# General Relativity: Section 5.1 Homogeneity & Isotropy Analysis

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• Alpha2021
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Alpha2021
From the section[5.1] of 'Homogeneity and Isotropy' from General Relativity by Robert M. Wald (pages 91-92, edition 1984) whatever I have understood is that -

##\Sigma_t## is a spacelike hypersurface for some fixed time ##t##. The hypersurface is homogeneous.

The metric of whole space is ##g## and the form of the metric on hypersurface ##\Sigma_t## is ##h##. Thus if ##g## is the metric of dimension ##4##, then ##\Sigma_t## has dimension ##3##. Next, Riemann curvature tensor ##R_{ab}{}^{cd}## is defined from the metric ##h## (Or, from the metric ##g##, I am not sure about this). Now, if there is an antisymmetric tensor ##A_{ij}## defined on ##\Sigma_t##, this tensor is transformed by ##R_{ab}{}^{cd}## as ##A'_{ij} = R_{ij}{}^{cd} A_{cd}##. ##R_{ij}{}^{cd}## itself is antisymmetric with respect to its two indices ##i## and ##j##. This transformation can be viewed as linear self-adjoint transformation. If we name this linear transformation as ##L## and vector space as ##W##, then ##L: W \to W##. This vector space ##W## can be spanned by eigenvector ##L##. The corresponding eigenvalues must be equal because of isotropy. Thus ##L## can be expressed as multiple of the identity operator i.e. ##L=K I##.

Then he suddenly claimed that

$$R_{ab}{}^{cd} = K \delta^c{}_a \delta^d{}_b$$

I have not understood how to claim this equation.

Last edited by a moderator:
Hi @Alpha2021 and welcome to PF!

For equations here we use LaTeX, and there is a LaTeX Guide link at the bottom left of the window you use to make posts. Please use it; it helps a lot to make your posts more readable when they have equations in them. I have edited your OP in this thread to use LaTeX.

Alpha2021
Alpha2021 said:
Riemann curvature tensor is defined from the metric ##h## (Or, from the metric ##g##, I am not sure about this).
In this particular case, Wald is using the Riemann tensor on the surface of homogeneity ##\Sigma_t##, meaning it is derived from the metric ##h## on that surface. He uses the notation ##{}^{(3)}R## for this reason.

Alpha2021 said:
I have not understood how to claim this equation.
You left out an essential part of that equation, the square brackets around the lower indexes on the RHS. The equation as Wald wrote it is (note that I am also using the ##{}^{(3)}R## notation described above):

$${}^{(3)} R_{ab}{}^{cd} = K \delta^c{}_{[ a} \delta^d{}_{b ]}$$

The square brackets denote antisymmetrization, meaning that the above equation expands to:

$${}^{(3)} R_{ab}{}^{cd} = K \left( \delta^c{}_a \delta^d{}_b - \delta^c{}_b \delta^d{}_a \right)$$

The expression in parentheses on the RHS is simply the identity operator in the vector space ##W## of 2-forms, i.e., the operator that takes any 2-form into itself. The antisymmetrization is necessary because we are dealing with 2-forms, which are antisymmetric (0, 2) tensors.

vanhees71, Alpha2021 and PeroK
@PeterDonis
Thanks a lot. I missed the latex option. I am sorry.

Last edited:
Hi @PeterDonis,
The first answer is clear to me.

About the second question - I am sorry for my mistake. But, I still have the confusion. I know Riemann tensor as
##R^{\eta }_{\; \mu \nu \alpha} = \frac{\partial \Gamma^{\eta}_{\mu \alpha}}{\partial x^{\nu}}+ \frac{\partial \Gamma^{\eta}_{\mu \nu}}{\partial x^{\alpha}} + \Gamma^{\eta}_{\kappa \nu }\Gamma^{\kappa }_{\mu \alpha} -\Gamma^{\eta}_{\kappa \alpha}\Gamma^{\kappa }_{\mu \nu} ##

Then how has it transformed to ##K\delta^c{}_{[a} \delta^d{}_{b]}##? What is its relation with equal eigen values?

Alpha2021 said:
how has it transformed to ##K\delta^c{}_{[a} \delta^d{}_{b]}##?
Because the constraints of homogeneity and isotropy mean that almost all of the components of the Riemann tensor of ##\Sigma_t## vanish. The only ones left are the ones in that equation.

Alpha2021 said:
What is its relation with equal eigen values?
Equal eigenvalues is one of the consequences of homogeneity and isotropy--specifically isotropy, which means that there can't be any direction in space that is different from the others. That requires all eigenvalues of ##{}^{(3)} R##, considered as a linear transformation on the space ##W## of 2-forms, to be equal; if they weren't, the different eigenvalues would mean different directions in space had different properties.

Alpha2021
Alpha2021 said:
@PeterDonis
You're welcome!

Alpha2021

## 1. What is the concept of homogeneity in general relativity?

Homogeneity in general relativity refers to the idea that the universe looks the same at every point in space. This means that the distribution of matter and energy is uniform, and there are no preferred locations or directions in the universe.

## 2. How does homogeneity relate to the cosmological principle?

The cosmological principle states that the universe is both homogeneous and isotropic on large scales. This means that not only is the universe the same at every point, but it also looks the same in all directions. Homogeneity is a key component of the cosmological principle and is a fundamental assumption in the study of the large-scale structure of the universe.

## 3. What is the significance of isotropy in general relativity?

Isotropy in general relativity means that the universe looks the same in all directions. This is important because it allows us to make predictions about the universe based on observations in any direction. Without isotropy, we would not be able to generalize our understanding of the universe beyond our own location.

## 4. How is homogeneity and isotropy analyzed in general relativity?

Homogeneity and isotropy are analyzed through the use of mathematical models, such as the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric, which describes a homogeneous and isotropic universe. Observational data, such as the cosmic microwave background radiation, is also used to test the assumptions of homogeneity and isotropy in the universe.

## 5. What are some possible implications of violating homogeneity and isotropy in general relativity?

If homogeneity and isotropy were found to be violated in the universe, it would challenge our current understanding of the universe and the laws of physics. It could also have implications for the validity of the cosmological principle and our ability to make accurate predictions about the universe. However, so far, observations have supported the assumptions of homogeneity and isotropy on large scales.

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