Can anyone help explain this to me?

[Peron]

In the wiki entry on time, it says

According to these theories, the concept of time depends on the spatial reference frame of the observer, and the human perception as well as the measurement by instruments such as clocks are different for observers in relative motion. The past is the set of events that can send light signals to the observer, the future is the set of events to which the observer can send light signals.

 

http://en.wikipedia.org/wiki/Time#Physical_definition

 

Can anyone better help explain what physicists mean by time in general relativity?

[ajb]

You may mean:

 

1) A choice of coordinate "x^{0} = t". That is one picks a decomposition of space-time into space and time. However, this choice will not be canonical. There are space-times known as globally hyperbolic. These, in some sense have the best causal structure and allow for one to formulate things quite precisely as space and time.

 

2) All massive physical particles follow what are known as time-like paths. We will not need the condition of a geodesic as we will allow for external forces.

 

I'll assume you know the basic of general relativity, in particular the idea of local coordinates and the metric. Let [math](M,g)[/math] be a Lorentzian manifold.

 

A path (aka curve) is map

 

[math]\gamma: \mathbb{R}\rightarrow M[/math].

 

In local coordinates represent this map as

 

[math]\gamma^{*}x^{\mu} = x^{\mu}(\lambda)[/math]. (with the usual abuse of notation)

 

Here [math]\lambda[/math] is a coordinate on the real line.

 

A path is said to be time-like if

 

[math]\frac{dx^{\mu}}{d \lambda} \frac{dx^{\nu}}{d \lambda} g_{\nu \mu} < 0[/math].

 

It is a fact that physical massive particles follow such paths.

 

We can then define the "proper time functional" as

 

[math]\tau = \int d \lambda \sqrt{- \frac{dx^{\mu}}{d \lambda} \frac{dx^{\nu}}{d \lambda} g_{\nu \mu}}[/math].

 

It is the above that one may also refer to as time. It is the time as measured by a clock moving along the path.

Edited April 26, 2010 by ajb