MTW Ex 21.15 - Curvature independent of Lapse and Shift functions

[TerryW]

Homework Statement

I'm completely baffled as to where to start on this!

Relevant Equations

See attached image of MTW Ex 21.15

Can anyone out there give me a hint as to where to start with this problem?

I've been looking at it for a while and can't see a way forward.

What exactly is "the curvature itself" here?BTW I think the dynamic initial value equations 21.116 and 21.117 are incorrect. MTW should have inserted to ADM equivalents of these equations.

Any help would be appreciated.Regards
TerryW

 

[member 553083]

The best place to start with this problem is by understanding what the curvature itself is. Curvature is a measure of how much a surface or space is curved. It can be defined as the rate at which a line deviates from being straight. Once you understand what the curvature is, you can then begin to work on solving the dynamic initial value equations. You may find it helpful to review the ADM (Arnowitt–Deser–Misner) formalism, which is a set of equations used to describe the dynamics of curved spacetime in general relativity. This formalism can be used to solve the dynamic initial value equations. Additionally, you may find it helpful to consult other resources, such as textbooks or online tutorials, that provide more detailed solutions to the equations.