what is relation between time and space in general relativity?

there is a relation between time and space in special theory of relativity: $$t^2c^2-L^2=\tau^2.c^2$$ what is relation between time and space in general relativity?

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– David Zaslavsky♦ Jun 27 '12 at 21:07

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– user1504 Jun 27 '12 at 21:29
-1: This is not a question. Are you asking for the generalization of the formula you gave? This is not a reasonable question--- the formula you gave is a pythagorean theorem in space-time. The generalization is the metric tensor, but it isn't even clear in which way you are looking for an answer. – Ron Maimon Jun 28 '12 at 1:26

2 Answers

The remarkable property of spacetime in GR is that it is locally that of SR. Or, more technically, tangent to every event in the curved spacetime of GR is an SR spacetime. What this means is that, to first order, the line element at any event can be put into the (differential) form of SR in some coordinate system:

$c^2 dt^2 - dL^2 = c^2 d\tau ^2$

The departure from the flat SR spacetime shows up at 2nd order; curvature is characterized by the 2nd order derivatives of the metric.

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he Einstein field equations (EFE) or Einstein's equations are a set of 10 equations in Albert Einstein's general theory of relativity which describe the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy. First published by Einstein in 1915 as a tensor equation, the EFE equate spacetime curvature (expressed by the Einstein tensor) with the energy and momentum within that spacetime (expressed by the stress–energy tensor).

There is also the linearized EFE which are used for simplifying many general relativity problems as well as discussing gravitational radiation. These equations can be found at: http://en.wikipedia.org/wiki/Linearized_gravity#Linearised_Einstein_field_equations.

A small side note. Although the Einstein field equations were initially formulated in the context of a four-dimensional theory, some theorists have explored their consequences in n dimensions. The equations in contexts outside of general relativity are still referred to as the Einstein field equations. The vacuum field equations (obtained when T is identically zero) define Einstein manifolds.

References: http://en.wikipedia.org/wiki/Einstein_field_equations

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