physicists are often far too hand-wavy and cavalier in their explanation of this paradox. Even Einstein vacillated on whether or not he thought GR was necessary to resolve the paradox.

Personally, I find the "just integrate over the paths through space-time" explanation to be OK, but it totally brushes off important subtleties. How does the universe 'know' that one twin took the longer path if there is no preferred frame or fabric from which to measure? Why can't SR be applied as usual during each inertial leg of the trip and pieced together? Where is the math that shows that when trying to stitch those pieces together with the non-inertial piece, that the twin-asymmetric time dilation enters discontinuously as the pieces are joined?

Just saying that the paradox is resolved because one twin is in a non-inertial reference frame for a short period of time is a cop-out -- the other twin can jump around while she is waiting for her sister and be in a non-inertial frame too.

Thanks for bearing with me -- it gets frustrating trying to get a good explanation from someone on this topic; if you stick to your guns you risk being condescended to as an anti-SR crackpot (I don't think you've done that yet, but unfortunately I've learned to expect it).

Cheers,

Orin

Phil

Hi Bee,

With all the talk about the “Equivalency Principle” as to what be its significance and roll in the context of General Relativity I thought I’d offer what the discover himself had to say on the matter sourced from his book “Relativity-The Special and The General Theory”. At first I thought I would simply type it all out from the copy I have then found it all online listed chapter by chapter. So here find chapter twenty entitled "The Equality of Inertial and Gravitational Mass as an Argument for the General Postulate of Relativity."

The first thing that you will notice is that despite the popular perception the thought experiment was not that of a person in an elevator yet rather a person confined to a large chest being hauled upwards by a rope at a constant acceleration. A mute point I know, yet something I thought I’d make note of.

Best,

Phil

Hi Cecil,

You asked: "EP is an assumption built into GR somehow, but how?"

You could say that the EP is built into GR because the fact that the space time is locally and only locally Minkowskian i.e. locally flat (i.e. EP) reveals that the space time geometry is Riemannian because that is the fundamental property of a Riemannian space. GR is based on Riemannian Geometry (curved space time etc).

BR

Sexy....sexy.....and the status quo think scientists are a stuffy, hard-up lot. Humans all of ya.

undestanding potatoes: go to the mid-west.

Hi Cecil again,

I forgot

As for the string theory i don't quite understand the relevance or what you mean. You are comparing apples and oranges here but in general the string can propagate in a flat or in a curved D (higher than 4) dimensional background space-time. The metric is just a background field. So i would say that the last problem of String theory is the EP:-)

Bee (or anyone else) do you see a relevance between the two?

BR

Hi Cecil,

What Bee and others have said is absolutely correct and yet perhaps a novice’s perspective may be of some help. In the pure sense GR is not a theory about gravity as being an aspect or rather consequence of a force, yet rather simply the straightest (in the Lagrangian sense) path available being taken in relation to an inertial body within various places in space-time; which means it describes the action of gravity without assigning it as being a force.

Therefore, it could also be maintained that no matter how the gravitational field (architecture of space-time) is created it is therefore not a direct result of a force. That is not to say that in the case of acceleration force is not being applied, it simply means that the effects of the gravitational field one experiences is not found to be attributable as a aspect of that force yet rather the field that forms in relation to it; which is a subtle yet important difference. It could be said in the strictest sense that any thought as having gravity being assigned attributable to a particle (graviton or string) is inconsistent with this and yet Einstein himself knew full well that GR was not the final word in this regard.

Best,

Phil

I have always read this as "Neither twin is aware the other clock is faster/slower until they compare"

Isn't this objective comparison the logical equivalent of a third observer and obviously a third observer could determine which twin is in motion.

Hi Orin,

Isn't SR supposed to be valid in inertial frames? Well, both twins are in inertial frames for the first and last legs of the trip, and therefore according to SR for those legs of the trip the aging should be symmetric. And according to SR the time dilation during the non-inertial turnaround can be made arbitrarily small.You as many others seem to assume if you make the spacetime region in which acceleration occurs arbitrarily small then this will eventually lead to an arbitrarily small time dilatiation. But in that limit the acceleration becomes arbitrarily large - if you squeeze all acceleration into one moment, you need to have infinite acceleration there. The proper time on the curve is continuous in that limit, it doesn't suddenly drop to zero because now you have two straight lines joint by a point. There is nothing 'hand-wavy' about it, it's a mathematically well defined limiting procedure (for a curve however that is unphysical, so why are we even talking about it?).

Personally, I find the "just integrate over the paths through space-time" explanation to be OK, but it totally brushes off important subtleties. How does the universe 'know' that one twin took the longer path if there is no preferred frame or fabric from which to measure?Because the universe knows how to distinguish an accelerated from a non-accelerated frame. SR tells you that all

inertialframes are relative. You can measure acceleration, take a spring scale with you. The twin who turns around is physically distinguishable from the one who is not accelerated. Best,B.

Hi Cecil,

EP is an assumption built into GR somehow, but how?I explained that in my post: The EP tells you the laws of physics are described in the form of tensor equations that in local frames reduce to those of Special Relativity, i.e. have local Lorentz invariance. I have no clue what your problem is with string theory, as far as I know it is Lorentz invariant. However, as I've said now repeatedly, the EP alone isn't sufficient to give you the right field equations. Best,

B.

Hi Bcarpent,

I think the introduction of a third observer is an unnecessary complication that obscures the issue. It's really not that complicated: the one twin is accelerated, it's not an inertial frame. The one ages more than the other. It is a mystery to me why people are still discussing it. There is no paradox with the twins. Period. End of story.

I really wonder if the twin paradox was never mentioned in introductionary courses whether somebody would even find the situation paradoxical.

Best,

B.

Hi Bee, Giotis:

Thanks for the replies. I guess I am not making myself very clear. So let me try again. The EP, i.e., gravitational mass = inertial mass for all forms of matter, is a postualte that has been verified to an accuracy of what 10^-8 10^-9? Now someone not nowing this fact or supposition could examine the GR equation and deduce this principle. Right or wrong? If right what condition in the GR equation implies the EP?

The only object that could do this seems to be the stress-energy tensor.

ST postulates fields that couple to certain matter but not others. In fact Sean Caroll just has a post concerning "dark matter" only coupling to a fifth force which would be a violation of EP.

The EP does not seem to be a scared principle such as Lorentz invariance (which I think would imply a constant speed of light).

Bee as you say the EP by itself was not sufficient for AE to complete GR and I understand this. (Well I understand that EP is not sufficient but have no clue what the other requirements are. After all it took AE a while to get it right.) But somehow he incorporated the EP into the theory. So working backwards how does one extract the EP from GR?

Hi Cecil,

So working backwards how does one extract the EP from GR?You take the field equations, take the limit of a test particle for the stress-energy tensor (i.e. a point particle that does not significantly disturb the background), you will find the particle moves on a geodesic. The curve is independent of the mass of the particle. It is locally in free fall, i.e. locally SR, that's what the EP says. Best,

B.

Hi Phil,

I can't let poor Galileo be passed over without a comment! You said (much) earlier:

"I’m also awestruck that one person introduced two such principles in a single lifetime. The first being of course that the laws of nature are the same or rather indistinguishable for all observers in uniform motion, the consequence of which literally forces one to SR and the one you explained which if insisted upon makes GR almost equally inescapable."

The principle of relativity goes back at least to Galileo: check Galileo's "Dialogue Concerning the Two Chief World Systems" which got him into a lot of religious hot water. His thought experiment about being below decks in a ship is still excellent today, and led him to relativity well before Albert.

Also, we are far from "literally led to SR" unless we propose what the particular coordinate transformation between the inertial frames is: Galilean, Lorentzian (Now we get SR), or something else.

I tend to have sympathy for "Dr Who" here as regards the equivalence principle. It helped Einstein build the final product, but is now perhaps better regarded as scaffolding that can be taken away. (An analogy might be Maxwell's mechanical visualisation of EM which helped him towards his equations (and apparently annoyed the French who had a bias for pure mathematical abstraction at the time...)).

The EP does remain valid in an appropriate local sense as opposed to Mach's principle which was another powerful guiding principle for Einstein during this time but whose ultimate contribution is dubious.

Incidentally I think that the history of Einstein's development of GR and the subsequent controversy and misunderstandings (many of them Einstein's) should be essential (and fun!) reading for anyone with an interest in Physics.

Now, all this talk about GR has got me over excited –I’m off for a cold shower...

-James

Hi James,

You are of course correct that I overlooked Galileo in my statement as you served to remind. Actually Einstein himself in hindsight wished he had referred to his concept as Invariant Theory rather then relativity. What I should have said is if you couple this with the speed of light as being a constant regardless of the observer’s reference frame then SR becomes practically mandated. Now I realize that one can go the other route as Bell has suggested and take Lorentz, Fitzgerald, Lamour and Poincare in combination to end up in essentially the same place in regards to prediction as long as one is content to resign themselves to admit that somehow nature conspirers to have us unable to perceive the ether while insisting there exists an actual rest frame. For me however this simply appears to be almost entirely phenomenologically based, somewhat ad hoc and more importantly lacks a conciseness and precision that should also have bearing on the issue.

This in part is what I meant by saying that principle based theories are largely frowned upon these days. I’m not insisting I know which is true, yet will admit if this is made certain one of my primary reasons for being interested and enjoying physical science will be lost. To end up with having only found that the world is as is, because that’s how it is, to be as it is, just doesn’t cut it for me. Instead I am hoping that with one or perhaps a few more happy thoughts something more profound and compelling will be discovered.

I also don’t understand how you can so casually dismiss equivalence principle as being nothing more then simply scaffolding, when on its own, without GR it lends insight by providing a method of asking and answering questions simply, with less required complexity of explanation and calculation. This doesn’t just relate to trivial things like the action of helium balloons in moving vehicles, yet to truly important things, such as understanding the nature of black holes.

Best,

Phil

"The twin paradox is not a paradox, so the explanation seems to go, because it doesn't take into account General Relativity. That's plain wrong."

This wrong statement is actually in some physics Textbooks like Tipler, which was our expPhys textbook.

Hi Bee,

You as many others seem to assume if you make the spacetime region in which acceleration occurs arbitrarily small then this will eventually lead to an arbitrarily small time dilatiation. But in that limit the acceleration becomes arbitrarily large - if you squeeze all acceleration into one moment, you need to have infinite acceleration there.Please show me the equation from SR that shows how the acceleration comes into it at all (I don't have any textbooks handy). As I mentioned before (and you didn't correct me), I recall the equation simply being the integral (with respect to proper time) over the gamma factor. Yes, a mathematically well-defined limiting procedure shows that the integral can be made arbitrarily small. This was 'standard SR' that I learned in school using Hartle's textbook. If this is wrong, please let me what equation should be used instead that involves an acceleration term.

Thanks,

Orin

If we are talking about architecture, I would say the process in relation to Einstein was more from the perspective of a keystone and not a cornerstone:)

This concept of Equivalence has been most troubling for me. Of course I would like to thank our hosts for the historical views. I find this most appealing in introduction.

Where in nature could we find such a principal at work, if we see the universe in relations to Lagrangian? Thought experiments and examples of balloons. Nice.

Experimentally, we push the limits to a QGP, and find elements within it that are Relativistic in nature?

Of course we understand the "toposense" may be appealing to an "exotic sensation," but it is just part of the accepts of a model in apprehension and a step forward from a realization.

Curvature parameters in a universe? Then how so such an application to such a energy realization?

Imagine then such a relation existing in the micro-perspective and one in relation to cosmology at large.

So two aspects of temperature in extreme that can hold the very same recognition of relativistic interpretations?

Is such a channel created(equilibrium) in the cosmos? Like a "light house effect" from M87?

Of course, I may have mistaken this principle, and for that, I would be sorry.

Best,

" My physics teacher notably told me when getting tired of my questions that there are maybe three people in the world who understand General Relativity"

1)One fool may ask more questions that thousand clever can answer;

2)Good student know to learn from his teacher.

Bellamy:” understanding potatoes: go to the mid-west.”

We say in Israel:”Rosh batata”.

Orin, Wikipedia again. The computation is done there for the following trajectory:

Let clock K be associated with the "stay at home twin". Let clock K' be associated with the rocket that makes the trip. At the departure event both clocks are set to 0.

Phase 1: Rocket (with clock K') embarks with constant proper acceleration a during a time A as measured by clock K until it reaches some velocity v.

Phase 2: Rocket keeps coasting at velocity v during some time T according to clock K.

Phase 3: Rocket fires its engines in the opposite direction of K during a time A according to clock K until it is at rest with respect to clock K. The constant proper acceleration has the value −a, in other words the rocket is decelerating.

Phase 4: Rocket keeps firing its engines in the opposite direction of K, during the same time A according to clock K, until K' regains the same speed v with respect to K, but now towards K (with velocity −v).

Phase 5: Rocket keeps coasting towards K at speed v during the same time T according to clock K.

Phase 6: Rocket again fires its engines in the direction of K, so it decelerates with a constant proper acceleration a during a time A, still according to clock K, until both clocks reunite.

Orin,

I'm traveling and have no textbooks at hand either. (In fact, I apparently don't even have my passport at hand, which is really bad.) If you want to compute the time dilatation of the accelerated observer, you'll have to introduce an accelerated observer, see keyword 'Rindler observer' and compute his eigentime. The gamma factor won't do. It typically comes with hypergeometric functions, the Wikientry Arun links to looks roughly okay. Best,

B.

Hi Arun,

The wikipedia article you site exemplifies my argument perfectly. For each phase of the trip during which the twins are in inertial frames (phases 2 and 5), the calculation is treated asymmetrically, even though the situation is symmetric. In other words, it is flatly assumed without justification that one twin ages less than the other. For the other phases of the trip, when the traveling twin accelerates, the equation they give for the time dilation plainly gives zero in the limit of large acceleration (assuming constant acceleration, v0 = aA, and a->infinity).

These are the arguments and the equations involving the integral over the gamma factor that I learned as an undergraduate, which seem to me to be lacking in credibility.

Bee's comment after yours seems to get to my point -- I think there must be aspects of SR I have yet to learn that explain the twin paradox in a proper fashion.

Cheers,

Orin

Hi Bee,

Take a look at the wikipedia entry again and see if you agree with me -- it is exactly the integral over the gamma factor I described (well, they integrate over the inverse of the gamma factor). Those integrals go to zero in the limit of infinity acceleration.

It sounded like the 'eigentime' and 'Rindler observer' keywords you mention may lead to an answer I can accept, but so far they look like what I've done in the past, calculating the integral for each twin along a path through space-time. Essentially this is the non-piece-wise equivalent of what is shown in the wikipedia entry. For the same reasons I described earlier, I find this method of computation hand-wavy.

I guess I would be OK if it were said plainly that valid SR time-dilation calculations are the integrals over complete space-time paths, and that it is plainly wrong to talk about the relativity between two inertial frames in a consistent way. This is because it appears that you can have different time-dilation between inertial frames depending on whether they have accelerated in the past. If you give someone a homework problem that says "frame A is moving relative to frame B, what is the time..." the student cannot give an unambiguous answer without knowing the origin of the frame's relative motion -- did frame A accelerate from B, or vice-versa?

Do you get me now?

Cheers,

Orin

Orin,

I'm on the way to the airport. I didn't even read the Wikientry, and though I'm really sorry, I presently don't have the time to. It wouldn't be the first time a Wikientry is wrong (it's probably retyped from a textbook anyway). Yes, why don't you check the Rindler observer and his eigentime in the meanwhile (the Rindler observer is frequently used for the Unruh effect). Best,

B.

Hi Orin,

After a quick look the Wiki calculation seems fine to me. For the accelerated parts it uses the usual Rindler's transformations to calculate proper time. Why is that wrong?

BR

Orin,

The wikipedia article you site exemplifies my argument perfectly. For each phase of the trip during which the twins are in inertial frames (phases 2 and 5), the calculation is treated asymmetrically, even though the situation is symmetric. In other words, it is flatly assumed without justification that one twin ages less than the other.No, the entire computation is treated "asymmetrically" from a single inertial frame, which is that of the non-travelling twin.

For the other phases of the trip, when the traveling twin accelerates, the equation they give for the time dilation plainly gives zero in the limit of large acceleration (assuming constant acceleration, v0 = aA, and a->infinity).You realize that if a -> infinity and A remains finite, the travelling twin achieves light speed and actually ages not at all in the phases 2 and 5; where as nontravelling twin definitely ages?

If a -> infinity and A -> 0 keeping the phase 2, 5 speed v constant, then the travelling clock shows the elapsed time of

Delta t prime = 2T / sqrt(1 + v^2/c^2)

and the stay at home clock shows the elapsed time of

Delta t = 2T

Since the period of acceleration A (as measured by the stay at home twin) tends to zero, yes, the stay at home clock obviously registers zero interval for the period of acceleration.

----

You keep using only the word "time-dilation".

Please be advised that there are two special relativity effects here that are responsible for the twins' different ages -

the first of course is time dilation - a clock in an inertial frame that is moving with respect to my inertial frame appears to run slow.

the second is that simultaneity is also defined only relative to an inertial frame.

This is because it appears that you can have different time-dilation between inertial frames depending on whether they have accelerated in the past. If you give someone a homework problem that says "frame A is moving relative to frame B, what is the time..." the student cannot give an unambiguous answer without knowing the origin of the frame's relative motion -- did frame A accelerate from B, or vice-versa?In special relativity as in galilean relativity, acceleration is an absolute. If a frame accelerated for any period of time, it is not an inertial frame of reference, period.

In special relativity, inertial frames are eternal and unchanging, and space-time is described by the relationships that the infinite set of inertial frames have with each other. A particular observer has a history and could have changed inertial frames many times (which means the observer's history includes acceleration).

To return to this:

...different time-dilation between inertial frames depending on whether they have accelerated in the past.If "the frame" accelerated in the past - or at any time - it is not an inertial frame.

Your statement is as though you're saying you have two rectangular coordinate systems x-y, and u-v in the Euclidean plane, rotated with respect to each other, but the x-u angle is not the same in all places on the plane - and therefore how on earth do we compute anything? Well, what you have is not a pair of rectangular coordinate systems.

Eigentime - or in English usage, proper time - is a geometrical quantity, just like ordinary length in Euclidean geometry. It is independent of coordinate systems; but is often easiest computed in a coordinate system.

The geometry of the Euclidean plane is that the shortest distance is the straight line. The geometry of Lorentzian space-time is that the longest eigentime is the "straight line" - i.e., trajectory of an unaccelerated particle. To one who understands the geometry of Euclidean space, there is no paradox in that the straight line between two points is shorter than two sides of a triangle or indeed any curve joining the two points. To one who understand the geometry of space-time, there is no paradox in that the proper-time between two space time events at a timelike interval is the longest in the trajectory of an inertial observer that participates in both events, and any other observer traces a trajectory with smaller proper time.

Arun,

I respectfully think that you are not reading my argument very carefully.

You realize that if a -> infinity and A remains finite, the travelling twin achieves light speed and actually ages not at all in the phases 2 and 5; where as nontravelling twin definitely ages?Of course. This is why I first wrote that v0 = aA. As a->infinity, A->0.

If a -> infinity and A -> 0 keeping the phase 2, 5 speed v constant, then the travelling clock shows the elapsed time of

Delta t prime = 2T / sqrt(1 + v^2/c^2)

and the stay at home clock shows the elapsed time of

Delta t = 2T

Yes, that is, as I said, if you assume that one frame is inertial while the other is not. Which is in contradiction with the fact that both frames are inertial during the periods for which the equations were derived.

Since the period of acceleration A (as measured by the stay at home twin) tends to zero, yes, the stay at home clock obviously registers zero interval for the period of acceleration.And this was my point -- one can't flippantly say that the twin paradox is due to the acceleration, when the effect due to the acceleration can be made arbitrarily small. What you can say is that because of the acceleration you can decide to call one frame inertial and the other not. I have a problem with this too. What if neither frame is inertial? What if the stay-at-home twin jumps around in joy and makes her frame non-inertial? All I'm saying is that there are subtleties that must be addressed.

You keep using only the word "time-dilation".

Please be advised that there are two special relativity effects here that are responsible for the twins' different ages

You are speaking to the choir here. If I was sloppy with my wording I apologize.

In special relativity as in galilean relativity, acceleration is an absolute. If a frame accelerated for any period of time, it is not an inertial frame of reference, period....

If "the frame" accelerated in the past - or at any time - it is not an inertial frame.

You are making it sound like neither twin's frame could ever possibly be inertial -- they have both accelerated in the past.

I think what you are trying to say is that, starting from when the twins depart from one another, you can't talk about an inertial frame for just one part of the trip (even if it is inertial during that leg) -- a frame must stick with the twin the entire trip and it is only an inertial frame if it is for the entire trip. This is exactly what I was saying to Bee in my last post to her ("I guess I would be OK if it were said plainly that...").

Let me try to make this clear. If you just have twins traveling relative to one another in inertial frames, according to SR you cannot say that there is asymmetric aging. But if one twin decides to turn around, then the previous leg of the trip where we just said there is no asymmetric aging now contributes to it. You can't have it both ways without knowing beforehand whether or not the twin's frame will become non-inertial.

Let's be very clear. I'm not arguing against SR, and I do believe I have a pretty good understanding of it (although one confusion -- I've never heard of 'eigentime' before -- according to your next post this is another word for proper time). But I am arguing against a kind of sloppiness that really irks me.

Cheers,

Orin

Arun,

The geometry of the Euclidean plane is that the shortest distance is the straight line.I agree with everything you say here.

This is consistent with the proviso I described in my post to Bee, which I will quote below:

"I guess I would be OK if it were said plainly that valid SR time-dilation calculations are the integrals over complete space-time paths, and that it is plainly wrong to talk about the relativity between two inertial frames in a consistent way. This is because it appears that you can have different time-dilation between inertial frames depending on whether they have accelerated in the past. If you give someone a homework problem that says "frame A is moving relative to frame B, what is the time..." the student cannot give an unambiguous answer without knowing the origin of the frame's relative motion -- did frame A accelerate from B, or vice-versa?"

Cheers,

Orin

The ratio by which the twins aged when they are again local is the same in all reference frames:

ratio = sqrt(t^2 - x^2 - y^2 - z^2)/t (c=1)

Inertial frames with relative velocities pursue different spacetime paths. The longest path accrues the least time. The is no anomaly until clocks are locally compared. Acceleration is irrelevant.

Three identical clocks are off (or not fabricated) and zeroed. Each clock has a short toggle jigger switch sticking out. Load them (or their parts) into separate spaceships.

CLOCK 1: Our clock. It sits in our inertial reference frame with its jigger sticking out. Touch the jigger and "off" goes "on" or "on" goes "off." Clock 1 is "off" and zeroed.

CLOCK 2: In a spaceship coasting at 0.999c relative to our inertial frame. Clock 2 is "off." It was built after all acceleration ceased, and set to zero. It skims past Clock 1, jiggers touch, Clocks 1 and 2 are "on" (local by touching). Elapsed time accrues in each.

CLOCK 3: As with Clock 2, but 180 degrees opposite direction. Clock 3 is zeroed and "off." It was built after all acceleration ceased, and set to zero. Clocks 2 and 3 later touch jiggers. Clock 2 is "off" Clock 3 is "on." Write down the elapsed time in Clock 2. Clock 3 coasts toward Clock 1.

CLOCK 1: Clock 3 and Clock 1 touch jiggers. All clocks are off. No clock has accelerated while "on" or even while existing. Write down all elapsed times.

BOTTOM LINE: Gather all three slips of paper. Or get results by radio. The numbers are unchanged by transmission. Compare elapsed times. #2+#3 sums to ~4.5% of #1's elapsed time. Thus the Twin Paradox (Triplets) without any running clock having existed during acceleration.

Orin,

Yes, that is, as I said, if you assume that one frame is inertial while the other is not. Which is in contradiction with the fact that both frames are inertial during the periods for which the equations were derived.You can do the computation in the multiple frames of the travelling twin, too; the problem is tying it all together, because you have to synchronize clocks across all these frames. It is much more difficult to do. You will get the same answer. From the point of view of the travelling twin, yes, less time elapses for the stay at home twin during phases 2 and 5. But that is more than compensated for by the time that elapses on the stay at home twin's clock as per the travelling twin during the acceleration phases.

It is simply much easier to do the computation in one particular inertial frame.

Sigh, I'll try one last time.

You are making it sound like neither twin's frame could ever possibly be inertial -- they have both accelerated in the past.All we need for this is that the twins start out in the same inertial frame and end up in the same inertial frame and one remains in the same inertial frame during the interval from start to end. The twin that changes inertial frames during this start to end process ends up younger.

I think what you are trying to say is that, starting from when the twins depart from one another, you can't talk about an inertial frame for just one part of the trip (even if it is inertial during that leg) -- a frame must stick with the twin the entire trip and it is only an inertial frame if it is for the entire trip. This is exactly what I was saying to Bee in my last post to her ("I guess I would be OK if it were said plainly that...").As I wrote, the inertial frames are eternal. The travelling twin's frame is switching between inertial frames. I don't understand what you mean by "a frame must stick with the twin the entire trip"; the twin is carrying rulers and clocks and always has a frame of reference.

Let me try to make this clear. If you just have twins traveling relative to one another in inertial frames, according to SR you cannot say that there is asymmetric aging.Each measures 100 ticks on the atomic clock he is carrying, and marks a space-time event, the 100tickversary. Each says the 100tickversary of the other twin happened later than his own.

But if one twin decides to turn around, then the previous leg of the trip where we just said there is no asymmetric aging now contributes to it. You can't have it both ways without knowing beforehand whether or not the twin's frame will become non-inertial.One twin starts to turn around at his 100tickversary. As per the turn around twin, the other's 100tickversary hasn't yet occurred. But when he finishes turning around, he finds not only the other's 100tickversary is over, but considerable additional time has elapsed on the other's clock.

Let's be very clear. I'm not arguing against SR, and I do believe I have a pretty good understanding of it (although one confusion -- I've never heard of 'eigentime' before -- according to your next post this is another word for proper time). But I am arguing against a kind of sloppiness that really irks me.There is no sloppiness here. You can draw space-time diagrams. Or you can do an equivalent exercise in the Euclidean plane, where it is not paradoxical to anyone that the length of the projection of a line segment onto another (non-parallel) line is less than the length of the line segment (this is symmetric and is the equivalent of no asymmetrical aging), and yet, "paradoxically", the triangle inequality holds.

Arun,

Thanks for trying to explain this to me, despite our mutual frustration.

There is no sloppiness here. You can draw space-time diagrams. Or you can do an equivalent exercise in the Euclidean plane, where it is not paradoxical to anyone that the length of the projection of a line segment onto another (non-parallel) line is less than the length of the line segment (this is symmetric and is the equivalent of no asymmetrical aging), and yet, "paradoxically", the triangle inequality holds.I am not arguing against the triangle inequality. I am arguing against the construction of the triangle.

I don't think that constructing a triangle (like the one shown on the twin paradox wikipedia page) is consistent with a rigorous (perhaps pedantic) interpretation of SR. I thought that SR does not accept the idea that there is any ontological validity to the idea of an underlying space-time; that all coordinates and coordinate-based variables must be measured relative to other objects. If this is true, then the universe has no underlying space-time on which paths can form a triangle -- the only information available is relative motion. If the relative motion is alone taken into account, there is nothing to distinguish one twin's inertial frame from the other, and the effect of the turnaround can be made arbitrarily small, so therefore the situation would be symmetric. Perhaps this view of mine about SR is wrong, but I had been under the impression that GR reintroduced the space-time as an independent entity. If all of you are under a different impression, then I withdraw my complaint about the explanation of the twin paradox.

Cheers,

Orin

vgprmbnPhil said: "In contrast today the whole idea of large guiding principles or characters of nature is more or less frowned upon."

Would that it were so! Unfortunately, it isn't. Pauli is supposed to have said: "Elegance is for tailors." I would like to say: "Principles are for priests". Also: "History is for historians." We don't see historians teaching physics to their students, so why do we have to teach history to physics students --- invariably getting it all wrong into the bargain? And what are "principles" anyway? Judging by the examples we have before us, I would like to define a "Principle" as "an irrelevant, misleading, or [as in Mach's principle] wrong statement which was historically important as great men struggled with their misunderstandings and which lives on to haunt later generations."

The long discussion here of the Twins is a perfect illustration of the harm that "Principles" can do. If you just look at the spacetime diagram you see that one of the twins has a straight worldline, and the other one is bent. Bent and straight are really different, even in California. Technically: one curve is a geodesic, one is not. No amount of blather about coordinates or "principles of relativity" etc etc etc can change that fact: you *cannot* map one onto the other while respecting the geometry of Minkowski space. So they are different, and the bent one has [of course] a different length. End of story.

The symmetries of Minkowski space [which is all the Lorentz transformations are] are an interesting but ultimately not very important aspect of its geometry. By over-emphasising their importance one is blocked from really understanding GR [where, generically, such things do not exist or exist in a much reduced form, eg in FRW geometries] and one is led straight into all kinds of unnecessary confusions.

Finally, at the risk of causing more confusion, it is NOT actually necessary for both twins to accelerate: they might be living in a torally compactified space! Then they meet again without anyone doing any accelerating. Yet one of them is still older. This is not a paradox of course, and the solution is easy to understand *if* you can get over all these hang-ups about "relativity", all of which are historical oddities of no real importance, relics of the past which do far more harm than good.

Hi Orin,

“If this is true, then the universe has no underlying space-time on which paths can form a triangle -- the only information available is relative motion. If the relative motion is alone taken into account, there is nothing to distinguish one twin's inertial frame from the other.”

“I had been under the impression that GR reintroduced the space-time as an independent entity. If all of you are under a different impression, then I withdraw my complaint about the explanation of the twin paradox.”

I am under impression that you are the only one here who close to understanding SR and GR. However, the proper math framework to describe all that is not formulated yet.

Hi Dr Who,

I acknowledge your view point and yet at the same time not your arguments or reasoning. I’m also aware that to debate the issue would more or less prove to be fruitless. In place of it I will simply state that in my own perception most of the current theorists agree with your perspective rather then mine and as a result find themselves in the position we are largely in today; that is one with fewer sign posts and logical restrictions which result in more and more untestable divergent theories, providing less and less predictive ability/utility. I believe there is a correlation to be found here, while you do not and as such unfortunately only time will tell. This of course would then be history, which from your perspective serves simply as a record, while in mine it may also be understood as a lesson.

Best,

Phil

Dear Phil,

I am willing to concede that a genius like Einstein has to be allowed to make mistakes in the course of his struggles. What I am not happy about is people re-enacting those mistakes here and now. I can imagine Einstein in Prague in 1912 writing up his notes, and those notes will be full of rubbish about Mach's principle and "general covariance", and that's fine by me. What is not fine is some obnoxious undergraduate at Charles University dusting off those notes 80 years later and thinking that by reading them he will really understand GR. He doesn't. All I am saying is that our understanding of these things has progressed since 1912, and if we insist on treating Einstein's ideas of that time as gospel we are unlikely to be able to go beyond them. Indeed, I would say that our current slow rate of progress is due to being mired in these archaic misunderstandings and the things that have grown out of them ["gravity is nothing but spin-2 excitations propagating on flat spacetime", etc etc etc --- all that nonsense.]

Anyway no hard feelings I hope.

Dr Who,

You are awfully sure of your self, and yet even Einstein himself vacillated regarding the nature of the solution to the twin paradox. You may be completely right in that Einstein was very much 'all about the principles', and maybe he, like others, clung too tightly to the principles that lead him to the theory in the first place. Nonetheless, you make sweeping statements that utterly trivialize many of the arguments and concerns of even the founder of SR. You may be right, but I think, who are you to condescend to Einstein?

You say:

No amount of blather about coordinates or "principles of relativity" etc etc etc can change that fact: you *cannot* map one onto the other while respecting the geometry of Minkowski space. So they are different, and the bent one has [of course] a different length. End of story.You are picturing space-time like it is some kind of sheet of paper on which you can draw world lines and compare their projections on one-another. This represents an implicit assumption. If instead all that fundamentally can be said is the statement "twin A meets twin B at proper time X1, twin B accelerates at proper time X2, twin A meets twin B at proper time X3" then you have to do some real thinking to be sure that you can declare that the situation is equivalent to yours. If you try to resolve the paradox by rigorous calculation with only the above information I contend that implicit non-well-motivated assumptions enter into your argument. Nonetheless, I do expect there to be *a* solution to the paradox that assumes nothing but the relative motion of A and B and derives asymmetric aging as a consequence of the acceleration phase with a contribution that cannot be made arbitrarily small by making the turnaround arbitrarily fast.

This may be as easy as recasting the resolution of the twin paradox to be purely due to the continuous rotation of the 'plane of simultaneity' at the turnaround point. Put this in mathematical language in the form of a continuous function of the twin's mutual clock comparisons along lanes of simultaneity as a function of each's proper time, and I will be happy.

Cheers,

Orin

Dr Who,

I can imagine Einstein in Prague in 1912 writing up his notes, and those notes will be full of rubbish about Mach's principle and "general covariance"I think you hit the nail on the head. And again you may very well be right in your assessment here, but you have to forgive me for reading Einstein's writings and getting certain impressions. These misunderstandings were certainly never cleared up in my physics education, nor in my textbooks, which I guess is a shame.

Cheers,

Orin

Hi Dr. Who,

In as what is being discussed is science, hard feelings has never served as being a recognized parameter :-) On the other hand I would say that conviction has. Also, it might be that my understanding of a principle and you own may differ. You seem to feel that a principle is some how or other simply dogma, while I see it as merely a testable and thereby falsifiable conviction or axiom if you will.

Take Fermat’s principle of least action as an example, which at first also was largely frowned upon for reasons similar to what you express. Yes, it was altered as to be later refined and expanded by those like Lagrange, Maupertuis, Euler and Leibniz, yet did this truly alter the implication or value of the insight as first raised?

The way I see such things is they should not be dismissed unless they have been proven to be false, as science so dictates and not simply because they are old and haven’t been extended for a period of time. To use your own argument when referring to Pauli’s dismissal I would say fashion should be left to the tailors and dressmakers while elegance as seen through the importance/relevance of symmetry, order and economy be maintained until proven worthless.

Best,

Phil

You may be right, but I think, who are you to condescend to Einstein?Orin, all of us who take Quantum Mechanics to be a complete theory condescend to Einstein.

That is at the essence of science and mathematics. While Einstein was at a pinnacle of achievement, he was not at the pinnacle of understanding.

I thought that SR does not accept the idea that there is any ontological validity to the idea of an underlying space-time; that all coordinates and coordinate-based variables must be measured relative to other objects. If this is true, then the universe has no underlying space-time on which paths can form a triangle -- the only information available is relative motion. If the relative motion is alone taken into account, there is nothing to distinguish one twin's inertial frame from the other, and the effect of the turnaround can be made arbitrarily small, so therefore the situation would be symmetric. Perhaps this view of mine about SR is wrong...Yes! (it is wrong).

The one correct statement is that there is nothing in spacetime on which to anchor coordinates - any more than there is in the Euclidean plane.

SR does not accept the idea that there is any ontological validity to the idea of an underlying space-time.Actually, to be frank, I don't know what the above sentence means, so I cannot say whether it is right or wrong. What I can say it is irrelevant to understanding how special relativity works.

Pinnacle acknowledges a process that is inherent in our exchange with nature.

Robert Lauglin:Physicists have always argued about which kind of law is more important - fundamental or emergent - but they should stop. The evidence is mounting that ALL physical law is emergent, notably and especially behavior associated with the quantum mechanics of the vacuum.Hi Arun,

“Orin, all of us who take Quantum Mechanics to be a complete theory condescend to Einstein.”

In as I agree and am certain that like GR QM will have to be accommodated within what proves to supersede them I’m happy to find I have a valid option.

“The one correct statement is that there is nothing in spacetime on which to anchor coordinates - any more than there is in the Euclidean plane.”

In as Witten and Smolin would share differing and opposing positions in regards to your discovery I think they also may be interested in reviewing your evidence.

Best,

Phil

String theorists like this emergent process in relation to the condensed matter theorist point of view?

While cosmology likes the interpretation of curvature in such a relativistic way we still have to deal with the incompatible nature of such an application at the most earliest times of our universe.

You have to look for this Equivalence process in nature. To discover it.

This process then is an "ancient one" in that an inductive/deductive approach "is" this relationship between the interior and the exterior world.

Arun,

Orin, all of us who take Quantum Mechanics to be a complete theory condescend to Einstein.We are not talking about QM here. We are talking about SR.

Mathematical ProblemsLecture delivered before the International Congress of Mathematicians at Paris in 1900

By Professor David Hilbert

While insisting on rigor in the proof as a requirement for a perfect solution of a problem, I should like, on the other hand, to oppose the opinion that only the concepts of analysis, or even those of arithmetic alone, are susceptible of a fully rigorous treatment. This opinion, occasionally advocated by eminent men, I consider entirely erroneous. Such a one-sided interpretation of the requirement of rigor would soon lead to the ignoring of all concepts arising from geometry, mechanics and physics, to a stoppage of the flow of new material from the outside world, and finally, indeed, as a last consequence, to the rejection of the ideas of the continuum and of the irrational number. But what an important nerve, vital to mathematical science, would be cut by the extirpation of geometry and mathematical physics! On the contrary I think that wherever, from the side of the theory of knowledge or in geometry, or from the theories of natural or physical science, mathematical ideas come up, the problem arises for mathematical science to investigate the principles underlying these ideas and so to establish them upon a simple and complete system of axioms, that the exactness of the new ideas and their applicability to deduction shall be in no respect inferior to those of the old arithmetical concepts.Arun,

Actually, to be frank, I don't know what the above sentence means, so I cannot say whether it is right or wrong. What I can say it is irrelevant to understanding how special relativity works.It basically means what you yourself said in your preceding comment:

The one correct statement is that there is nothing in spacetime on which to anchor coordinates - any more than there is in the Euclidean plane.I believe this to be relevant. I have described my concern in detail, which you have chosen to not respond to directly.

Hi Bee:

Just another comment concerning the EP. The weak EP equates inertial mass to gravitational mass, correct? Which implies universal free fall for all objects irresepctive of composition.

Now we also have E = MC^2 which I believe is a consequnce of SR, correct? The question is: IF, and I repeat, IF the WEP is not valid then what mass is used in the equivalence equation?

I have been fumbling around trying to articulate a basic question: If the WEP is not valid does this not have consequences beyond GR? The whole question of what is mass is kinda ill defined. How does one actually measure mass anyway? Energy?

Orin,

The one correct statement is that there is nothing in spacetime on which to anchor coordinates - any more than there is in the Euclidean plane.I believe this to be relevant. I have described my concern in detail, which you have chosen to not respond to directly.I believe this

isresponding to your point - just as in the Euclidean plane, the lack of a preferred coordinate system doesn't mean we can't draw a triangle.Hmmm... just thinking out loud here.

The implications of "Gauss's mountain" was a preview of the departure of what the triangle was to mean. Geometries in non-euclidean then became recognizable.

Greater then, or less then, a 180 degrees.

Experimentally, this became a basis for exploration which implanted experimental choice departures from euclidean space, which is flat.

This was a mathematical adventure of "pure thought" toward the process of interpreting this mathematics in the natural world.

How could one say the experimental process was first when such thought had to exist? It had to already exist in nature for us to test the theorization by definition.

So we emulate the process by experimentation. Oh sweet "spooky action at a distance?"

Please feel free to correct the logic/illogical....under the "arch application.":)

Scattering amplitudes and entanglement were progressive techniques. How one may see this light now used, going to a fifth dimensional perspective.

Welcher Weg (which way?)

Cecil Kirksey: Even though my involvement in that question about EP operating on different 'types' of matter was about 15 yrs ago (I've not kept up with the latest research), I don't think the answer is clear yet. See my old comment on cosmicvariance and a more recent post by Mark at the same blog.

Descendants?

Reflections on Relativity

In any case, it seems reasonable to agree with Buhler, who concludes in his biography of Gauss that "the oft-told story according to which Gauss wanted to decide the question [of whether space is perfectly Euclidean] by measuring a particularly large triangle is, as far as we know, a myth."Bee, your outline of the Equivalence Principle fits in with my understanding (based on middle-brow explanations) that a "uniform" and parallel gravity field is just like the environment in an elevator with constant (proper) acceleration. Even then I came to appreciate the need to adjust for "Rindler coordinates" arising from the Lorentz contraction of the accelerating framework. That led to hyperbolic motion of the frame and position-dependent proper acceleration:

a_0 = X/c^2. The question I'm developing to is whether the field around an extended planar mass is like that of the "elevator" in EP explanations.

Hence I liked to play with various scenarios in Rindler "elevators" to see if odd things happened in g-fields. One thing I came up with (original? - in 1979) was that a gyroscope being moved rapidly across the floor of a RE would precess (due to the gyroscope orientation being "Thomas precessed" relative to the standard of floor orientation.) Has anyone heard about that?

I always assumed (makes an ASS out of U and ME!) that RE field would also define g-field environment above a very extended (not infinite!) planar mass distribution. But I got into a big argument in the Cosmic Variance thread at http://cosmicvariance.com/2007/11/13/arxiv-find-universal-quantum-mechanics/#comments about issues like gravitomagnetism and the manner which objects, starting out very past and parallel to the planar mass, fall to the floor. I thought I could use the intuitive EP/RE understanding, but was told by Greg Egan there that I could not (if you can skim that thread it would help to get the point.) Significant "money quote" from Greg:

>

The metric I get from the weak-field approximation described in Misner, Thorne & Wheeler’s Chapter 18, applied to a planar mass distribution of area mass density sigma, for a square mass of half-side-length H, is:

ds^2 = (-1+h) dt^2 + (1+h) (dx^2 + dy^2 + dz^2)

where h = ...

...

This space-time is not flat. It has a Riemann curvature tensor with components that are first-order in sigma (as opposed to the Einstein and Ricci tensors, which are second-order in sigma, i.e. zero in the linearised approximation — as they must be in order to be vacuum solutions).

> [end quote since I don't have time for HTML games]

OK, quick scoop and challenge: Is it really true that the g-field above an extended planar mass (intuitively expected to be like the elevator field to the extent we don't move vertically enough to get the hyperbolic variations in "g") really can't be modeled by the Einstein/Rindler elevator field of common understanding, after all? If so, what implications and notable exceptions and outcomes? Thanks, and for your patience also.

Hi Neil,

Well, I think you've answered your own question. A spacetime either has a curvature or it hasn't. I don't know how you want to 'model' a spacetime with curvature by one without. As to the differences, I don't know, I'd compute the geodesics and see what they do. Best,

B.

Hi Cecil,

For one, the way I have stated the EP it doesn't speak about inertial and gravitational masses. As to your question about the definition of gravitational mass, for that you'd need the field equations. To begin with please note it is generally not only a mass that gravitates but the stress-energy tensor, it is just for testparticles in the appropriate limit that one has this simple case. If you couple something to gravity, the field equations will tell you how it couples, and that then how tells you what the gravitational mass (energy) is. Roughly speaking you'd have to compare the graviational to the kinetic energy tensor (which is usually the same due to the EP). I don't see how the definition of gravitational mass would have any consequences except for gravity. Best,

B.

Anonymous,

I believe this is responding to your point - just as in the Euclidean plane, the lack of a preferred coordinate system doesn't mean we can't draw a triangle.Ah, yes. I now realize I misinterpreted Arun's statement. What I am saying:

My interpretation of SR was that 'lack of a prefered coordinate system' went beyond what that of the Euclidean plane, in the sense that you

can'tdraw a triangle. I guess the property of not being able to draw a triangle could itself be taken as a the expression of my interpretation of space-time in SR. This interpretation goes hand in hand with Einstein's original 'Machian' ideas and motivation for SR, and one of the reasons I have always found SR beautiful. In other words, reality is defined by the motion of objects relative to each other, not relative to some 'objective' coordinate system. Apparently, as Arun points out, this view may be dated and now naive. I don't know better, as every textbook on SR I have read seems to indicate that the interpretation of SR has not improved since its inception and does nothing to dispel the notion that Einstein was right about the motivations for his theory.Cheers,

Orin

Hi Orin,

What Arun and Dr. Who patiently told you is correct. You apparently also did not quite understand what I initially said. I said the proper time on the curve is continuous in the limit when you squeeze all the acceleration into one point, not about the limit of the point itself. This limiting situation is not identical to having one twin moving away from Earth and somebody else who never was accelerated moving back towards Earth again in some triangle.

Dr Who actually said it much better actually than I did: the one curve is a geodesic, the other one isn't. Period. They are not equivalent and can't be made such. The one observer is inertial, the other one isn't. The connection of both lines in that point you dislike does matter. As I said already above though I don't know why you're hanging yourself up on this unphysical limit with infinite acceleration in one point - just because the limit looks unintuitive? There is nothing 'hand-wavy' here, there are no 'subleties' and there is no paradox.

Your 'argument' that sweeping away an apparent paradox that was interesting a century ago is 'condescending' to the founder of the theory is ridiculous. For one, you use it to make your 'worries' more serious, a tactic that's called 'appeal to authority' which doesn't have any scientific merit, so please omit it. Second, if we were to argue with the worries and misunderstandings of deceased people we'd never make any progress.

Best,

B.

hmm, though maybe it should have been 'proper time of the curve' and not 'on the curve'? Sorry if that was confusingly formulated.

Hi Bee,

The connection of both lines in that point you dislike does matter. As I said already above though I don't know why you're hanging yourself up on this unphysical limit with infinite acceleration in one point - just because the limit looks unintuitive? There is nothing 'hand-wavy' here, there are no 'subleties' and there is no paradox.Wow. No. This is exactly what I meant earlier when I said that if one tries to press some of these points they end up being condescended to with dismissive blanket statements like "there are no subtleties". Also, as I have repeatedly said, I don't in the slightest think there is a paradox at all. I'm pressing a point about an explanation for the paradox, not the paradox itself. Finally, no I'm not pressing the point about the infinite acceleration because the limit looks unintuitive -- it doesn't look unintuitive to me -- I am doing it for simplicity, so as to remove the mathematical contribution due to the accelerating leg of the trip.

Your 'argument' that sweeping away an apparent paradox that was interesting a century ago is 'condescending' to the founder of the theory is ridiculous. For one, you use it to make your 'worries' more serious, a tactic that's called 'appeal to authority' which doesn't have any scientific merit, so please omit it.I disagree. First of all, did you read Arun's posts? They seemed condescending to me (of course you're not a good person to be appealing to given that I think your last post was rather condescending), just from anyone's perspective. I brought Einstein into my statement to be diplomatic -- I was basically saying, "if I were Einstein, would you really formulate your arguments so flippantly?" Perhaps I was appealing to authority, but more to make a personal point about discourse rather than content.

Anyways, this is frustrating, and the conversation should probably be dropped. I can only say that it is unfortunate given that I simply have a genuine curiosity about this matter that I would like to satiate as best I can.

Cheers,

Orin

Hi Orin,

Well, I am genuinely sorry if I misunderstood you. It seemed to me you were trying to say there's something funny with the explanation we've been trying to give you. If not, then I think the issue is settled anyway. Thanks for your interesting comments. Best,

B.

OK thanks Bee. For anyone who wants to try, this is perhaps the more cogent question:

whyis the allegedly curved g-field (if G. Egan was right ...) of an extended planar mass not like the elevator flat field?PS: thanks andrew for excerpt from Lewis Caroll about the EP. Perhaps even more "amazing", Edgar Allen Poe first came up with the modern-like explanation of how to avoid the Olbers' paradox in 1848.

From http://en.wikipedia.org/wiki/Olbers%27_paradox

Edgar Allan Poe was the first to solve Olbers' paradox when he observed in his essay Eureka: A Prose Poem (1848):"Were the succession of stars endless, then the background of the sky would present us a uniform luminosity, like that displayed by the Galaxy – since there could be absolutely no point, in all that background, at which would not exist a star. The only mode, therefore, in which, under such a state of affairs, we could comprehend the voids which our telescopes find in innumerable directions, would be by supposing the distance of the invisible background so immense that no ray from it has yet been able to reach us at all."[1]

However, IIRC there is an odd mathematical paradox in which one can evade the Olbers paradox by having ever more attenuated "hierarchies" of stars and galaxies, so the integral of star surface interception going out to infinity actually converges - true?

OK, better worded: Why is the field from an extended planar mass curved (if it is, and in space-time in the net since of course it is still a parallel field) rather than the flat field of the Rindler elevator? tx

Orin: "This interpretation goes hand in hand with Einstein's original 'Machian' ideas and motivation for SR, and one of the reasons I have always found SR beautiful. In other words, reality is defined by the motion of objects relative to each other, not relative to some 'objective' coordinate system."

Orin, this is GR you are talking about not SR. In GR this is formulated with the principle of general covariance i.e. GR is invariant under active diffeomorphisms. This means that it doesn't need a background or a frame if you like. You could say that the Gravitational field is the background. BUT SR still needs an inertial frame extended throughout all space time to work. During the turnaround of the twin he changes inertial frame. That's the whole point.

I think that is cause of all the misunderstandings.

Bee: Could you please clarify the thing about the Wiki calculation. Do you find it wrong? To me it looks fine.

BR

There was no intent to be condescending and I only apologize for where I added to the confusion.

-Arun

Arun: "Sigh, I'll try one last time."

I don't know Arun, this seemed pretty intentional to me, to say the least.

Proper time to quote John Archibald Wheeler. Well, somebody's got to.

"Matter tells spacetime how to curve; spacetime tells matter how to move."

As to Poe and Dodgson, I'll add that Jonathan Swift wrote that Mars had two small moons, one so close that it rises in the West and sets in the East, many years befor Asaph Hall discovered Phobos and Deimos.

Finally, I'm biased because I've loved my wife since I met her about 24 years ago, but perhaps the sexiest thing about her is that she is a Physics professor.

-- Prof. Jonathan Vos Post

http://magicdragon.com

“Orin, all of us who take Quantum Mechanics to be a complete theory condescend to Einstein.”

I don’t know what you have in mind, but if you claim “non-relativistic” QM at that time (without Principle of Local Gauge Invariance) was complete and that QT is complete today you are hopelessly naïve.

“That is at the essence of science and mathematics.”

I guess you condescend to D.Hilbert too.

Robert Lauglin:”Physicists have always argued about which kind of law is more important - fundamental or emergent - but they should stop. The evidence is mounting that ALL physical law is emergent, notably and especially behavior associated with the quantum mechanics of the vacuum.”

A. Einstein started his “Note about QT” (Electrons et photons, 5-th Solvay) with the statement that he didn’t contribute essentially to QM. Obviously it is self requirement rank 1א physicist (L.D. classification). One should contribute at least on level M. Gell-Mann or/and R.P. Feynman to understand that he is not self appointed God.

Hi Orin,

“We are not talking about QM here. We are talking about SR.”

No. E. Schrödinger: “Matter stands much the same with another system, the electromagnetic field. Its laws are "relativity personified", a non-relativistic treatment being in general impossible.” See also F.J. Dyson “Feynman’s proof of the Maxwell equations”, Am. J. Phys., 58, 209 (1990).

P.S. I am not anonymous at 9:13 AM, August 18, 2008.

D.Hilbert:” On the contrary I think that wherever, from the side of the theory of knowledge or in geometry, or from the theories of natural or physical science, mathematical ideas come up, the problem arises for mathematical science to investigate the principles underlying these ideas and so to establish them upon a simple and complete system of axioms, that the exactness of the new ideas and their applicability to deduction shall be in no respect inferior to those of the old arithmetical concepts”.

In other words it is the famous question of A.Einstein:”What really interests me is whether God had any choice in the creation of the world.”

Both are talking about what is known in Geometry as the set of the Principal Postulates. I consider their analog in Physics the following:

1) Postulate of Unitarity: determination of the Physics as the general theory of fields-W.R.Hamilton and E. Schrödinger wave mechanics;

2) Postulate of Communication/Causality: physical determination of geometry;

3) Postulate of Relativity: definition of the action and the inertial systems, determination of the connection between the physics and geometry;

4) Postulate of Local Gauge Invariance: determination of the fundamental interactions (mutually interacting physical systems);

5) Postulate of Least Action: determination of the dynamical behavior of the physical system (physical analog of the Fifth Postulate).

Thus the answer to A. Einstein question is Yes. EP is the empirical fact and must

be derived deductively within the adequate theory of gravitational interaction.

I claim that the presented set is a complete set. The completeness will be verified by the actual realization the unification of all fundamental interactions program. The Postulate of Relativity is an origin, cradle of modern physics. The completeness will be proved through the attempts to reduce it.

Thank you all.

Dr Who.

Please write down a prediction of General Relativity that cannot be derived from the Spin 2 propagating in a flat background formalism.

You cannot of course, because the two are equivalent, as they must be. In fact, entire textbooks written by Nobel Laureates treat the subject.

Which language you choose to express physics in, is inconsequential. Sometimes geometrical langauge is more useful (for instance to look at horizons), and other times pure field theory is more useful (gravitational waves, numerical solutions etc)

Reductionism paved the way for Emergence. Emergence from reductionism, is creatively free.

A proposition then:These two avenues exist in the idea of "The Equivalence Principle?" At the "same time."

No time to read all these comments, but an excellent place to learn about the twin paradox is in Schutz's book.

Also, take a look at the nice front cover (and explanation inside) of Koks' book.

No time to read all these comments, but an excellent place to learn about the twin paradox is in Schutz's book.

Also, take a look at the nice front cover (and explanation inside) of Koks' book.

Hi

Bee,

very nice explanation.

Btw since you write excellent articles you should

blog on the speed of gravity controversy

in 2003 I'd like to know your take on it.

http://www.blogodoom.com/images/girls_with_math_on_backs.jpg

My choice of general relativity continues to be correct.

Hi Bee,

You said:

unless your spacetime allows closed timelike loops you will have to accelerate one of the twins to get them to meet again which breaks the symmetryWhat do closed timelike loops do? I am trying to figure out the Magellan variation of the paradox, the one with cylindrical universe. Can you please help?

Thanks!

Aditi

A closed timelike loop allows you to go back to a point in space

and time. That means it is a curve of non-zero length on which you'd have your own eigentime, but you could re-visit a time in the past. It causes the usual time-travel conundrums and is generally considered to be something you better avoid in your theory. Best,B.

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