Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 309 / 4

Contract ${\nabla }_a{T}^{ab}=0$ for the perfect fluid in general relativity with $u_b$. Using $u_b{u}^b=-1$ and $u_b{\nabla }_a{u}^b=0$ gives the relativistic perfect-fluid energy equationthe local first law of thermodynamics. Project instead with $g^a{}_b+u^a{u}_b$ to obtain the relativistic Euler equation

Direct differentiation gives, for example,Differentiating ${\sigma }^1,{\sigma }^2$ similarly and substituting the definitions yields $d{\sigma }^1={\sigma }^2\wedge {\sigma }^3$ and $d{\sigma }^2={\sigma }^3\wedge {\sigma }^1$. Hence these Maurer-Cartan forms on the three-sphere obey

Since $d{e}^0=0$ and $d{e}^i=\frac{1}{2}{ϵ}^i{}_{jk}{e}^j\wedge e^k$, Cartan's first structure equation givesCartan's second structure equation then givesUsing ${\Theta }^i{}_j=\frac{1}{2}{R}^i{}_{jkl}{e}^k\wedge e^l$ yieldswith no time-index curvature components. Contraction givesand therefore

In the orthonormal frame, the comoving perfect fluid in general relativity has $T_{00}=\rho$ and $T_{ij}=p{\delta }_{ij}$. The Einstein field equations with cosmological constant therefore giveThusChoosinggives $p=0$ and $\rho =1/(16\pi )>0$. The metric is then the Einstein static universe supported by pressureless matter and positive vacuum energy.