广义相对论 General Relativity PHYS5390M01

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$$g\left(\lambda^{3} \gamma, \lambda \pi_{A^{\prime}}\right)=\lambda^{-1} g\left(\gamma, \pi_{A^{\prime}}\right) \forall \lambda \in C-{0} .$$
This enables one to define the spinor field
$$p^{A} \equiv p^{A A^{\prime} B^{\prime}} \pi_{A^{\prime}} \pi_{B^{\prime}} \quad,$$
and the patching function
$$f^{A} \equiv p^{A} g\left(p_{B} \omega^{B}, \pi_{B^{\prime}}\right),$$
and the function
$$F\left(x^{a}, \pi_{A^{\prime}}\right) \equiv g\left(i p_{A} x^{A C^{\prime}} \pi_{C^{\prime}}, \pi_{A^{\prime}}\right) .$$

PHYS5390M01COURSE NOTES ：

and such that
$$\psi_{A^{\prime} B^{\prime} C^{\prime}}=\mathcal{D}{A A^{\prime}} \gamma{B^{\prime} C^{\prime}}^{A}$$
The second potential is a spinor field symmetric in its unprimed indices
$$\rho_{C^{\prime}}^{A B}=\rho_{C^{\prime}}^{(A B)}$$
subject to the equation
$$\mathcal{D}^{C C^{\prime}} \rho_{C^{\prime}}^{A B}=0$$
and it yields the $\gamma_{B^{\prime} C^{\prime}}^{A}$ ‘ potential by means of
$$\gamma_{B^{\prime} C^{\prime}}^{A}=\mathcal{D}{B B^{\prime}} \rho{C^{\prime}}^{A B}$$
If we introduce the spinor fields $v_{C^{\prime}}$ and $x^{B}$ obeying the equations
$$\begin{gathered} \mathcal{D}^{A C^{\prime}} \nu_{C^{\prime}}=0 \ \mathcal{D}{A C^{\prime}} \chi^{A}=2 i \nu{C^{\prime}} \end{gathered}$$

广义相对论|PH30101 General relativity代写

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the corresponding null geodesics are
\begin{aligned} &\gamma_{X}: x^{A A^{\prime}} \equiv x_{0}^{A A^{\prime}}+\lambda \bar{X}^{A} X^{A^{\prime}} \ &\gamma_{Y}: x^{A A^{\prime}} \equiv x_{1}^{A A^{\prime}}+\mu \bar{Y}^{A} Y^{A^{\prime}} \end{aligned}
If these intersect at some point, say $x_{2}$, one finds
$$x_{2}^{A A^{\prime}}=x_{0}^{A A^{\prime}}+\lambda \bar{X}^{\mathcal{A}} X^{A^{\prime}}=x_{1}^{A A^{\prime}}+\mu \bar{Y}^{A} Y^{A^{\prime}}$$
where $\lambda \mu \in R$. Hence
$$x_{2}^{A A^{\prime}} \bar{Y}{A} X{A^{\prime}}=x_{0}^{A A^{\prime}} \bar{Y}{A} X{A^{\prime}}=x_{1}^{A A^{\prime}} \bar{Y}{A} X{A^{\prime}}$$

PH30101 COURSE NOTES ：

This enables one to define the spinor field
$$p^{A} \equiv p^{A A^{\prime} B^{\prime}} \pi_{A^{\prime}} \pi_{B^{\prime}}$$
and the patching function
$$f^{A} \equiv p^{A} g\left(p_{B} \omega^{B}, \pi_{B^{\prime}}\right)$$
and the function
$$F\left(x^{a}, \pi_{A^{\prime}}\right) \equiv g\left(i p_{A} x^{A C^{\prime}} \pi_{C^{\prime}}, \pi_{A^{\prime}}\right)$$