# Momentum and spacetime curvature

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4 hours ago, md65536 said:

It should be possible to contrive an example where the Newtonian forces are small, and the purely GR aspects dominate and give results that are the exact opposite of what Newtonian gravity predicts.

How about an interior metric, such as FLRW - under the right circumstances, the distance between points in such a spacetime will increase over time (metric expansion), whereas under Newton the interior of energy-momentum distributions will always contract, but never expand (assuming there’s no effects other than gravity of course).

Or how about something like a gravitational geon - a topological construct that is held together purely by gravitational self-energy, without the presence of any other energy-momentum sources at all? This particular solution relies entirely on GR self-interaction effects - under Newton, a completely empty spacetime without any gravitational sources cannot contain (or maintain) gravity.

Any type of radiative spacetime should qualify too, since Newtonian gravity has no radiative degrees of freedom. At most you can have varying gravitational forces, but the oscillations of the force vector would be “longitudinal” (ie in the radial direction) and of a dipole nature, whereas in GR the effects are transverse and quadrupole.

Or anything with angular momentum, since Newton can’t model frame dragging effects, and thus will give wrong trajectories for free-falling bodies around rotating objects. You can also give angular momentum to an interior spacetime, and get something like the Gödel metric - it contains a number of peculiar effects that I don’t think Newton would be able to replicate, and certainly not based on just invariant mass.

5 hours ago, md65536 said:

The energy of the beams is frame-dependent, and as total momentum approaches zero, so does the energy.

I’m not so sure about this - the energies and momenta are certainly frame-dependent, but their sums should never cancel. Since E=hf, there is no physically realisable frame in which either beam is seen to have f=0 by the other beam, so I think you will always get a non-vanishing net energy of each beam with respect to the other. So I think in Newton, the beams should always attract according to an inverse-square law, irrespective of their relative direction of motion. Note that what we are asking about is the attraction between the beams (ie of each beam to the other), not how an external test-particle is attracted to the two-beam system as a whole (which would involve the system’s invariant mass in Newtonian gravity).

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Posted (edited)
On 3/16/2023 at 5:42 AM, Markus Hanke said:

How about an interior metric, such as FLRW [...]

Definitely a Newtonian approximation has an error amount and it seems completely useless in some of these cases.

An example I was thinking of is if you have 3 equal masses in an equilateral triangle say arranged in a v shape, and a test mass balanced in the center. Then you increase the masses of all 3 with the mass at the bottom very slightly more than the others. With Newtonian gravity the test mass would be expected to accelerate downward. But suppose that the increases of mass on the top 2 were from rotation, with each spinning opposite of the other so that the test mass would instead fall directly upward due to frame dragging.

That might work fine or not??? because as explained in this thread the effects are non-linear. You can't just add up or overlay 2 separate frame-dragging effects. So there may be some additional energy in the interaction of the 2 spinning masses (it might even be that to get the test mass to move upward, it still needs greater mass increase "upward"???), or some other thing I can't imagine. I guess either "there is an answer given just the information above, but it's more complicated than suggested" or "the answer changes depending on factors other than just which of the 3 masses is given more rotation or energy." I have no idea which.

On 3/16/2023 at 5:42 AM, Markus Hanke said:

Or how about something like a gravitational geon - a topological construct that is held together purely by gravitational self-energy, without the presence of any other energy-momentum sources at all? This particular solution relies entirely on GR self-interaction effects - under Newton, a completely empty spacetime without any gravitational sources cannot contain (or maintain) gravity.

This is an example of how we're talking about different things.

Mass in Newtonian gravity is based on how it's measured, such as how a mass fits gravity equations. It's not based on a prediction of what Newton's laws say it should weigh. For example we wouldn't say "The Newtonian mass of an atom is a lot less than the measured mass of an atom because it doesn't include the contribution of binding energy." Geons have mass, and Newtonian gravity can approximate its effect.

Suppose there's some elementary particle that has a measured mass, and behaves approximately as Newtonian gravity predicts. Say, for argument only, along the lines of Wheeler's speculation, that it is discovered the particle is actually a geon. We wouldn't say, "According to Newtonian gravity, this particle can now be considered massless and is predicted not to gravitate, contrary to measurements."

Mass in Newtonian gravity necessarily includes contributions of things that are only explained by GR. Meanwhile mass in GR does not have a single definition that works perfectly in all cases. So I guess if we're talking about mass and assuming that GR is the best model of spacetime, we're necessarily mixing concepts that are not purely GR. But, we still use mass and can make sense of statements about it, I guess because there is no alternative that can better convey the same information as concisely and meaningfully.

Well, this is all rather pointless, except that I think the concept that energy in the COM frame is equivalent to mass is generally useful even if it can be problematic.

On 3/16/2023 at 5:42 AM, Markus Hanke said:

I’m not so sure about this - the energies and momenta are certainly frame-dependent, but their sums should never cancel. Since E=hf, there is no physically realisable frame in which either beam is seen to have f=0 by the other beam, so I think you will always get a non-vanishing net energy of each beam with respect to the other. So I think in Newton, the beams should always attract according to an inverse-square law, irrespective of their relative direction of motion.

Sure, there's always energy in any given frame (even where it's too low to measure), but for beams with the same direction, none of it can be defined as rest mass, because there's no frame where the net momentum is zero.

However... if the beams aren't infinitely long, adding a single massive particle "to the system" then lets you define a rest mass that includes the beams. What would happen if you simply included a mass? My first intuition about it is certainly wrong.

With a mass, the COM frame doesn't depend on where the mass is. Would adding a small mass arbitrarily far away change the behavior of the beams? That certainly seems counter-intuitive. Or instead of a mass, you could just add another beam or pair of beams aimed in the opposite direction. If those were added very far away, would that make the 2 beams aimed in a common direction start attracting each other?

I think the answer is no, even though the system now has a rest mass. Indeed, if you have some beams close together in one direction, and some far away in another direction, that's approximately a description of the original experiment, except with a change of scale. The 2 beams aimed in one direction should bend toward the beams moving in the opposite direction, not toward each other.

So either intuition has broken down here, or the intuitive explanation is that it must be that when separate bits of energy combine to make up a rest mass, the density distribution of that mass is not necessarily the same as the density distribution of the energy that makes up it. The mass is located in some way between the oppositely directed beams, not inside the beams themselves.

(By this point, I don't know if it's a better argument that intuition fails, or that intuition can be improved to make sense of GR, but the standard explanation involving how the pp-waves of the different beams superimpose differently, might be a more intuitive and more useful way to think about it than this, and might better suggest what kind of mass distribution should be expected.)

Edited by md65536
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4 hours ago, md65536 said:

That might work fine or not???

I don’t know, to be honest. In GR, the n-body problem is notoriously complicated, because you cannot ignore the non-linear effects; and once you add angular momentum to the situation, I wouldn’t dare to make any predictions based on intuition. This is a scenario that would require a numerical simulation.

4 hours ago, md65536 said:

Geons have mass, and Newtonian gravity can approximate its effect.

Yes, this is true at some distance from the geon - in fact, this spacetime is asymptotically flat, so sufficiently far from the geon the situation becomes completely Newtonian, it looks just like any other gravitating body. But my point was that the geon itself cannot exist in Newtonian theory - it’s just a region of non-trivial spacetime curvature that is held stable purely by non-linear self-interaction effects (there are no other forms of energy-momentum). Newtonian gravity has no such effects, nor does it have any concept of spacetime curvature, so the concept of a “geon” would be meaningless within that theory. You couldn’t describe the geon itself using Newtonian gravity - only its effects on external test particles that are sufficiently distant.

4 hours ago, md65536 said:

Meanwhile mass in GR does not have a single definition that works perfectly in all cases.

It’s even more than that - it’s the question “how much energy-momentum in total does a given region of spacetime contain” that hasn’t got a straightforward answer that all observers can agree on. Fundamentally this is because you need to account for contributions from gravity itself, but in GR these contributions are not localisable, so they are difficult to work with.

But in GR this isn’t really a big problem, because this concept of mass is not really used anywhere. You don’t necessarily need a concept of mass in order to solve the field equations, and obtain particle trajectories. This is unlike in Newtonian theory, where you have no choice but to use “mass”, because it explicitly appears in all the equations.

5 hours ago, md65536 said:

(By this point, I don't know if it's a better argument that intuition fails, or that intuition can be improved to make sense of GR, but the standard explanation involving how the pp-waves of the different beams superimpose differently, might be a more intuitive and more useful way to think about it than this, and might better suggest what kind of mass distribution should be expected.)

The other option is to not look at this in terms of mass at all, but treat it purely as a geometric problem. This is what I tend to do. The advantage of such an approach is that it is pure GR, so it works in all cases, because there is a set procedure you can go through to work out the maths. The disadvantage is of course that even in some seemingly simple scenarios you may not be able to develop a good intuition of what happens, because the geometry is just too complicated to be sure. Your V-shaped arrangement of test particles above is a good example for this.

So perhaps we can say that a mass-based Newtonian intuition will work fine in some cases, but some other cases require a fully geometric GR approach. The question is of course how do you know the difference, and unfortunately there’s no general answer to that. For me personally, any spacetime that contains degrees of freedom such as angular momentum, charge, multipole moments, stresses, strains etc etc, I wouldn’t treat using Newtonian-based intuition, because I have been stung with this too many times in the past. This is why - and you might have noticed that over the years - when I answer people’s GR questions on this forum, I am very careful about giving intuition-based predictions about what happens in specific scenarios; in fact I try to avoid doing so altogether, unless I have the mathematical abilities to actually check the prediction, or the scenario is already known and has been analysed in the literature. GR is just a very subtle beast, and it’s easy to be wrong about things.

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Posted (edited)
3 hours ago, Markus Hanke said:

GR is just a very subtle beast, and it’s easy to be wrong about things.

I'd like to hear your advice about the next textbook on GR for me. I know the principles and the main results and don't want to go again through lengthy introductions and the basics. My goal is rather to delve into the subtleties and technicalities, calculations, examples, and exercises. What would you recommend? Thanks in advance.

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11 hours ago, Markus Hanke said:

You couldn’t describe the geon itself using Newtonian gravity - only its effects on external test particles that are sufficiently distant.

I think that applies to all elementary particles in Newtonian gravity.

16 hours ago, md65536 said:

it must be that when separate bits of energy combine to make up a rest mass, the density distribution of that mass is not necessarily the same as the density distribution of the energy that makes up it.

I think I've been tricked into showing that I was wrong all along. Even with a Newtonian approximation of GR, adding energy somewhere does not necessarily simply add a Newtonian gravitational effect there. Even when the added energy adds rest mass to a system, it might not behave as if the mass was added where the energy was.

Back to the original problem of 2 masses, it's possible it works approximately well when the masses pass, ie. with the system treated as a particle that gains rest mass as measured from a distance. However, when adding energy to each of the 2 masses, the behavior is not the same as if you increased their individual rest masses.

To try to get away from the Newtonian view, mass is not some kind of "stuff" that exists independently, it's just the measurement.

11 hours ago, Markus Hanke said:

when I answer people’s GR questions on this forum, I am very careful about giving intuition-based predictions about what happens in specific scenarios; in fact I try to avoid doing so altogether, unless I have the mathematical abilities to actually check the prediction

Of course I'd rather have simple answers but I suppose that's best, especially when bad intuitive explanations can stick around for decades, being repeated in pop sci media etc.

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11 hours ago, md65536 said:

I think that applies to all elementary particles in Newtonian gravity.

Yes, good point

11 hours ago, md65536 said:

Even with a Newtonian approximation of GR, adding energy somewhere does not necessarily simply add a Newtonian gravitational effect there.

Yes, with “necessarily” being the key term. It does work sometimes and gives the right intuition; but at other times it doesn’t work. The problem is that generally speaking there is no easy way to tell which is which - that’s why I think it’s dangerous to mix Newton with GR.

I'd like to hear your advice about the next textbook on GR for me. I know the principles and the main results and don't want to go again through lengthy introductions and the basics. My goal is rather to delve into the subtleties and technicalities, calculations, examples, and exercises. What would you recommend? Thanks in advance.

That’s a difficult to answer question, unfortunately. My own GR understanding came about as the sum total of a large number of different texts, and there’s really no single book that has all the subtleties in it. One text that was instrumental for me personally is one that is unfortunately not available in English (to the best of my knowledge), which is T Fliessbach, Allgemeine Relativitätstheorie. It’s the only text I have seen that explicitly and step-by-step goes through the entire procedure of solving the Einstein equations for Schwarzschild and FLRW from scratch - you can see exactly where sources come into this, how boundary conditions appear, where things like the mass term in the metric originate etc. It found that very illuminating, because it shows how many of our Newtonian intuitions about gravity just don’t hold. But like I said, unfortunately there seems to be no English version.

Then there’s of course MTW - ignoring the introductory parts, it’s the only text I know of that goes into the question of why the field equations look like they do. It goes into the topological considerations and conservation laws that underlie the entire machinery of GR, and explains the actual meaning of things like the Bianchi identities very well. I have not seen some of this material in any other text I know of. This is a 1500 pages tome, and evenly divided between introductory topics and more advanced stuff, but I think anyone interested in GR will take something away from this book, irrespective of what level they are on.

For the formal maths the gold standard is of course Wald, though to be honest as being an amateur I found it to be above my pay grade. I can see this would be of great benefit to someone who actually has a background in maths, or at least has studied the entire subject matter at university level. It’s a great reference for theorems, proofs etc though.

If you can give me some time, I will have to think about your question some more, and have a look through my personal library. It’s been years since I’ve really gone through these books - I’ve taken the key points out and compiled extensive notes for myself, which is what I am mainly working off these days. I might add some more recommendations here later

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@GenadyStephani’s Exact Solutions to the Einstein Field Equations is an absolute must if you want to go above and beyond the basics - not only does it classify the known exact solutions according to different schemes, it also explains the general features of these classes of spacetimes, and their mathematical treatment. The book also gives an overview over which general methods are available to find solutions to the equations, and what forms these solutions may take. So it’s definitely much more than “just” a reference catalogue. But be warned - this is definitely not a beginner’s text, it’s mathematically fairly demanding in places.

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Thanks a lot, @Markus Hanke. Please let me know if you have more thoughts on this question. No hurry.

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