- #1

- 267

- 2

**Question on wording for paper (re-explained in a second post)**

In a paper (undergrad thesis type paper) on GR, I have the statement:

*A vanishing Ricci Tensor, i.e., [tex] R_{\mu \nu}=0 [/tex]*

is not enough to explicitly define a flat space-time. In the case where the manifold being investigated has only three-dimensions, proving that the Ricci tensor vanishes

is

is not enough to explicitly define a flat space-time. In the case where the manifold being investigated has only three-dimensions, proving that the Ricci tensor vanishes

**is**enough to assume a flat space. However, in a four dimensional space-time, the case of [tex] R_{\mu \nu}=0 [/tex]is

**not**sufficient proof of flatness. In a four-dimensional space-time, there are in fact geometries containing curvature that contain vanishing Ricci tensors. These correspond physically to gravitational waves.QUESTION:

This isn't a direct quote (I don't have the file on this computer), but my question is if I can generalize the statement about four dimensions to any dimension higher than three?

e.g., can I say "The use of a vanishing Ricci tensor as proof of flat space is valid only in three dimensions, and cannot be generalized to dimensions of four or higher."??

I feel really uncomfortable any time I attempt to explain mathematics with "words." I'm always afraid I'm going to choose the wrong words and make a totally erroneous statement.

If it makes a difference in the response, section immediately prior to this statement talks about the vanishing of the Christoffel symbols (or Affine connection....again, semantics has me uncomfortable) in a flat space causing the covariant derivative to equal the normal derivative. It also includes the expression

[tex]

R_{\mu \nu}=\Gamma ^{\alpha}_{\ \mu \alpha ,\nu}-\Gamma ^{\alpha}_{\mu \nu ,\alpha}=0 [/tex]

I'm trying to say that just showing that [tex]

R_{\mu \nu}=0

[/tex] isn't enough to "prove" a non-curved space-time in four dimensions......and want to know if I can generalize that to any dimension higher than 3. (Perhaps I'm wrong on three dimensions too....I'm stating that [tex] R_{rs}=0 [/tex] IS enough to assume flat space in three dimensions (and again....I don't know if this is generalizable to dimensions less than 3 as well)

I am just an undergraduate working on a thesis type paper and the professor I'm working with is gone for the week. I don't want to bother him with any more emails than necessary. lol

*Okay....I give up.....I can never get the "R" to show up in LaTeX form. It happens every time I try to post math. I don't know what I'm doing wrong, but the third equation is supposed to start with (R_uv), not (uv). My apologies. If I'm doing something stupid, can a mod (maybe the one that changed my post to include the R last week) tell me what it is?

Maybe I should keep it as my calling mark?

I'll be known as that "stupid 'no-R' guy."

Last edited: