How Can You Simplify General Relativity Equations Without Using Tensors?

[Vardonir]

Homework Statement

Essentially, my adviser just told me to get the equation, then expand it so that it doesn't have any tensors. That's it. Just get rid of the tensors.

He said that this would be normally found in some textbook...? I've searched a lot of books, but I never saw anything that didn't have the equations in tensor form.

Homework Equations

I got G_μv = 8πT_μv.

Reading further, I also got the following:
G_μv = R_μv - (1/2)g_μvR
R = g^αβR_αβ

And I also found the Ricci tensor in terms of the Christoffel symbols but that's too long to type.

The Attempt at a Solution

What I tried doing is just plug in values of the greek indices and treat each one as a separate equation. However, when I use g_αβ = diag(-1, 1, 1, 1), the left hand side always becomes zero.

Is there any other value of the metric tensor that I can use?

 

[HallsofIvy]

The whole point of the Einstein tensor is that it is trivial in flat space, where there is no gravity. What you get for the Einstein tensor depends upon the metric tensor which, in turn, depends upon the distribution of mass in space. What situation are you assuming? You are trying to find the Einstein tensor for what space?

 

[Vardonir]

The whole point of the Einstein tensor is that it is trivial in flat space, where there is no gravity. What you get for the Einstein tensor depends upon the metric tensor which, in turn, depends upon the distribution of mass in space. What situation are you assuming? You are trying to find the Einstein tensor for what space?

I wasn't really given that much information, aside from "find the equations in terms of x, y, z, and t." I'm guessing Euclidean space, then.

 

[Muphrid]

I wouldn't plug in anything for the metric. Just leave the formulas in terms of $g_{tt}, g_{tx}$ and so on.

 

[paranormal]

I would first clarify with my adviser what they mean by "getting rid of tensors" in the equations. It is important to understand the reasoning behind this request and to make sure that the resulting equations are still accurate and valid.

Assuming that the goal is to express the equations in a more simplified form, one approach could be to use the Einstein field equations in the form of Gμv = 8πTμv and then use the metric tensor gμv to express the Ricci tensor Rμv in terms of the Christoffel symbols. This would result in a set of equations that do not explicitly contain tensors, but still represent the same physical relationships.

Alternatively, one could also use a different coordinate system or a different metric tensor, such as the Minkowski metric, to simplify the equations. However, it is important to note that these simplifications may also result in a loss of information or accuracy in the equations.

Overall, it is important to carefully consider the implications of "getting rid of tensors" in the equations and to ensure that the resulting equations are still valid and representative of the physical phenomena being studied.