Is GR superior to Newtonian gravity?

[zonde]

Can we use GR to model real world gravitating objects without any reference to Newtonian gravity? There are a lot of interactions between massive particles involving exchange of energy. In Newtonian gravity we have gravitational potential which fits into the framework of classical physics quite nicely. It does not seem like there is similar concept in GR.
GR is very good when we describe large distinct objects moving in gravitational field of other large distinct objects. But it does not seem quite adequate for internal dynamics of gravitating body because concepts like binding energy is somewhat orthogonal to GR.
Does it seems right what I am saying?

 

[Ibix]

You certainly can model the interiors of extended objects in GR. I wrote an Insight article on one simple model. The Oppenheimer-Snyder black hole includes a model of the star before it collapses into a black hole. And modelling of neutron star mergers emitting gravitational waves has been done for LIGO.

I think, as is usually the case, if you have a fairly middle-of-the-road case like a planet then Newton is sufficient and mathematically much simpler. Why bother including the contribution of pressure and internal energy to gravity when it's a correction on the fifteenth decimal place, or whatever?

Binding energy is certainly a concept in GR - if two distant objects fall together under their mutual gravity and collide plastically the kinetic energy converted to heat in the impact is calculable (in principle!). And the energy you need to supply to separate them to infinity again is well defined. It just doesn't have an interpretation in terms of released GPE.

 

[Dale]

Can we use GR to model real world gravitating objects without any reference to Newtonian gravity?

Yes. There is a pretty substantial literature on this topic.

In Newtonian gravity we have gravitational potential which fits into the framework of classical physics quite nicely.

Anything that you can do with the Newtonian gravitational potential can be done with the weak field metric in GR.

Does it seems right what I am saying?

Not to me. Relativistic stellar dynamics is an advanced topic, but the weak field metric is covered in every introductory source I have ever seen.

 

[PeterDonis]

concepts like binding energy is somewhat orthogonal to GR.

Not at all. In GR, in fact, it's simpler than in Newtonian mechanics, since it's just the difference between the invariant mass of a system's constituents when they are not bound, and the invariant mass of the bound system. In other words, it's the energy that has to be released in order to assemble the bound system from its unbound constituents. The reason it's simpler in GR is that you can make use of mass-energy equivalence (note that I phrased things in terms of "mass" above); in Newtonian mechanics you can't, and that complicates things conceptually.

 

[zonde]

You certainly can model the interiors of extended objects in GR. I wrote an Insight article on one simple model.

Nice article. But you just plug in parameters for stress-energy tensor. The point is that parameters for stress-energy tensor are input for GR not output.

Anything that you can do with the Newtonian gravitational potential can be done with the weak field metric in GR.

Yes. So what about strong field? With what do you replace Newtonian gravitational potential in strong field regime?

Not at all. In GR, in fact, it's simpler than in Newtonian mechanics, since it's just the difference between the invariant mass of a system's constituents when they are not bound, and the invariant mass of the bound system. In other words, it's the energy that has to be released in order to assemble the bound system from its unbound constituents. The reason it's simpler in GR is that you can make use of mass-energy equivalence (note that I phrased things in terms of "mass" above); in Newtonian mechanics you can't, and that complicates things conceptually.

Yes, Newtonian physics does not adjust for mass difference from binding energy. In regime where this difference can't be ignored it simply makes Newtonian physics wrong I would say.
But it seems you shy away from going just one step further. Bound system is not a black box and entities in bound system locally still obey the same physics laws as entities in unbound systems. Given equivalence principle that means invariant mass should be reduced and it can be reduced only by universal proportionality factor for local physics laws to keep their form. And that proportionality factor just happens to be redshift factor.

 

[pines-demon]

Ever heard of Newton-Cartan theory?

 

[Ibix]

Nice article.

Thank you.

The point is that parameters for stress-energy tensor are input for GR not output.

In general you specify a material model in the form of differential equations relating the components of the stress-energy tensor, and initial conditions that describe your specific scenario. This is exactly the same input as Newtonian physics. My example is just very simple, with unvarying material properties so there's no clear distinction between initial conditions and the material model.

Cosmological models, for instance, relate the density to the scale factor as $\rho\propto a^{-n}$, where $n$ depends on the type of stuff ($n=3$ for matter, $n=4$ for radiation, for example) and relate density to pressure by $\rho=wp$, where $w$ again depends on your type of stuff. The resulting system of equations is soluble analytically for "pure" cases, but more realistic cases with a mix of matter and radiation and dark energy requires you to resort to a numerical integrator.

 

[zonde]

In general you specify a material model in the form of differential equations relating the components of the stress-energy tensor, and initial conditions that describe your specific scenario. This is exactly the same input as Newtonian physics.

Imagine such material model:
There are bunch of neutrons bound by gravity. These neutrons are degenerate meaning all gravitational binding energy has been removed from the system and they occupy lowest possible energy states allowed by Pauli exclusion principle.
Will they arrange themselves similarly to electrons in some heavy atom? Like two lowest energy neutrons right at the center of gravitational potential in some sort of "s1 orbital" and other neutrons in higher energy orbitals spanning larger and larger volume around the center of gravitational potential? Let's ignore other fundamental forces for the sake of the question.

 

[zonde]

Ever heard of Newton-Cartan theory?

As I understand it's Newtonian gravity rewritten in terms of GR approach. It does not seem relevant for the questions I am trying to ask. But I could be wrong of course.

 

[Dale]

But you just plug in parameters for stress-energy tensor. The point is that parameters for stress-energy tensor are input for GR not output

Of course. The same is true in Newtonian gravity. Why do you express this as an objection?

Yes. So what about strong field? With what do you replace Newtonian gravitational potential in strong field regime?

Newtonian gravity doesn’t work in the strong field regime.

You are trying to claim some superiority of Newtonian gravity over GR, so your objections here make no sense. They are either things that are the same with Newtonian gravity or things where Newtonian gravity fails.

About the only things that actually is better for Newtonian gravity is that its equations are simpler to calculate. Everywhere that Newtonian gravity makes accurate predictions, so does GR. You can take the same inputs, plug them in to GR’s more complicated equations, and get the same output. But the reverse is not true. In many cases GR makes accurate predictions but Newtonian gravity does not.

 

[Ibix]

There are bunch of neutrons bound by gravity. These neutrons are degenerate meaning all gravitational binding energy has been removed from the system and they occupy lowest possible energy states allowed by Pauli exclusion principle.
Will they arrange themselves similarly to electrons in some heavy atom?

No idea. I think you'd need to solve the Tolman-Oppenheimer-Volkov equation and plug that back through the EFEs to get the background spacetime, then do some QFT on that curved background to get the states of the neutrons. That's a quasi-classical model because we don't have a true quantum theory of gravity. The QFT aspects of it are also a bit beyond me, I'm afraid.

 

[zonde]

Of course. The same is true in Newtonian gravity. Why do you express this as an objection?

As I see Newtonian gravitational potential is an input in determining mass-energy arrangement inside gravitating body and it fits right in with other fundamental forces.
I am sort of skeptical that GR has something to substitute for gravitational potential. Probably that's why I express it as an objection.

You are trying to claim some superiority of Newtonian gravity over GR, so your objections here make no sense.

Not exactly. Newtonian gravity clearly does not work in strong field regime. I am rather sort of suspecting absence of adequate model in that area. But I'm certainly open to possibility that I'm wrong.

 

[Dale]

As I see Newtonian gravitational potential is an input in determining mass-energy arrangement inside gravitating body and it fits right in with other fundamental forces.

How do you think that is that different from GR?

I am sort of skeptical that GR has something to substitute for gravitational potential.

It literally has the gravitational potential. The weak field metric in GR is $$ ds^2= -(1+2U)dt^2+(1-2U)(dx^2+dy^2+dz^2)$$ where $U$ is the Newtonian gravitational potential. It isn't a substitute for the gravitational potential, it is the gravitational potential. Any correct calculation you can get using the Newtonian potential in Newtonian gravity you can use the same potential in GR and get the same correct calculation.

I am rather sort of suspecting absence of adequate model in that area.

If you are concerned about strong gravitational fields then Newtonian gravity is irrelevant. Newtonian gravity is already empirically excluded in strong fields, and therefore differences between GR and Newtonian gravity in strong fields do not imply an inadequacy of GR.

 

[PeterDonis]

With what do you replace Newtonian gravitational potential in strong field regime?

You have it backwards. GR doesn't "replace" the Newtonian potential with anything. The Newtonian potential is something that emerges in the Newtonian approximation. In the strong field regime where that approximation breaks down, there is no reason to expect anything to "replace" the Newtonian potential.

 

[PeterDonis]

I am rather sort of suspecting absence of adequate model in that area.

I have no idea why you would think this, since GR is precisely such a model.

 

[PeterDonis]

There are bunch of neutrons bound by gravity. These neutrons are degenerate meaning all gravitational binding energy has been removed from the system and they occupy lowest possible energy states allowed by Pauli exclusion principle.

I suggest spending some time reading Shapiro & Teukolsky, which discusses in some detail the physics of neutron stars, including quantum effects.

 

[pines-demon]

As I understand it's Newtonian gravity rewritten in terms of GR approach. It does not seem relevant for the questions I am trying to ask. But I could be wrong of course.

Well it is the right approximation of GR for the problems you are dealing with.