Skip to main content Accessibility help
×
Home
Hostname: page-component-5f95dd588d-5p2mf Total loading time: 0.301 Render date: 2021-10-28T22:05:55.775Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }
Coming soon

8 - The gravitational field equations

M. P. Hobson
Affiliation:
University of Cambridge
G. P. Efstathiou
Affiliation:
University of Cambridge
A. N. Lasenby
Affiliation:
University of Cambridge
Get access

Summary

Let us now follow Einstein's suggestion that gravity is a manifestation of spacetime curvature induced by the presence of matter. We must therefore obtain a set of equations that describe quantitatively how the curvature of spacetime at any event is related to the matter distribution at that event. These will be the gravitational field equations, or Einstein equations, in the same way that the Maxwell equations are the field equations of electromagnetism.

Maxwell's equations relate the electromagnetic field F at any event to its source, the 4-current density j at that event. Similarly, Einstein's equations relate spacetime curvature to its source, the energy–momentum of matter. As we shall see, the analogy goes further. In any given coordinate system, Maxwell's equations are second-order partial differential equations for the components Fµν of the electromagnetic field tensor (or equivalently for the components Aµ of the electromagnetic potential). We shall find that Einstein's equations are also a set of second-order partial differential equations, but instead for the metric coefficients gµν of spacetime.

The energy–momentum tensor

To construct the gravitational field equations, we must first find a properly relativistic (or covariant) way of expressing the source term. In other words, we must identify a tensor that describes the matter distribution at each event in spacetime.

Type
Chapter
Information
General Relativity
An Introduction for Physicists
, pp. 176 - 195
Publisher: Cambridge University Press
Print publication year: 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Send book to Kindle

To send this book to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Send book to Dropbox

To send content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Dropbox.

Available formats
×

Send book to Google Drive

To send content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Google Drive.

Available formats
×