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

• I
• a dull boy
In summary, the theorem can be used to calculate the general curvature of a spacetime, but it does not always apply to pseudo-Riemannian manifolds and the manifold must be compact.f

#### a dull boy

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

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

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?

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.

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

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