Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
  1. Home
  2. Search

Refine search

Refine search

Access:
Content type:
Publication date:
Apply   From and To filters
Subject: Show more
Journals Show more
Publishers: Show more
Societies Show more
Series: Show more
Collections: Show more

Actions for selected content:

Select all | Deselect all
  • View selected items
  • Export citations
  • Download PDF (zip)
  • Save to Kindle
  • Save to Dropbox
  • Save to Google Drive

Save Search

You can save your searches here and later view and run them again in "My saved searches".

Please provide a title, maximum of 40 characters.
Cancel
×

1 results

  • 7 - The Cauchy problem in General Relativity

  • Stephen W. Hawking, University of Cambridge, George F. R. Ellis, University of Cape Town
  • Book:
    The Large Scale Structure of Space-Time
    Published online:
    17 February 2023
    Print publication:
    16 February 2023, pp 226-255

  • Summary

    In this chapter we give an outline of the Cauchy problem in general relativity and show that, given certain data on a space like three-surface there is a unique maximal future Cauchy development D+() and that the metric on a subset of D+() depends only on the initial data on J–() ∩. We also show that this dependence is continuous if has a compact closure in D+(). This discussion is included here because of its intrinsic interest, its use of some of the results of the previous chapter, and its demonstration that the Einstein field equations do indeed satisfy postulate (a) of §3.2 that signals can only be sent between points that can be joined by a non-spacelike curve.

    In §7.1, we discuss the various difficulties and give a precise formulation of the problem. In §7.2 we introduce a global background metric ĝ to generalize the relation which holds between the Ricci tensor and the metric in each coordinate patch to a single relation which holds over the whole manifold. We impose four gauge conditions on the covariant derivatives of the physical metric g with respect to the background metric ĝ.