- #1

Alpha2021

- 11

- 1

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.

##\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.

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