# Gravitational field in GR

## Main Question or Discussion Point

He who calls himself "Nabeshin" stated:

"It seems to me the distinction can be best seen in the following: Differential geometry is mathematics, and this will tell us what the geodesics on a given manifold are. So if we're just finding geodesics on manifolds, maybe it's the manifold corresponding to schwarzchild, we're just doing mathematics. But when I say "I'll restrict myself to Lorentzian manifolds, and I'll have that particles move on timelike geodesics" all of a sudden I'm doing physics. I've restricted myself to a subclass of mathematical theories which I have hypothesized correspond to observable reality."
What he meant to say is that when we think in terms of particles and how it moves on timelike geodesics, then it's physics. While thinking in terms of what the geodesics on a given manifold are in the differential geometry is pure mathematics."

How about the gravitational field as as tensor field. Do we use it to correspond to Newtonian gravitational field? But since gravity is pure spacetime geometry. Why do we need the idea of gravitational field as a tensor field? Why not do away with it totally and just focus on geometry?

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From what I know, and I apologize if I am repeating something that you already know; is that we need tensors because the curvature of spacetime is very complex since our universe is 4 dimensional. For example, a Riemanian tensor is basically a matrix that contains values that correspond to the amount of space time curvature in an area of spacetime. For our universe the Riemanian tensor has 16 values. We need the tensors so that we will be able to predict how much spacetime will curve.

If I am wrong on any of this I hope that someone will correct me.

bcrowell
Staff Emeritus
Gold Member
How about the gravitational field as as tensor field. Do we use it to correspond to Newtonian gravitational field? But since gravity is pure spacetime geometry. Why do we need the idea of gravitational field as a tensor field?
There is no tensor field in GR corresponding to the Newtonian gravitational field. Proof: Suppose that there was such a tensor field. The tensor transformation law never takes a zero tensor to a nonzero tensor. But by the equivalence principle, the gravitational field can be made zero or nonzero based on a choice of coordinates.

Why not do away with it totally and just focus on geometry?
We do.

There is no tensor field in GR corresponding to the Newtonian gravitational field. Proof: Suppose that there was such a tensor field. The tensor transformation law never takes a zero tensor to a nonzero tensor. But by the equivalence principle, the gravitational field can be made zero or nonzero based on a choice of coordinates.

We do.
But in another thread. Someone called Atyy said the following which implied there is a newtonian gravitational field equivalent in the EFE:

"The technical answer to the OP, going a different route from clocks is that the gravitational field is not the manifold. The gravitational field is simply a tensor field on the manifold with a certain gauge structure. Other forces such as the electromagnetic force, and matter such as electrons, are represented by other fields. The interaction between the gravitational field and the electron field is represented by a term in the Lagrangian, just as the interaction between the electric field and electrons is represented by another term in the Lagrangian."

So why did he mean "The gravitational field is simply a tensor field on the manifold with a certain gauge structure". Did he mean the newtonian gravitational field? if not. What?

bcrowell
Staff Emeritus
Gold Member
So why did he mean "The gravitational field is simply a tensor field on the manifold with a certain gauge structure". Did he mean the newtonian gravitational field? if not. What?

bcrowell
Staff Emeritus
Gold Member
My interpretation of the post is that when he says "gravitational field," he doesn't actually mean an analog of the Newtonian gravitational field g. It's fairly common in relativity to use "field" to mean something more general and vague.

My interpretation of the post is that when he says "gravitational field," he doesn't actually mean an analog of the Newtonian gravitational field g. It's fairly common in relativity to use "field" to mean something more general and vague.
Is it right that the gravitational potential still exist in GR perhaps as the tensor? If not, what is the gravitational potential equivalent in GR?

pervect
Staff Emeritus
Gravitational potential does not really exist in a general geometry in GR. Something equivalent can exist in special cases - for instance a static geometry, or more generally a stationary geometry.

However, it's not usually called "gravitational potential". For instance, if you look up "gravitational potential" in the index of Wald's book "General Relativity", you won't find any entries for the term.

If you look at Wald's chapter on energy in general relativity, however, a couple of notions of energy in General Relativity will be discussed, the easiest of which is related to the existence of time-like Killing vectors.

I am reminded that some authors DO refer to the metric tensor as a sort of "gravitational potential" - it's probably not the sort of gravitational potential the original poster is thinking of, though. I belive this usage comes from a variational treatment. See for instance http://www.cft.edu.pl/~kijowski/Odbitki-prac/GR_gauge.pdf.

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atyy
But in another thread. Someone called Atyy said the following which implied there is a newtonian gravitational field equivalent in the EFE:

"The technical answer to the OP, going a different route from clocks is that the gravitational field is not the manifold. The gravitational field is simply a tensor field on the manifold with a certain gauge structure. Other forces such as the electromagnetic force, and matter such as electrons, are represented by other fields. The interaction between the gravitational field and the electron field is represented by a term in the Lagrangian, just as the interaction between the electric field and electrons is represented by another term in the Lagrangian."

So why did he mean "The gravitational field is simply a tensor field on the manifold with a certain gauge structure". Did he mean the newtonian gravitational field? if not. What?
By "gravitational field" I meant the metric tensor field. In the approximation where we have test particles undergoing geodesic motion, and in the slow motion, weak field limit, the metric tensor field is analogous to the Newtonian gravitational potential. See Eq 6.26 of http://nedwww.ipac.caltech.edu/level5/March01/Carroll3/Carroll6.html

OTOH, by "gravitational field" bcrowell meant the gradient of the Newtonian potential.

The idea of freely falling test particles having worldlines that are geodesics of the metric tensor field is what is normally meant by gravity (metric tensor field) being geometry.

The manifold by itself without the metric tensor field has no geometry (in the GR sense) since angle between test particle worldlines and proper time along a test particle worldline cannot be defined unless there is a metric tensor field. Also, spacetime curvature is related to the second derivative of the metric tensor field.

However, I favour a "non-geometrical" view of the metric tensor field, since test particles and ideal clocks that read proper time are not fundamental objects in GR. Technically the gauge structure of the metric tensor field still defines a geometry, but then so does the gauge structure of the electromagnetic 4-potential.

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