In GR gravity, does "equal and opposite" still hold and what does that mean?

[FactChecker]

TL;DR Summary

In GR, if two massive objects distort spacetime, are the effects still "equal and opposite" in some sense? In what sense?

In GR, if two massive objects distort spacetime, are the effects still "equal and opposite" in some sense? In what sense? It seems that for "equal" to hold, the equations of GR must force the effect of one mass, A, on spacetime at the location of another mass, B, to equal the converse (the effect of mass B on spacetime at location A). But the masses may not be equal. It hurts my head. And what would "equal and opposite" even mean in GR?
Is this a characteristic of force fields?

 

[PeterDonis]

are the effects still "equal and opposite" in some sense?

Why do you think this would be the case? Remember that in GR, gravity is not a force.

 

[Nugatory]

"Equal and opposite" is a non-relativistic formulation; there is an implicit "at the same time" in the statement that the force on A from B is equal and opposite to the force on B from A.

Classical electrostatics, as when we're applying Coulomb's law, finesses this issue because it's static - nothing changes so the forces are the same no manner when we look and "at the same time" makes no difference.

This doesn't work with electrodynamic problems in which the forces are changing over time. Instead we introduce the notion of the electromagnetic field: charged particle A moves; it is subject to a force from its interaction with the local electromagnetic field; it also changes the field; the change propagates at the speed of light and eventually reaches charged particle B; particle B is subject to a force from its interaction with the changing local field. Everything is calculated in terms of the charges and fields at a single point so there is no "at the same time" problem and electrodynamics works as a relativistic theory.

Classical gravity doesn't have this out because Newton's law for the force between two masses is based on where they are right now, at the same time. There's no way of recasting this in relativistic terms. This conflict between classical gravitation and special relativity was a major problem and a strong motivation for developing general relativity; the cure was to replace the notion of gravitational forces between bodies with inertial motion in curved spacetime.

So to answer your question: No, "equal and opposite" does not hold in relativity, both because it is inherently incompatible with relativity and because it is about forces, which aren't part of GR. When people say "gravity isn't a force in GR" they aren't just being pedantic about what forces are, they are telling you that "gravitational force" is the wrong model and thinking in those terms will confuse and mislead you.

[Note - an "are" => "aren't" typo was fixed hours late above.]

 

[Mister T]

Even outside of GR, Newton's Third Law does not always hold. This is why modern physics has given primacy to conservation of momentum in the hierarchy of physical law.

 

[FactChecker]

Thanks all! I really like those answers. As a casual amateur, I wondered why the equations of GR would even make it nearly "equal and opposite". But conservation of momentum might be the answer. Have I stumbled into a consequence of Noether's theorem?

 

[PeterDonis]

conservation of momentum might be the answer

As far as looking for the best a relativistic theory can do at coming close to the concept of Newton's Third Law, yes, it is the answer. Or, to look at it the other way around, Newton's Third Law is actually just a consequence of conservation of momentum in the approximation where Newton's Laws are a sufficiently accurate description of the physics (meaning when gravitational fields are very weak and all relative speeds are very small compared to the speed of light).

 

[FactChecker]

As far as looking for the best a relativistic theory can do at coming close to the concept of Newton's Third Law, yes, it is the answer. Or, to look at it the other way around, Newton's Third Law is actually just a consequence of conservation of momentum in the approximation where Newton's Laws are a sufficiently accurate description of the physics (meaning when gravitational fields are very weak and all relative speeds are very small compared to the speed of light).

Somehow I find this very beautiful and inspirational in its basic simplicity.

 

[pervect]

TL;DR Summary: In GR, if two massive objects distort spacetime, are the effects still "equal and opposite" in some sense? In what sense?

In GR, if two massive objects distort spacetime, are the effects still "equal and opposite" in some sense? In what sense? It seems that for "equal" to hold, the equations of GR must force the effect of one mass, A, on spacetime at the location of another mass, B, to equal the converse (the effect of mass B on spacetime at location A). But the masses may not be equal. It hurts my head. And what would "equal and opposite" even mean in GR?
Is this a characteristic of force fields?

I'm not sure what you mean by "equal and opposite", and it sounds like you are not sure either? Could you give an example of what you mean in Newtonian terms. What comes to mind is equal and opposite forces on two masses m1 and m2, with Newton's law of gravitation, but I'm not sure that's what you meant.

You can probably same something that's similar and approximately true in GR if you have a conserved momentum (for instance an asymptotically flat space-time), and no gravitational radiation (which can carry away momentum). Additionally you may have to add a constraint that you are in the linear realm of the field equations, because linearity is necessary for a total effect to be the sum of the effect of all the parts, i.e. in the linear realm, the effect of several masses on m1 is the sum of the effect of each mass on m1. In the non-linear realm, you can't in general write the total effect as such a linear sum.

That said, even with all these assumptions, I don't have any coordinate independent notion of what the "effect" might be, just the coordinate dependent notion of the rate of change of the conserved momentum with respect to coordinate time.

 

[FactChecker]

I'm not sure what you mean by "equal and opposite", and it sounds like you are not sure either? Could you give an example of what you mean in Newtonian terms. What comes to mind is equal and opposite forces on two masses m1 and m2, with Newton's law of gravitation, but I'm not sure that's what you meant.

Yes, that was what I meant.

You can probably same something that's similar and approximately true in GR if you have a conserved momentum (for instance an asymptotically flat space-time), and no gravitational radiation (which can carry away momentum). Additionally you may have to add a constraint that you are in the linear realm of the field equations, because linearity is necessary for a total effect to be the sum of the effect of all the parts, i.e. in the linear realm, the effect of several masses on m1 is the sum of the effect of each mass on m1. In the non-linear realm, you can't in general write the total effect as such a linear sum.

That is what I vaguely had in mind as the complication that made me wonder how "equal and opposite" could even be defined.

That said, even with all these assumptions, I don't have any coordinate independent notion of what the "effect" might be, just the coordinate dependent notion of the rate of change of the conserved momentum with respect to coordinate time.

I see. I'm afraid it is really just too complicated for a casual amateur like me to understand in any concrete way. That is ok with me. I am satisfied that the complexity I imagined is real and not just the result of my ignorance.
Thanks.

 

[Mister T]

Newton's Third Law involves action at a distance and is not compatible with modern field theories, going back to Faraday and then Maxwell's work on electromagnetism.

 

[Orodruin]

Newton's Third Law involves action at a distance and is not compatible with modern field theories, going back to Faraday and then Maxwell's work on electromagnetism.

Not entirely accurate. However, you need to let go of the action at a distance and instead consider the forces between charges and field. The force of the charge on the field is the same magnitude and opposite direction of the force of the field on the charge.

In relativity this is essentially captured by the total stress-energy tensor being divergence free (meaning the divergences of distinct contributions to it being equal in size and opposite in sign).

Edit: In short, Newton’s third law is essentially telling you momentum is conserved and the stress-energy tensor being divergence free captures this as well as energy conservation.

 

[FactChecker]

Not entirely accurate. However, you need to let go of the action at a distance and instead consider the forces between charges and field. The force of the charge on the field is the same magnitude and opposite direction of the force of the field on the charge.

I think this might get to the heart of my question in a way that I had not considered.

In relativity this is essentially captured by the total stress-energy tensor being divergence free (meaning the divergences of distinct contributions to it being equal in size and opposite in sign).

Edit: In short, Newton’s third law is essentially telling you momentum is conserved and the stress-energy tensor being divergence free captures this as well as energy conservation.

Interesting. Thanks!

 

[Orodruin]

I think this might get to the heart of my question in a way that I had not considered.

Well … not really if I understood your question correctly as pertaining to gravity. For other interactions, yes - the principle exists and is local. But gravity is not locally exchanging stress energy from one system to another.