Negative Mass in General Relativity?

[Image Source: Ginva.com]

Science News ran a piece the other week about a paper that has appeared in PRD titled “Negative mass bubbles in de Sitter spacetime”. The Science News article is behind a paywall, but don’t worry I’ll tell you everything you need to know.

The arxiv version of the paper is here. Since I’m quoted in the Science News piece saying something to the extent that I have my reservations but think it’s a promising direction of study, I have gotten a lot of questions about negative masses in General Relativity lately. So here a clarification.

First one has to be careful what one means with mass. There are three types of masses: inertial mass, passive gravitational mass, and active gravitational mass. In General Relativity these masses, or their generalization in terms of tensors respectively, are normally assumed to be identical.

The equality of inertial and passive gravitational mass is basically the equivalence principle. The active gravitational mass is what causes space-time to bend; the passive gravitational mass is what couples to the space-time and determines the motion of particles in that background. The active and passive gravitational masses are identical in almost all theories I know. (The Schrödinger-Newton approach is the only exception that comes to mind). I doubt it is consistent to have them not be equal, but I am not aware of a proof for this. (I tried in the Schrödinger-Newton case, but it’s not as trivial as it looks at first sight.)

In General Relativity one further has to distinguish between the local quantities like energy-density and pressure and so on that are functions of the coordinates, and global quantities that describe the space-time at large. The total mass or energy in some asymptotic limit are essentially integrals over the local quantities, and there are several slightly different ways to define them.

The positive mass theorem, in contrast to what its name suggests, does not state that one cannot have particles with negative masses. It states instead, roughly, that if your local matter is normal matter and obeys certain plausible assumptions, then the total energy and mass are also positive. You thus cannot have stars with negative masses, regardless of how you bend your space-time. This isn’t as trivial a statement as it sounds because the gravitational interaction contributes to the definition of these integrated quantities. In any case, the positive mass theorem holds in space that is asymptotically flat.

Now what they point out in the new paper is that for all we know we don’t live in asymptotically flat space, but we live in asymptotic de-Sitter space because observational evidence speaks for a positive cosmological constant. In this case the positive mass theorem doesn’t apply. Then they go on to construct a negative mass solution in asymptotic de Sitter space. I didn’t check the calculation in detail, part of it is numerical, but it all sounds plausible to me.

However, it is somewhat misleading to call the solution that they find a negative mass solution. The cosmological constant makes a contribution to the effective mass term in what you can plausibly interpret as the gravitational potential. Taken together both, the effective mass in the potential is positive in the region where this solution applies. The local mass (density) is also positive by assumption. (You see this most easily by looking at fig 1 in the paper.)

Selling this as a negative mass solution is like one of these ads that say you’ll save 10$ if you spend at least $100 – in the end your expenses are always positive. The negative mass in their solution corresponds to the supposed savings that you make. You never really get to see them. What really matters are the total expenses. And these are always positive. There are thus no negative mass particles in this scenario whatsoever. Further, the cosmological constant is necessary for these solutions to exist, so you cannot employ them to replace the cosmological constant.

It also must be added that showing the existence of a certain solution to Einstein’s field equations is one thing, showing that they have a reasonable chance to actually be realized in Nature is an entirely different thing. For this you have to come up with a mechanism to create them and you also have to show that they are stable. Neither point is addressed in the paper.

Advertisement break: If you want to know how one really introduces negative masses into GR, read this.

In the Science News article Andrew Grant quotes one of the authors as saying:

“Paranjape wants to look into the possibility that the very early universe contained a plasma of particles with both positive and negative mass. It would be a very strange cosmic soup, he says, because positive mass gravitationally attracts everything and negative mass repels everything.”

This is wrong. Gravitation is a spin-2 interaction. It is straightforward to see that this means that like charges attract and unlike charges repel. The charge of gravity is the mass. This does not mean that negative gravitational mass repels everything. Negative gravitational mass repels positive mass but attracts negative mass. If this wasn’t so, then you’d run into the above mentioned inconsistencies. The reason this isn’t so in the case considered in the paper is that they don’t have negative masses to begin with. They have certain solutions that basically have a gravitational attraction which is smaller than expected.

In summary, I think it’s an interesting work, but so far it’s an entirely theoretical construct and its relevance for the description of cosmological dynamics is entirely unclear. There are no negative mass particles in this paper in any sensible interpretation of this term.

Gender disparity? Yes, please.

[Image Source: Papercards]

Last month, a group of Australian researchers from the life sciences published a paper that breaks down the duration of talks at a 2013 conference by gender. They found that while the overall attendance and number of presentations was almost equally shared between men and women, the women spoke on the average for shorter periods of time. The main reason for this was that the women applied for shorter talks to begin with. You find a brief summary on the Nature website.

The twitter community of women in science was all over this, encouraging women to make the same requests as men, asserting that women “underpromote” themselves by not taking up enough of their colleagues’ time.

Other studies have previously found that while women on the average speak as much as men during the day, they tend to speak less in groups, especially so if the group is predominantly male. So the findings from the conference aren’t very surprising.

Now a lot of what goes around on twitter isn’t really meant seriously, see the smiley in Katie Hinde’s tweet. I remarked one could also interpret the numbers to show that men talk too much and overpromote themselves. I was joking of course to make a point, but after dwelling on this for a while I didn’t find it that funny anymore.

Women are frequently told that to be successful they should do the same as men do. I don’t know how often I have seen advice explaining how women are allegedly belittling themselves by talking, well, like a woman. We are supposed to be assertive and take credit for our achievements. Pull your shoulders back, don’t cross your legs, don’t flip your hair. We’re not supposed to end every sentence as if it was a question. We’re not supposed to start every interjection with an apology. We’re not supposed to be emotional and personal, and so on. Yes, all of these are typically “female” habits. We are told, in essence, there’s something wrong with being what we are.

Here is for example a list with public speaking tips: Don’t speak about yourself, don’t speak in a high pitch, don’t speak too fast because “Talking fast is natural with two of your best friends and a bottle of Mumm, but audiences (especially we slower listening men) can’t take it all in”. Aha. Also, don’t flirt and don’t wear jewelry because the slow men might notice you’re a woman.

Sorry, I got sick at point five and couldn’t continue – must have been the Mumm. Too bad if your anatomy doesn’t support the low pitches. If you believe this guy that is, but listen to me for a moment, I swear I’ll try not to flirt. If your voice sounds unpleasant when you’re giving a talk, it’s not your voice, it’s the microphone and the equalizer, probably set for male voices. And do we really need a man to tell us that if we’re speaking about our research at a conference we shouldn’t talk about our recent hiking trip instead?

There are many reasons why women are underrepresented in some professions and overrepresented in others. Some of it is probably biological, some of it is cultural. If you are raising or have raised a child it is abundantly obvious that our little ones are subjected to gender stereotypes starting at very young age. Part of it is the clothing and the toys, but more importantly it’s simply that they observe the status quo: Childcare is still predominantly female business and I yet have to see a woman on the garbage truck.

Humans are incredibly social animals. It would be surprising if the prevailing stereotypes did not affect us at all. That’s why I am supportive of all initiatives that encourage children to develop their talents regardless of whether these talents are deemed suitable for their gender, race, or social background. Because these stereotypes are thousands of years old and have become hurdles to our selfdevelopment. By and large, I see more encouragements for girls than I see for boys to follow their passion regardless of what society thinks, and I also see that women have more backup fighting unrealistic body images which is what this previous post was about. Ironically, I was criticized on twitter for saying that boys don’t need to have a superhero body to be real men because that supposedly wasn’t fair to the girls.

I am not supportive of hard quotas that aim at prefixed male-female ratios. There is no scientific support for these ratios, and moreover I witnessed repeatedly that these quotas have a big backlash, creating a stigma that “She is just here because” whether or not that is true.

Thus, at the present level women are likely to still be underrepresented from where we would be if we’d manage to ignore social pressure to follow ancient stereotypes. And so I think that we would benefit from more women among the scientists, especially in math-heavy disciplines. Firstly because we are unnecessarily missing out of talent. But also because diversity is beneficial for the successful generation and realization of ideas. The relevant diversity is in the way we think and argue. Again, this is probably partly biological and partly cultural, but whatever the reason, a diversity of thought should be encouraged and this diversity is almost certainly correlated with demographic diversity.

That’s why I disapprove of so-called advice that women should talk and walk and act like men. Because that’s exactly the opposite from what we need. Science stands to benefit from women being different from men. Gender equality doesn’t mean genders should be equal, it means they should have the same opportunities. So women are more likely to volunteer organizing social events? Wtf is wrong with that?

So please go flip your hair if you feel like it, wear your favorite shirt, put on all the jewelry you like, and generally be yourself. Don’t let anybody tell you to be something you are not. If you need the long slot for your talk go ahead. If you’re confident you can get across your message in 15 minutes, even better, because we all talk too much anyway.


About the video: I mysteriously managed to produce a video in High Definition! Now you can see all my pimples. My husband made a good camera man. My anonymous friend again helped cleaning up the audio file. Enjoy :)

Frequently Asked Questions

[Image source: Stickypictures.]

My mom is a, now-retired, high school teacher. As teenager I thought this was a great job and wanted to become a teacher myself. To practice, I made money giving homework help but discovered quickly I hated it for a simple reason: I don’t like to repeat myself. I really don’t like to repeat myself.

But if I thought spending two years repeating how to take square roots - to the same boy - was getting me as close to spontaneous brain implosion I ever wanted to get, it still didn’t quite prepare me for the joys of parenthood. Only the twins would introduce me to the pleasure of hearing Jingle Bells for 5 hours in a row, and re-reading the story about Clara and her Binky until the book mysteriously vanished and will not be seen again unless somebody bothers to clean behind the shoe rack. “I told you twice not to wash the hair dryer,” clearly wasn’t my most didactic moment. But my daughter just laughed when the fuse blew and the lights went off. Thanks for asking, we got a new dryer.

And so I often feel like I write this blog as an exercise in patience. Nobody of course bothers to search the blog archives where I have explained everything. Sometimes twice! But today I will try to be inspired by Ethan who seems to have the patience of an angel, if a blue one, and basically answers the same questions all over and over and over again. So here are answers to the questions I get most often. Once and forever I hope...

1. Is string theory testable?
   
   The all-time favorite. Yes, it is. There is really no doubt about it. The problem is that it is testable in principle, but at least so far nobody knows how to test it in practice. The energy (densities) necessary for this are just too high. Some models that are inspired by string theory, notably string cosmology, are testable with existing experiments. That it is testable in principle is a very important point because some variants of the multiverse aren’t even testable in principle and then it is indeed highly questionable whether it is still science. Not so though for string theory. And let me be clear that I mean here string theory as the candidate theory of everything including gravity. Testing string theory as means to explain certain strongly coupled condensed matter systems is an entirely different thing.
   
   
2. Do black holes exist?
   
   Yes. We have ample evidence that supermassive black holes exist in the centers of many galaxies and that solar-sized black holes are found throughout galaxies. The existence of black holes is today generally accepted fact in the physics community. That black holes exist means concretely that we have observational evidence for objects dense enough to be a black hole and that do not have a hard surface, so they cannot be a very dim stars. One can exclude this possibility because matter hitting the surface of a star would emit radiation, whereas the same would not happen when the matter falls through the black hole horizon. This horizon does not have to be an eternal horizon. It is consistent with observation, and indeed generally believed, that the black hole horizon can eventually vanish, though this will not happen until hundreds of billions of years into the future. The defining property of the black hole is the horizon, not the singularity at its center, which is generally believed to not exist but for which we have no evidence one way or the other.
   
   
3. Why quantize gravity?
   
   There is no known way to consistently couple the non-quantized theory of general relativity to the quantum field theories of the standard model. This only works in limiting cases. The most plausible way to resolve this tension is to quantize gravity too. It is in principle also possible that instead there is a way to couple quantum and classical theories that has so far been missed, or that the underlying theory is in some sense neither classical nor quantum, but this option is not favored by most researchers in the field today. Either way, the inconsistency in our existing theories is a very strong indication that the theories we have are incomplete. Research in quantum gravity basically searches for the completion of the existing theories. In the end this might or might not imply actually quantizing gravity, but Nature somehow knows how to combine general relativity with quantum field theory, and we don’t.
   
   
4. Why is it so hard to quantize gravity?
   
   It isn’t. Gravity can be quantized pretty much the same way as the other interactions. It’s just that the theory one arrives at this way cannot be a fundamental theory because it breaks down at high energies. It is thus not the theory that we are looking for. Roughly speaking the reason this happens is that the gravitational equivalent of a particle’s charge is the particle’s energy. For the other known interactions the charge and the energy are distinct things. Not so for gravity.
   
   
5. Is quantum gravity testable?
   
   Again, yes it is definitely testable in principle, it’s just that the energy density necessary for strong quantum gravitational effects is too high for us to produce. Personally I am convinced that quantum gravity is also testable in practice, because indirect evidence can prevail at much lower energy densities, but so far we do not have experimental evidence. There is a very active research area called quantum gravity phenomenology dedicated to finding the missing experimental evidence. You can check these two review papers to get an impression of what we are presently looking for.

The underappreciated value of boring truths

My primary reaction to any new idea on the arXiv is conviction that it’s almost certainly wrong, and if I can’t figure out quickly why it’s wrong, I’ll ignore it because it’s most likely a waste of time. In other words, I exemplify the stereotypical reaction of scientists which Arthur Clarke summed up so nicely in his the three stages of acceptance:

1. “It’s crazy — don’t waste my time.”
2. “It’s possible, but it’s not worth doing.”
3. “I always said it was a good idea.”

Maybe I’m getting old and bold rather than wise and nice, but when it comes to quantum gravity phenomenology, craziness seems to thrive particularly well. My mother asked me the other day what I tell a journalist who wants a comment on somebody else’s work which I think is nonsense. I told her I normally say “It’s very implausible.” No, I’m not nice enough to bite my tongue if somebody asks for an opinion. And so, let me tell you that most of what gets published under the name of quantum gravity phenomenology is, well, very implausible.

But quantum gravity phenomenology is just an extreme example of a general tension that you find in theoretical physics. Consider you’d rank all unconfirmed theories on two scales, one the spectrum from exciting to boring, the other the spectrum from very implausible to likely correct. Then put a dot for each theory in a plane with these two scales as axes. You’d see that the two measures are strongly correlated: The nonsense is exciting, and the truth is boring, and most of what scientists work on falls on a diagonal from exiting nonsense to boring truths.

If you’d break this down by research area you’d also find that the more boring the truth, the more people work on nonsense. Wouldn’t you too? And that’s why there is so much exciting nonsense in quantum gravity phenomenology - because the truth is boring indeed.

Conservative wisdom says that quantum gravitational effects are tiny unless space-time curvature is very strong, which only happens in the early universe and inside black holes. This expectation comes from treating quantum gravity as an effective field theory, and quantizing it perturbatively, ie when the fluctuations of space-time are small. The so quantized theory does not make sense as a fundamental theory of gravity because it breaks down at high energies, but it should be fine for calculation in weak gravitational fields.

Most of the exciting ideas in quantum gravity phenomenology assume that this effective limit does not hold for one reason or the other. The most conservative way to be non-conservative is to allow the violation of certain symmetries that are leftover from a fundamental theory of quantum gravity which does not ultimately respect them. Violations of Lorentz-invariance, CPT invariance, space-time homogeneity, or unitarity are such cases that can be accommodated within the effective field theory framework, and that have received much attention as possible signatures of quantum gravity.

Other more exotic proposals implicitly assume that the effective limit does not apply for unexplained reasons. It is known that effective field theories can fail under certain circumstances, but I can’t see how any of these cases play a role in the weak-field limit of gravity. Then again, strong curvature is one of the reasons of failure, and we do not understand what the curvature of space-time is microscopically. So sometimes, when I feel generous, I promote “implausible” to “far-fetched”.

John Donoghue is one of the few heroically pushing through calculations in the true-but-boring corner of quantum gravity phenomenology. In a recent paper, he and his coauthors calculated the quantum contributions to the bending of light in general relativity from 1-loop effects in perturbatively quantized gravity. From their result they define a semi-classical gravitational potential and derive the quantum corrections to Einstein’s classical test of General Relativity by light deflection.

They find a correction term that is suppressed by a factor ℏ G/b² relative to the classical result, where b is the impact parameter and G is Newton’s constant. This is the typical result you’d expect from dimensional reasons. It’s a loop correction, it must have an extra G in it, it must have an inverse power of the impact parameter so it gets smaller with distance, thus G/b² is a first guess. Of course you don’t get tenure for guessing, and the actual calculation is quite nasty, see paper for details.

In the paper the authors write “we conclude that the quantum effect is even tinier than the current precision in the measurement of light deflection”, which is an understatement if I have ever seen one. If you are generous and put in a black hole of mass M and a photon that just about manages to avoid being swallowed, the quantum effect is smaller by a factor (m_p/M)² than the classical term, where m^p is the Planck mass. For a solar mass black hole this is about 70 orders of magnitude suppression. (Though on such a close approach the approximation with a small deflection doesn’t make sense any more.) If you have a Planck-mass black hole, the correction term is of order one – again that’s what you’d expect.

Yes, that is a very plausible result indeed. I would be happy to tell this any journalist, but unfortunately news items seem to be almost exclusively picked from the ever increasing selection of exciting nonsense.

I will admit that it is hard to communicate the relevance of rather technical calculations that don’t lead to stunning results, but please bear with me while I try. The reason this work is so important is that we have to face the bitter truth to find out whether that’s really all that there is or whether we indeed have reason to expect the truth isn’t as bitter as it said on the wrapping. You have to deal with a theory and its nasty details to figure out where it defies your expectations and where your guesses go wrong. And so, we will have to deal with effective quantum gravity to understand its limits. I always said it was a good idea. Even better that somebody else did the calculation so I can continue thinking about the exciting nonsense.

Bonus: True love.

And the winners are...

The pile of money whose value you have been guessing came out to be 68.22 Euro and 0.5 Deutsche Mark, the latter of which I didn't count. Hoping that I didn't miss anybody's guess, this means the three winning entries are:

• Rbot: 72
• Rami Kraft: 62
• droid33: 58.20

Congratulations to the winners! Please send an email to hossi[at]nordita.org with your postal address and I will send the books on the way.

Make a guess, win a book.

The twins' piggy banks are full, so I've slaughtered them. Put in your guess of how much they've swallowed and you can win a (new) copy of Chad Orzel's book "How to Teach Quantum Physics to Your Dog". (No, I'm not getting paid for this, I have a copy I don't need and hope it will make somebody happy.) You can put in your guess until Monday, midnight, East Coast Time. I will only take into account guesses posted in the comments - do not send me an email. I am looking for the amount in Cent or Euro, not the number of coins. The winners will be announced Tuesday morning. Good luck!

The paradigm shift you didn’t notice

Inertia creeps.

Today, for the first time in human history a scientist has written this sentence – or so would be my summary of most science headlines I read these days. Not only do the media buy rotten fish, they actually try to resell them. The irony is though that the developments which really change the way we think and live happen so gradually you wouldn’t ever learn about them in these screaming headlines.

HIV infection for example still hasn’t been cured, but decades of hard work turned it from a fatal disease into a treatable one. You read about this in longwinded essays in the back pages where nobody looks, but not on the cover page and not in your news feed. The real change didn’t come about by this one baby who smiles on the photo and who was allegedly cured, as the boldface said, but by the hundreds of trials and papers and conferences in the background.

These slow changes also happen in physics. Quantum measurement is a decoherence process rather than collapse. This doesn’t break the ground but slowly moves it. It’s an interpretational shift that has spread through the community. Similarly, it is now generally accepted that most infinities in quantum field theory do not signal a breakdown of the theory but can be dealt with by suitable calculational methods.

For me the most remarkable shift that has taken place in physics in the last decades is the technical development and, with it, acceptance of renormalization group flow and effective field theories. If this sounds over your head, bear with me for I’m not going into the details, I just want to tell you why it matters.

You have certainly heard that some quantum field theories are sick and don’t make sense – they are said to be non-renormalizable. In such a theory the previously mentioned infinities cannot be removed, or they can only be removed on the expense of introducing infinitely many free parameters which makes the theory useless. Half a century ago a theory with this disease was declared dead and went where theories go to die, into the history aisle.

Then it became increasingly clear that such non-renormalizable theories can be low-energy approximations to other theories that are healthy and renormalizable. The infinities are artifacts of the approximation and appear if one applies the approximation outside its regime of validity.

These approximations at low energies are said to be “effective” theories and they typically contain particles or degrees of freedom that are not fundamental, but instead “emergent”, which is to say they are good descriptions as long as you don’t probe them with too high energy. The theory that is good also at high energies is said to be the “UV completion” of the effective theory. (If you ever want to fake a physics PhD just say “in the IR” instead of “at low energy” and “UV” instead of “high energy”.)

A typical example for an effective theory is the nuclear force between neutrons and protons. These are not fundamental particles – we know that they are made of quarks and gluons. But for nuclear physics, at energies too small to test the quark substructure, one can treat the neutrons and protons as particles in their own right. The interaction between them is then effectively mediated by a pion, a particle that is itself composed of two quarks.

Fermi’s theory of beta-decay is a historically very important case because it brought out the origin of non-renormalizability. We know today that the weak interaction is mediated by massive gauge-bosons, the W’s and the Z. But at energies so low that one cannot probe the production and subsequent decay of these gauge bosons, the weak interaction can be effectively described without them. When a neutron undergoes beta decay, it turns into a proton and emits an electron and electron-anti-neutrino. If you do not take into account that this happens because one of the quark constituents emits a W-boson, then you are left with a four-fermion interaction with a coupling constant that depends on the mass of the W-boson. This theory is not renormalizable. Its UV completion is the standard model.

Upper image: One of the neutron's quark constituents interacts via a gauge boson with an
electron. Bottom image: If you neglect the quark substructure and the boson-exchange, you get a four-fermion interaction with a coupling that depends on the mass of the boson and which is non-renormalizable.

So now we live and work with the awareness that any quantum field theories is only one in a space of theories that can morph into each other, and the expression of the theory changes with the energy scale at which we probe the physics. A non-renormalizable theory is perfectly fine in its regime of validity. And thus today these theories are not declared dead any longer, they are declared incomplete. A theory might have other shortcomings than being non-renormalizable, for example because it contains dimensionless constants much larger than (or smaller than) one. Such a theory is called unnatural. In this case too you would now not simply discard the theory but look for its UV completion.

It is often said that physicists do not know how to quantize gravity. This isn’t true though. Gravity can be quantized just like the other interactions; the result is known as “perturbatively quantized gravity”. The problem is that the theory one gets this way is non-renormalizable, which is why it isn’t referred to as quantum gravity proper. The theory of quantum gravity that we do not know is the UV-completion of this non-renormalizable perturbative quantization. (It cannot be non-renormalizable in the same way as Fermi’s theory because gravity is a long-range interaction. We know that gravitons, if they have masses at all, have tiny masses.)

But our improved understanding of how quantum field theories at different energies belong together has done more than increasing our acceptance of theory with problems. The effective field theory framework is the tool that binds together, at least theoretically, the different disciplines in physics and in the sciences. No longer are elementary particle physics and nuclear physics and atomic physics and molecular physics different, disconnected layers of reality. Even though we cannot (yet) derive most of the relations between the models used in these disciplines, we know that they are connected through the effective field theory framework. And at high energies many physicists believe it all goes back to just one “theory of everything”. Don’t expect a big headline announcing its appearance though. The ground moves slowly.