## Sunday, February 14, 2010

### Free Falling

My recent post on the black hole information loss problem drifted off into the basics of General Relativity, and I thought it would be worthwhile to repeat some essentials, most notably what "freely falling" means. To start with, we need the Equivalence Principle which we previously discussed here:
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.

The commonly used example is if you were standing in an elevator, you couldn't distinguish between the elevator being pulled up in constant acceleration (by, say, a flying pig) far away from any gravitating masses, or standing on the surface of a planet, being pulled down by gravity. Similarly, you couldn't distinguish between your elevator being not accelerated in empty space, and being freely falling in a gravitational field. That indeed was, so the story goes, what sparked Einstein's idea of the Equivalence principle:
“For if one considers an observer in free fall, e.g. from the roof of a house, there exists for him during his fall no gravitational field---at least in his immediate vicinity.”~A.Einstein [via]

The easiest way to understand how the non-accelerated observer in Special Relativity becomes a freely falling observer in a curved background is to generalize Newton's first law: “In the absence of a force, a body either is at rest or moves in a straight line with constant speed.” (:-)) Now in a curved background, speaking of a “straight line” is not particularly meaningful. Instead, one uses curves of minimal length, so called “geodesics” which is the straight-forward generalization of the flat space's straight lines. The first law then turns into: In the absence of a force, a body moves on a geodesic. And that's what we mean by freely falling.

Doesn't sound too complicated, but there's one thing people tend to get confused about. If space-time is not flat, the motion on a geodesic describes the motion in a gravitational field already. It thus seems like there's a force acting on the body, isn't that what we've learned in high school, things falling from towers etc? No! That's exactly what we mean with “gravity is not a force.” Gravity is a property of space-time. You are “freely falling” as long as there's no force acting on you that pushes you off the geodesic.

Right now, sitting in a chair trying to figure out what I'm telling you, you are not freely falling. There's a force acting on you, which is a combination of the electromagnetic interaction and the Pauli exclusion principle that prohibits you from falling through the Earth. You are not moving on a geodesic. The guy who fell off the roof was moving on a geodesic - no force acting, no acceleration - until he hit the pavement. That's where he was accelerated, which corresponds to a force acting. That's what makes the confrontation with the pavement so unpleasant.

[If you know a little bit about the mathematics, the Equivalence Principle says one can always locally chose coordinates in which space-time is flat and you can get the first derivatives of the metric to vanish too. This means in particular that in this “freely falling” coordinate system, the Christoffel-symbols vanish. That is only possible because the Christoffel-symbols are not tensors. In this freely falling coordinate system, the equation of motion is just that the derivative of the momentum with respect to the proper time vanishes, i.e. it's the same as Special Relativity - the Equivalence Principle at work. You can actually start from there and, using covariance, obtain the equation of motion in all other coordinate systems.]

If you look outside the “local” surrounding mentioned in the Equivalence Principle you can however distinguish flat from curved space. You can do that for example by measuring the distance to nearby geodesics over time. The change of this distance gives you something called “geodesic deviation” which is related to the curvature tensor. Whether or not this geodesic deviation vanishes is a coordinate independent statement: A tensor that vanishes in one coordinate system does so in all coordinate systems. In your elevator this means if you measure very precisely you could figure out that in a gravitational field there's a slight difference between particles moving at the top and at the bottom of your elevator.

Okay, if you've made it till here, there's one question remaining: how do we get back the “gravitational force” that we're used to talk about? Well, you can go and define something like a force in a particular coordinate system, for example the coordinate system labeling distances to the Earth's surface (or, more conveniently, distance to its center). Using such a particular coordinate system allows to separate the terms in the geodesic equation between the time derivative of the momentum (dot p) and “everything else,” which you can then go and interpret as a “force.” Also known as “the Newtonian limit” for obvious reasons, this allows to identify a derivative of the metric tensor in Einstein's field equations with the Newtonian potential.

[You can find this derivation in any textbook on the topic, in Sean Carroll's lecture notes page 105/106 or in my master's thesis, page 28 (in German).]

Zero G flights make active use of the Equivalence Principle. Basically, when you're inside the plane you get the floor pulled away below your feet and are freely falling: there is no force acting on you. You don't need the plane for the fall. You need the plane to slowly push you off the geodesic again and prevent you from colliding with planet Earth. Not to mention that said planet has an atmosphere which, at some hundred miles per hour, causes quite some friction that would prevent actual geodesic motion. But if being a theoretical physicist has any advantage then it's to disregard friction on cue ;-)

1. Hi Bee,

It has always puzzled me how seriously we have to take the principle of equivalence? E.g., suppose we have scalar matter Phi. Then does the principle of equivalence say that there is no coupling Phi * Trace(Ricci)? Of course, this term is non-zero only in the presence of matter - but it could be dark matter which doesn't directly interact with Phi matter. If such a term is not forbidden, then Phi matter would behave differently from other matter in a freely falling frame, at least, so I think.

Similarly, if one started calculating loop diagrams in a quantum GR, then a lot of terms are going to show up that like the example above are potential violaters of the principle of equivalence. So, is it that the principle of equivalence is only roughly valid, to the first order in the Planck mass?

Tx!
-A

2. Hi Bee,

A very nice article on what the equivalency principle serves as being in GR and also what gravity represents as being within it. It reminds me of the old saying “a fall has never killed anyone, yet rather the sudden stop” :-)

Best,

Phil

3. Well, the principle of Equivalence is a postulate. It's a well confirmed one, but a postulate nevertheless. Yes, one can play around with modifications of GR that violate the equivalence principle. The term you suggest though seems to me would be some form of Brans-Dicke theory. I don't really see how quantum contributions would violate the equivalence principle as in: Gravity couples to the stress-energy? Best,

B.

4. This comment has been removed by the author.

5. Hi Bee,

From the point of view that GR is a classical theory where space-time is a homogenous and indivisible entity, while in QM all that exists is observed and considered as being discrete this forms to be the basic conflict between the two. That is from the strictest sense a graviton is not something that would be recognized as real within GR and if so considered then has gravity required to be a force rather then the consequence of the architecture of space-time. That is could it not be said that when attempts to incorporate hypothesises such as the equivalency principle within a totally QM framework you essentially are asking to have apples taken as oranges.

Best,

Phil

6. Dear Bee,

I didn't know that one can book zero-g flights commercially. Hm... 5000 Dollars, OK, not that cheap.

Interestingly, the How it works page carefully avoids mentioning the "free fall" of the airplane ;-)

Cheers, Stefan

7. Dear Bee,

If matter can directly couple to curvature, then since curvature cannot be banished by a suitable choice of frame, e.g., in the free falling frame, gravity would be detectable locally.

E.g., a phi^2 R term would change the mass of a phi particle when R was not zero. Extremely local physics would be able to detect gravity. Of course R is zero in a vacuum, so what's the problem? Well, we could have matter that is dark relative to the phi.

-A

8. Hi Stefan,

From the early days of the American space program the plane that made such flights for astronaut training was known as “The Vomit Comet”. It is little wonder why for attracting tourists they would have dropped such a name:-) Still today there is virtually no one who has found going into orbit as initially a very comfortable experience for the stomach. Isn’t it then interesting that the worst case of motion sickness one can have being resultant from the standpoint of GR has one being affected by no force at all, yet rather just be free to follow ones inertial path within space-time absent of any.

Best,

Phil

9. Hi Bee,

I know it’s off topic yet I just noticed you have just recently bordered the ouside of your blog in pink. It looks very nice and stands as testimony for Stefan of the widely held opinion that only real men can look good in the colour:-)

Best,

Phil

10. Hi Phil,

Yes, I was feeling very pink this morning :-) But don't worry, it's only a temporary feature. Best,

B.

11. Hi Phil,

regarding your earlier comment, it is actually not true that in a quantum theory "all that is observed" is necessarily discrete. Operators corresponding to observables can very well have a continuous spectrum, or a partly continuous spectrum. For a simple example think of the double-slit experiment. The position where the particle hits the screen is not discretized. Best,

B.

12. Dear Stefan,

Yes, I also came across the price tag. Somewhat above my budget, but if it drops by a factor 10 it might become affordable. Yes, I also noticed how carefully they were talking about parabolic curves, avoiding to plainly say: look, we just let you fall down and try to catch you before you hit the ground ;-p Best,

B.

13. Dear Arun,
Yes, in principle you can do that. What I was saying is that it would mean G was generically time- and position dependent and there are strong constraints on such modifications. Or in other words, you also need an equation of motion for your phi field, and you want it to get out of the way for all well-tested predictions of GR, which isn't easy. Best,

B.

14. Hi Bee,

So you were in he pink today as they say, which is to mean you are feeling good. That brings a suggestion then to have the border of the day reflect your mood. If then I see it is red I’ll know to be more careful with my comments ;-)

Oh yes I understand and appreciate what you say as QM not being solely to be taken as a theory resentitive of discreteness, yet also as the character lent to it by the wave function having a level of unity and holism. However in the normal way this is considered it is resultant of the incompleteness of our being able o determine reason for the action of what is observed rather than the result of some well defined physical aspect of reality.

Best,

Phil

15. Hi Phil,

B.

16. Nice article, Bee, thanks. How does one then explain that which was well known to David when he killed Goliath with his sling, namely: Centripetal force, which of course is too often confused with Centrifugal force?

Phil Warnell, you are aware yes that Verlinde feels gravitons are not real fundamental particles, but rather akin to acoustics' phonons?

Specific, Verlinde says at his blog:

... gravitons do not exist when gravity is emergent. Gravitons are like phonons. In fact, to make that analogy clear consider two pistons that close of a gas container at opposite ends. Not that the force on the pistons due to the pressure is also an example of an entropic force. We keep the pistons in place by an external force. When we gradually move one of the pistons inwards by increasing the force, the pressure will become larger. Therefore the other piston will also experience a larger force. We can also do this in an abrupt way. We then cause a sound wave to go from one piston to the other. The quantization of this sound wave leads to phonons. We know that phonons are quite useful concepts, which even themselves are often used to understand other emergent phenomena.

Similarly, gravitons can be useful, and in that sense exist as effective "quasi" particles. But they do not exist as fundamental particles.

... Erik Verlinde

17. Steven: Rotation is an accelerated motion, not sure what you're asking for. Gosh, I didn't know Verlinde writes a blog. I was just scratching my head over his paper. I'll check this out, thanks :-) Best,

B.

18. Steven: Can you find some sort of feed for his blog?

19. Properties of space time are generalizations of forces, not distinct entities from forces.

In fact one of the key features of the stress-energy tensors is that you can integrate over the pressure terms along a surface normal to a direction to find force in that direction.

And as for freely falling observers being approximately Minkowski, true to first order approximation on a local neighborhood, but then to second order approximation on a larger local neighborhood, the coordinates are approximately Rindler. One can carry the Taylor expansion on indefinitely by the property that the coordinate system is analytic on the local neighborhood.

If one where to take into account all the stress-energies acting on you sitting in a chair, including E-M, Pauli Exclusion, etc... then you to would be freely falling. What you physically feel as force is tidal force acting on your body: 150 pounds on the feet, 120 pounds on the knees, 100 pounds on the hips,...10 pounds on the neck.

One needs to be careful not to confuse the weightlessness, were all particles are accelerating close to uniformly and the tidal forces are small, with not accelerating at all. And NASA does not refer to space flight as 0G flight but rather micro-gravity flight.

Finally, I have one more bone to pick, if the curvature is so small near a black hole then why can't you use Newton mechanics with an appropriate choice of effective force in your calculations? After all Newton Mechanics is the unique low curvature limit of GR.

20. Hi Steven,

Perhaps I’m a little thick when it comes to such things, yet rather if what is considered is emergent or fundamental it still stands as being real. That is to find logical what Verlinde is proposing is for me to have something that seems obviously and necessarily to be undividable and whole to be in actually discrete. I have never understood why such wholeness cannot be considered as also a possibility, as for instance as the set of real numbers would be as compared to that of the naturals. That is to consider that it is possible to have continuity and holisism and that our way to define them simply be a consequence of our limit to conceive them as being so. That’s for me it always seemed more logical that the world presents as the resultant action between what one gets when an entity of discreetness finds itself within one being continuous and indivisible. Then again this may be just the fault of my own failing of comprehension and not a refection of Verlinde’s concept being invalid.

Best,

Phil

21. Bee I am really curious if you also observed the following parallel? Maybe you could do a future blog on it.

There is an interesting parallel between the electric field and the gravity field, a duality if you will. It leads to a 5th Maxwell equation.

In the electric field we have Coulomb’s Law F = q1q2/4Ï€ÎµR^2. The analogous entity in gravity is Newton’s Law of Gravitation F = Gm1m2/R^2. In the electric field the energy stored in the electric field of a spherical single plate capacitor with radius R and charge q is given by U = q^2/8Ï€ÎµR. The gravity analogue for the energy stored in the gravity field by a spherical mass m is U = Gm^2/2R.

The analogue of the electric field vector in gravity is a gravity field vector that I have named Newton Height or NH for short (or the vector field g if you prefer):

NH = – Gm/R^2 (direction opposite unit vector R)

The gravity analogue to Gauss’s Law for the Electric Field, Maxwell’s fifth equation, the Law for the Gravity Field:

∫ NH • dA (closed surface) = -- 4Ï€Gm

For the electric field we figured out the field strength and direction using Gauss’ Law and integrated the electric field energy density Î¼ = ½ ÎµE^2 over all of space from the surface of the charge to infinity in all directions to get the energy stored in the field. The energy density of a gravitational field is given by Î¼ = (1/8Ï€G) NH^2.

Then for Newton’s law of gravity we have F = mNH analogous to F = qE and F = − del U(gravity field) analogous to F = − del U(electric field).

22. Hi Steven,

Also I was going to mention that for anyone that would like to explore matters such as this more closely I would definitely recommend Dr. Harvey Brown’s Book ‘Physical Relativity: Space-time Structure from a Dynamical Perspective’. That said and although I personally don’t find his arguments show conclusively the nonexistence of space-time as an being an actually element of reality. it does serve to have all considered more comprehensivelynand broadly ,with therein being something that shouldn’t be denied due consideration.

Best,

Phil

23. Hi Bee,

Steven: Can you find some sort of feed for his blog?

Well, sure. I linked it in my last post. Click here to see it again, or copy'n'paste the following:

http://staff.science.uva.nl/~erikv/page20/page18/page18.html

It's not a "blog" in the usual sense that one he encourages replies and gives responses like yours, and it's a bit of an extension of what he started at Lubos' blog, "The Reference Frame (which encourages the immediate bombing of Iran)", but a blog he calls it nevertheless.

He does give his e-mail at that page's bottom for further research. ;-)

I hope that helps.

24. Hi Dave,

the analogy you cite between electrostatics and Newton gravity is a very nice example of, well, an analogy, in the precise sense that different physical situations are described by the same mathematical structure, and behave similar (it's called analogy, and not duality, so LastAncientGreek needn't be worried).

In the example, the common mathematical structure is that there is a potential (a field), which is described by the Poisson equation, and a force which is proportional to the gradient of the potential (well, that's why the field is called "potential"). Interpreting the constants in the formulas as charge or mass, you end up with the Coulomb law of electrostatics or Newton's law of gravity.

Best, Stefan

25. Bee:Sehr gut erklÃ¤rt. Ich habe alles verstanden.

What I like very much in your post is that you're pointing out the idea that a force is an interpretation of the general relativity theory. From your writing, it is clear that asking if gravitation is a force OR the consequence of your dashing forth through space-time geodesics is of weak interest. Both hold if considering a proper transformation of the theory.

There are rather known people in Physics who constantly want to state that things are so and not so, according to what they like or dislike.

To me, such a constanly "multi-viewpointed" argumentation should be the standard.

Best,

26. Albert Einstein (1879–1955)

One part of the theory of Relativity was inspired when a painter fell off a roof. Einstein found out that while the painter was falling freely, he felt weightless. This led Einstein to realize that gravity was a form of inertia, a result of the way things moved through space - and General Relativity was born.

Looking to secure and anchor the sensations of free fall, as a measure of earth's mantel, not to difficult to understand. A Sensorium?:)

Gravity is stronger closer to Earth, so GOCE was designed to orbit as low as possible while remaining stable as it flies through the fringes of our atmosphere. To help avoid drag and ensure that the gravity measurements are of true gravity, the satellite has to be kept stable in ‘free fall’. Any buffeting from residual air at this low altitude could potentially drown out the gravity data.

I mean psychologically to have any of us think that time could be measured in "human emotion with thought" is a hard thing to comprehend had one not understood the realm of thought as a "containment of experience."

Einstein's thought experiment with regards to a beautiful girl was not to categorize time to a hot element but to seek to hold relevance to how experience could be measured sometimes. Although again, as Einstein said "Gravity is not responsible for one falling in love?:)

It is not to unlikely that one would look at containment of experience as "radically the same" as once moved to a completely finer disposition of freedoms from our matter states?

Gravity is Talking, LISA will Listen

Gravity Experiments

Best,

27. “On Earth, buoyancy continuously deforms and moves fluids in complex manners, making it difficult to study how materials that solidify from the melt form semiconductors and other products,” said Dr. Aleksandar Ostrogorsky, the SUBSA principal investigator who also teaches and conducts research at the Rensselaer Polytechnic Institute in Troy, N.Y. “In microgravity, the fluids are almost stagnant, resembling solids. The absence of motion makes it easier to observe and mathematically describe what is occurring when the crystals are melted, and how the materials solidify to form a new crystal.” See:Gravity Free Environment on Earth

28. Phonon:

A particle of sound. The energy E of a phonon is given by the Einstein relation, E = hf. Here f is the frequency of the sound and h is Planck's constant. The momentum p of a photon is given by the de Broglie relation, p = h/Î». Here Î» is the wavelength of the sound

29. Einstein's elevator fails if the vacuum is divergently interactive with its contents. Billion lightyear pathlengths show no photon refraction, dispersion, dichroism, or gyrotropy; arXiv:0706.2031, 0801.0287. No measurable property of matter diverges: Eotvos experiments, Nordtvedt effect, PSR J1903+0327.

Ashtekar split GR into chiral halves re La Coupe du Roi. Parity-odd Chern-Simmons addition to parity-even Einstein Hilbert action. Test for a footed vacuum with shoes: Fully inverse geometric parity atomic mass distributions are single crystals in enantiomorphic space groups P3(1)21 versus P3(2)21 and P3(1) versus P3(2) - opposite shoes. (all experiments versus time of day)

Do opposite shoes vacuum free fall identically? (quartz or gamma-glycine)
Do opposite shoes have identical enthalpies of fusion to socks? (benzil)
Do Meissner-levitated opposite shoes spontaneously spin in opposite directions? (quartz)
Do molecular left-left, left-right, right-right paired propellers on a rigid shaft show spin population divergence?

Einstein's elevator can fail without contradiction of prior observations. Somebody should look.

30. Darn, I was about to make this nitpicking remark about no force acting on someone falling from a roof (in vacuum, right?) when I read your last remark about disregarding friction on cue. That reminds me of the ooold joke: "Consider a spherical cow,..."

31. Aaron: Sure you can expand the local neighborhood beyond the freely falling approximation, I wrote that in my post, didn't I? The Newtonian limit not a weak curvature approximation but a weak field approximation and it's usually for slowly moving objects. The tidal force is the gradient of the gravitational force. My head would have to be on top of a mountain to notice the difference. There's a reason why to good accuracy down here on Earth we use a constant gravitational field for every day calculations. What you are talking about is accumulating pressure, it stems from weight (more weight above my knees than above my head) not from tidal forces. Look, you must have read this some dozen times, the question what kills you when you fall into a black hole, the thing with "being pulled like a spaghetti" etc etc (Google somewhat, I'm sure you'll find it.) That are tidal forces, but you'll only physically notice them well inside the horizon. Best,

B.

32. Hi Dave,

Well, I was about to say the same as Stefan. Newtonian gravity and the Coulomb force law are indeed very similar and that's because the ingredients you use to derive them are the same. (That is incidentally why I believe most physicists fail to be very excited by Verlinde's paper.) Basically, all you need in a conserved charge (electric charge or mass), you need 3 dimensions (the potential depends on the number of dimensions), and you need a long-range interaction (the strong interaction for example does not give you a Coulomb-like force because of confinement). That's basically it. Assume spherical symmetry, integrate over a volume, use Gauss' law (you need the conserved charge on the other side of the equation), divide by surface, et voila F ~ 1/R^2. Best,

B.

33. Steven: Yes, thanks, I found the blog, but I can't find a feed (RSS, ATOM, something) on the site to pipe it into a my Google reader?

34. JÃ©rÃ´me: You are making a good point there. There are usually many equally good explanations for the same topic, but none of them is worth much without understanding, and each simplified explanation by itself is incomplete exactly because its simplified. Unfortunately, it happens quite often that people get stuck with a single sentence out of such a "simple" explanation, and then draw wrong conclusions. The sentence "gravity is not a force" is one of them. Another example is the "pair production at the black hole horizon" (you can't localize the particles well enough). Most popular science books offer one or the other simplified explanation that, if you don't know what they "really" mean (ie what are the equations) doesn't make sense. Eg the cosmological constant is something like antigravitation (fails to make clear the point is an usual equation of state), and so on. Maybe a "multi-viewpoint" as you suggest would be helpful to this end. Best,

B.

35. "Ahh! I've entered and suddenly exited my geodesic... and I can't get up!"

36. And where does the pressure come from? I have to re-iterate one must be much more careful with one's accounting. One can't pick and choose the terms to put into the stress-energy, except through arguments of magnitude.

So for the example of sitting at the desk, you have to include the stress-energies of the furniture, ground, etc...because the Van de Waals forces are of the same scale as gravitation. Fortunately Newtonian mechanics is an excellently limiting tool in this case.

But for a black hole, if you use Hawking's argument that the test particle is accelerating to maintain a stationary distance, then the source of that force needs to be incorporated into the stress-energies, which will change the thermodynamics, entropy, and information, in a mathematical sense.

37. Is there no architectural consideration to the elemental structure of planets that form "according to some higher order?"

Okay, some science fiction then:)Oui! Non?

In order to consider mining pursuits on the moon I needed to understand the makeup. My company has laid claim:)

Planets are round because their gravitational field acts as though it originates from the center of the body and pulls everything toward it. With its large body and internal heating from radioactive elements, a planet behaves like a fluid, and over long periods of time succumbs to the gravitational pull from its center of gravity. The only way to get all the mass as close to planet's center of gravity as possible is to form a sphere. The technical name for this process is "isostatic adjustment."

With much smaller bodies, such as the 20-kilometer asteroids we have seen in recent spacecraft images, the gravitational pull is too weak to overcome the asteroid's mechanical strength. As a result, these bodies do not form spheres. Rather they maintain irregular, fragmentary shapes.

I mean in the most simplest sense the surface tension of the water droplet in space seemed an appropriate example?:)

APOLLO measures the round-trip travel time of laser pulses bounced off the lunar retroreflectors to a precision of a few picoseconds, corresponding to about one millimeter of precision in range to the moon. Using this information, we will be able to gauge the relative acceleration of the earth and moon toward the sun (like a modern-day Leaning Tower of Pisa experiment) in order to ascertain the free-fall properties of earth's gravitational self-energy.

Did You know Newton was interested as well in the compositional makeup of the earth? It just seems some of us were more inclined to want to understand that order:)

Best,

38. Aaron: There's a pressure because there's forces acting, which is what I've said, you're not freely falling (rspt the human body is not pressureless dust). If you use the identification of gravity as a force that I've mentioned in my post, of course the "other forces" are of the same order of magnitude. Reg Hawking radiation, I haven't used any such argument. Best,

B.

39. This comment has been removed by the author.

40. Anthony: Yes, you already said that, and nobody here gives a damn about your opinion. So why don't you just shut up if you have nothing to say. Thanks,

B.

41. You tell him, Bee. anthony's from Croatia, btw. Or at least he logged in using Croatian as his default language. Took me a while to figure that out. 5 minutes. Maybe 4.

The problem with Black hole information is the paradox, not the loss. You're doing well there, Bee. Thanks.

42. Steven: Yes, thanks, I found the blog, but I can't find a feed (RSS, ATOM, something) on the site to pipe it into a my Google reader?

Well thanks for teaching me about RSS and ATOM, Bee, I was born in the 1950's so I'm not up on this stuff that learned folks like you have learned.

The way I got there was to go to the Wikipedia entry: "Gravity as an entropic force." At the bottom of the page is Verlinde's home page, and I just surfed from there.

Lisi surfs for real, I surf the internet. I used to love the real risk-taking stuff until I broke my leg once in 3 places. Thanks, but doing that once a lifetime is enough. Surfing the internet is much safer, and much more rewarding from an intellectual point of view.

43. You are still putting the stress-energy into a special category reserved only for gravitation, which it is not. Stress-energy is a generalization of classical mechanics to a relativistic setting that incorporates classical forces.

Please think through the formulation of the stress-energies of sitting at a desk, or in the simplified thought experiment a linear field in one direction and an opposing inverse quadratic Van der Waals field in the opposite direction. One can solve for the geodesics, and taking the lowest order terms at small velocities (unit tangents nearly parallel with the unit time vector) recover Newtons laws.

The point is, if you are identifying free fall with geodesics, then as long as you have accounted for all the stress energies then everything is in free fall all the time (just like in the classical setting, if you account for all the forces all the time then all paths obey Newton's Laws, Lagrangian, Hamiltonian, or whatever flavor of classical mechanics suites you best).

The question is then how much tidal force occurs causally across the system, nearly none in weightlessness, lots down here on earth.

Re: Anthony, it is true that we will not conceivably be able to physically test hypotheses concerning metrics containing black holes in the next century. However the work on smooth manifold thermodynamics will almost certainly yield novel homeotopies on smooth manifolds, in particular I think there is great hope in developing an alternative proof of PoincarÃ©'s Theorem by "heating" a smooth manifold to "melt out" its local topological features, that is instead of using the Ricci Flow, one evolves the manifold using entropy maximization, but that is very speculative and it is mathematics, not physics.

44. Aaron: Yes, I am talking about General Relativity, I think I stated that several times very clearly. I also clearly stated that yes, you can take the classical limit and identify a gravitational force. It seems to me you would like to put forward a different definition for "free fall" than the one commonly used which is content of this post. I think that's a step backwards, but a definition is just a definition. Best,

B.

45. Anthony: Bee, Black Hole Information Loss is a stupid problem.
a

Indeed, there are stupid people, who work hard on difficult issues, and people like you, who know everything and teach us with so beautiful arguments.

Believe me Anthony, you are essential to Science.

Worst,

46. Well thanks for teaching me about RSS and ATOM, Bee, I was born in the 1950's so I'm not up on this stuff that learned folks like you have learned.

@Steven: Most of those who are your age don't even use Internet. I find you are pretty good with new technologies anyway :)

47. No, the original point I was taking umbrage with was the loose use of the strict Topological definition of locally flat to justify that a freely falling observer has trivial orthogonal vacuum solutions to the divergence operators (Fourier terms). The solutions are global on the manifold, and thus non-trivial in curved manifolds.

48. Aaron: You are putting a lot of words in my mouth that I never used. I entirely agree with you that the notion of a particle is not a local construct. I never claimed it is. It was in fact you who had a problem with the "freely falling" observer in the black hole geometry, so what's your point? Best,

B.

49. /*...you couldn't distinguish between your elevator being not accelerated in empty space, and being freely falling in a gravitational field..*/

In fact you could do it easily, because gravity force has always center of action in our Universe, while acceleration force hasn't. Many short distance forces are violating equivalence principle apparently, because they depend on surface, not the mass of objects - for example human love, Casimir or van der Waals forces...

50. In a contentious Cosmic Variance thread, I and others were hashing over whether the gravity field above an "infinite" (FAPP) uniform layer of mass was the equivalent of from acceleration (i.e., the intuitive idea of an linear g-field, and analog of the uniform E field above the analogous planar charge.)

Well, I already know about Rindler coordinates and that the actual apparent field of a accelerating observer A can't be uniform proper g for all elements in that environment anyway. That is, they can't maintain equal g and remain in the same relative relations. Hence the field for A, is given as g = -c^2/X where X is proper distance as locally incremented from the Rindler horizon (yadda, you get the picture ...)

But according to some folks, the discrepancy goes beyond that to involve issues like the way the gravitational analog of magnetism would work (gravimagnetism; I prefer "gravitism".) IOW, if you move laterally in a g field, there should be an analog to the magnetic field. Apparently that gets really sticky and isn't so intuitive and clear as for EM, and depends of whether you're in a field just like acceleration or from a planar mass.
Thoughts?

51. In EM charge allows for the symmetry breaking so that a direction of torsion can be "chosen" for each sign of charge.

So how would a mass know which way to turn? If it did occur it would be a stunning symmetry breaking, and would show up in things like an asymmetry in the distributions of rotations and orientations of galaxies.

52. maybe, Uncle Al can help?

53. So...if there is an intrinsic handedness to the universe would that explain republican politics?

54. LOL....possibly an algorithm to political direction.:)

Matter defined states are always nice to see when deferring to ones constitution. You know, wo/men of gold or silver?

best,

55. This is not an US blog, so stop talking about American politics. Be less provincial, please.

56. Okay Stefan,

There seems to be some correlations in this topic of what can be defined as matter states and how one might arrive at that? Grvaity probe B, or the Jets?

Plato had a thought there that I would just reiterate is not just American in design.

* The Republic: "You must contrive for your future rulers another and a better life than that of a ruler and then you may have a well-ordered State; for only in the State which offers this will they rule who are truly rich not in silver and gold but in virtue and wisdom which are the true blessings of life."

Best,