# Is the total amount of gravity in the Universe conserved?

• I
In thinking about force symmetry and conservation laws, I think I am right that the total amount of color charge and electric charge in the Universe is conserved, but is gravity conserved? does a Universe at maximum entropy have the same gravity as one say, just after the big bang?

Thanks, Mark

Orodruin
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What do you mean by "amount of gravity"? Without specifying this, your question cannot be answered.

Ssnow
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but is gravity conserved?

you mean the total mass? I suggest to reformulate the question...

Is gravity conserved in the way that mass-energy is conserved? Did the early Universe have the same amount of gravity as it does today?

PeterDonis
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I think I am right that the total amount of color charge and electric charge in the Universe is conserved, but is gravity conserved?

"Gravity" is not a charge like color charge and electric charge. The closest analogue to a charge for gravity would be energy. As for whether that is conserved, read this:

http://www.preposterousuniverse.com/blog/2010/02/22/energy-is-not-conserved/

I am swimming in deep water here, but isn't it possible to define some notion of "gravity conservation" by considering the topological invariants of the spacetime in question ? In particular I am thinking of the ( generalised ) Gauss-Bonnet theorem here, which connects a measure of total curvature with the Euler characteristic of the manifold; the latter being a topological invariant means that there is a constraint as to how curvature can evolve over time. If you start with a particular gravitational set-up, then it is certainly possible to distort the spacetime manifold in various ways, but you can do so only in a manner that preserves the Euler characteristic.

In that sense, could one not define some concept of "total curvature" ( in the context of GR ) that is in fact a conserved quantity ? This is easily visualised by taking a sphere - you can twist, distort and deform that sphere in any which way you want, but no matter what you do, you will never be able to deform it into a flat plane, i.e. you will never be able to completely eliminate the curvature that was intrinsic to the manifold when you started. The same should be true for gravity - you cannot locally "eliminate" it, the most you can do is spread it out as gravitational radiation.

PeterDonis
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I am thinking of the ( generalised ) Gauss-Bonnet theorem

AFAIK this theorem only applies to Riemannian manifolds, not pseudo-Riemannian, so it would not apply to spacetime.

AFAIK this theorem only applies to Riemannian manifolds, not pseudo-Riemannian, so it would not apply to spacetime.

You are right, I overlooked that However, it seems that the theorem can be extended to cover pseudo-Riemannian manifolds as well, so long as they are orientable :

https://duetosymmetry.com/files/An. Acad. Brasil. Ci. 1963 Chern.pdf

PeterDonis
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it seems that the theorem can be extended to cover pseudo-Riemannian manifolds as well, so long as they are orientable

Good point @PeterDonis, I overlooked that as well 