Penrose equation for a vacuum spacetime

In summary, the conversation discusses the Bianchi identity and its various forms, including the identity in a suggestive form using the Riemann tensor. The conversation also mentions the possibility of using symmetries of the Riemann tensor to simplify the identity. A related arxiv paper is also mentioned.
  • #1
etotheipi
Homework Statement
Derive the Penrose equation for a vacuum spacetime$$\nabla^{\lambda} \nabla_{\lambda} R_{\mu \nu \rho \sigma} = 2 {R^{\kappa}}_{\mu \lambda \sigma} {R^{\lambda}}_{\rho \kappa \nu} - 2{R^{\kappa}}_{\nu \lambda \sigma} {R^{\lambda}}_{\rho \kappa \mu} - {R^{\kappa}}_{\lambda \sigma \rho} {R^{\lambda}}_{\kappa \mu \nu}$$
Relevant Equations
N/A
Okay so for this one we can consider the Bianchi identity again$$\begin{align*}

\nabla_{[\lambda}R_{\sigma \rho] \mu \nu} = 2 \nabla_{\lambda} R_{\sigma \rho \mu \nu} + 2\nabla_{\rho} R_{\lambda \sigma \mu \nu} + 2\nabla_{\sigma} R_{\rho \lambda \mu \nu} &= 0 \\

\nabla^{\lambda} \nabla_{\lambda} R_{\sigma \rho \mu \nu} &= \nabla^{\lambda} \nabla_{\rho} R_{\sigma \lambda \mu \nu} + \nabla^{\lambda} \nabla_{\sigma} R_{\lambda \rho \mu \nu} \\

\nabla^{\lambda} \nabla_{\lambda} R_{\sigma \rho \mu \nu} &= g^{\epsilon \lambda} \nabla_{\epsilon} \nabla_{\rho} R_{\sigma \lambda \mu \nu} + g^{\epsilon \lambda} \nabla_{\epsilon} \nabla_{\sigma} R_{\lambda \rho \mu \nu} \end{align*}$$Here I wrote it in a form that's suggestive of another identity$$(\nabla_a \nabla_b - \nabla_b \nabla_a){T^{c_1, \dots, c_k}}_{d_1,\dots,d_l} = \sum_{j=1}^{l} {R_{abd_j}}^e {T^{c_1, \dots, c_k}}_{d_1, \dots, e, \dots d_l}
- \sum_{i=1}^{k} {R_{abe}}^{c_i} {T^{c_1, \dots, e, \dots, c_k}}_{d_1, \dots, d_l}
$$Let's say we consider ##T = R##, i.e.$$\begin{align*}(\nabla_{\epsilon} \nabla_{\rho} - \nabla_{\rho} \nabla_{\epsilon}) R_{\sigma \lambda \mu \nu} &= {R_{\epsilon \rho \sigma}}^{e} R_{e\lambda \mu \nu} + {R_{\epsilon \rho \lambda}}^{e} R_{\sigma e \mu \nu} + {R_{\epsilon \rho \mu}}^{e} R_{\sigma \lambda e \nu}+ {R_{\epsilon \rho \nu}}^{e} R_{\sigma \lambda \mu e}\end{align*}$$Is there a way to tidy this up using some other symmetries of the Riemann tensor and obtain the desired result? Thanks
 
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  • #2
It seems there are no takers here so I searched for something related and found this arxiv paper which you’ve likely seen starting on page 2 thru 4 or so:

https://arxiv.org/pdf/gr-qc/9809074.pdf
 

1. What is the Penrose equation for a vacuum spacetime?

The Penrose equation for a vacuum spacetime is a mathematical formula developed by British mathematician Sir Roger Penrose to describe the curvature of spacetime in the absence of any matter or energy. It is a simplified version of the Einstein field equations and is used to study the properties of black holes and other regions of highly curved spacetime.

2. How is the Penrose equation derived?

The Penrose equation is derived from the Einstein field equations by setting the energy-momentum tensor to zero, which corresponds to a vacuum spacetime. This simplifies the equations and allows for a more straightforward analysis of the curvature of spacetime.

3. What does the Penrose equation tell us about black holes?

The Penrose equation helps us understand the properties of black holes, such as their event horizons and singularities. It also allows us to calculate the rate at which a black hole is rotating and the strength of its gravitational field.

4. Can the Penrose equation be applied to other spacetimes?

Yes, the Penrose equation can be applied to any vacuum spacetime, not just those surrounding black holes. It can also be used to study the properties of other highly curved regions of spacetime, such as the early universe or the interiors of neutron stars.

5. What are the implications of the Penrose equation for our understanding of the universe?

The Penrose equation has helped us gain a deeper understanding of the nature of spacetime and the effects of gravity on it. It has also played a crucial role in the development of theories such as general relativity and black hole thermodynamics, which have significantly advanced our understanding of the universe.

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