How Does a Diagonal Metric Affect the Symmetry and Tensor Equations in Space?

[Funzies]

Hey guys! I am considering a space with a diagonal metric, which is maximally symmetric.

It can be proven that in that case of a diagonal metric the following equations for the Christoffel symbols hold:
$$\Gamma^{\gamma}_{\alpha \beta} = 0$$
$$\Gamma^{\beta}_{\alpha \alpha} = -(1/g_{\beta\beta})\partial_{\beta}g_{\alpha\alpha}$$
$$\Gamma^{\beta}_{\alpha \beta} = \partial_{\alpha}\ln(\sqrt{|g_{\beta\beta}|})$$
$$\Gamma^{\alpha}_{\alpha \alpha} = \partial_{\alpha}\ln(\sqrt{|g_{\alpha\alpha}|})$$

Furthermore: for a maximally symmetric space we have for the Riemann tensor:
$$R_{\rho\sigma\mu\nu} = R/12(g_{\rho\mu}g_{\sigma\nu}-g_{\rho\nu}g_{\sigma\mu})$$, where R is the Ricci scalar.

Given these equations, I come across a contradiction. From the above equation for the Riemann tensor we easily see that if it has three different indices, it must be zero (IF the metric is diagonal). However, if I plug in the Christoffelsymbols into the definition of the Riemanntensor expressed in Christoffel symbols and their derivatives, I do not find that this is zero in general. Does anyone know what is going wrong here?

EDIT: For a maximally symmetric space, three indices cannot be unequal, but for a diagonal space this might not be the case. How is this all related?

 

[pervect]

If I put the following diagonal line element into GrTensor, for coordinates (t,x,y,z),

-p(t,x,y,z)*d[t]^2+q(t,x,y,z)*d[x]^2+r(t,x,y,z)*d[y]^2+s(t,x,y,z)*d[z]^2;

I get the following (just a partial set).

$$\Gamma^t{}_{tt} = \frac{1}{2} \frac{\partial_t \, p}{p}$$
$$\Gamma^x{}_{tt} = -\frac{1}{2} \frac{\partial_x \, p}{q}$$
$$\Gamma^t{}_{xt} = \frac{1}{2} \frac{\partial_x \, p}{p}$$

They look similar, except for a missing factor of 1/2. The ln's complicate things a bit - are they really that useful?

I was rather surprised to see an apparently non_zero term pop out for R_txty myself. I'm not sure if it's really nonzero, or just didn't simplify:

$$R_{txty} = 1/4\,{\frac {-2\, \left( {\frac {\partial ^{2}}{\partial x\partial y}} p \left( t,x,y,z \right) \right) p \left( t,x,y,z \right) q \left( t, x,y,z \right) r \left( t,x,y,z \right) + \left( {\frac {\partial }{ \partial y}}p \left( t,x,y,z \right) \right) \left( {\frac { \partial }{\partial x}}p \left( t,x,y,z \right) \right) q \left( t,x, y,z \right) r \left( t,x,y,z \right) + \left( {\frac {\partial }{ \partial x}}p \left( t,x,y,z \right) \right) \left( {\frac { \partial }{\partial y}}q \left( t,x,y,z \right) \right) p \left( t,x, y,z \right) r \left( t,x,y,z \right) + \left( {\frac {\partial }{ \partial y}}p \left( t,x,y,z \right) \right) \left( {\frac { \partial }{\partial x}}r \left( t,x,y,z \right) \right) p \left( t,x, y,z \right) q \left( t,x,y,z \right) }{p \left( t,x,y,z \right) q \left( t,x,y,z \right) r \left( t,x,y,z \right) }}$$

 

[bcrowell]

When you say "maximally symmetric space" and "diagonal space," you're talking about two different kinds of things. The symmetry is intrinsic, but the diagonal form of the metric is coordinate-dependent.

I think the only maximally symmetric spaces in 3+1 dimensions are Minkowski space, de Sitter space, and anti de Sitter space. Have you tried, for example, checking your calculations in the special case of de Sitter space?

From the above equation for the Riemann tensor we easily see that if it has three different indices, it must be zero (IF the metric is diagonal).

What if the different components of the metric are unequal?

EDIT: For a maximally symmetric space, three indices cannot be unequal, but for a diagonal space this might not be the case. How is this all related?

Are you thinking that the different components of the metric have to be equal because it's maximally symmetric? That's not true. The symmetry doesn't have to be manifest when you write the metric in certain coordinates.

 

[pervect]

Given the line element, I don't see any requirement that there be ANY Killing vectors, much less the maximum possible number (which would be a requirement to be maximally symmetric space, if I understand correctly).

 

[sardar]

I would first commend you for your exploration of a space with a diagonal metric and your efforts to understand its properties. This is an interesting topic and has potential implications in various fields such as physics and cosmology.

In response to your question, it is important to note that the equations you have mentioned are derived under certain assumptions and conditions. The equations for the Christoffel symbols, in particular, are valid for a diagonal metric and a maximally symmetric space. However, the equation for the Riemann tensor is a more general one and may not hold in all cases, especially when the assumptions for the Christoffel symbols are not satisfied.

In a diagonal metric, the Christoffel symbols are zero for unequal indices, as you have correctly pointed out. This is a consequence of the symmetry of the metric, which implies that the space is isotropic in all directions. However, when the indices are equal (as in the case of the Riemann tensor), this symmetry may not hold and the Christoffel symbols may not be zero. This does not necessarily lead to a contradiction, but it does highlight the limitations of the assumptions and equations we are working with.

Furthermore, the equation for the Riemann tensor that you have mentioned is valid for a maximally symmetric space, which is characterized by a constant curvature. In a diagonal metric, this may not always be the case. Therefore, it is important to consider the specific conditions and assumptions under which these equations are derived and apply them accordingly.

In conclusion, the apparent contradiction you have encountered may be a result of the specific conditions and assumptions under which the equations are derived. It is important to keep in mind that these equations are not absolute and may not hold in all cases. Further exploration and analysis may help to better understand the relationship between a diagonal metric and a maximally symmetric space.