What are the meanings of coordinates ?

[hiyok]

Hello, I have a question in GRT. Suppose one is going to obtain the gravitational field of a point mass. He at first sets up a coordinate system. E.G., he may imagine laying out three rods throughout the space and put a clock somewhere. This clock may be put still at the origin of the rods frame or somewhere else. Now my question is, what is the difference?
To put it in another way: suppose we use Swartzchild coordinate system, and then how do know by which clock the coordinate time is taken, just from the form of the metric?

 

[tiny-tim]

Hello hiyok!

(btw, it's Schwarzschild (meaning "black shield") )

… suppose we use Swartzchild coordinate system, and then how do know by which clock the coordinate time is taken, just from the form of the metric?

I don't understand …

stationary clocks at different distances from the origin will be going at different rates …

there's no such thing as "the" coordinate time.

 

[ZikZak]

You could check the metric to see under what conditions d(tau) = dt. In your case, t is the time read out on clocks arbitrarily distant from the gravitating body and at rest with respect to it. Verify this by letting dr=dtheta=dphi=0 and then letting r become large.

 

[hiyok]

Hello hiyok!

(btw, it's Schwarzschild (meaning "black shield") )


I don't understand …

stationary clocks at different distances from the origin will be going at different rates …

there's no such thing as "the" coordinate time.

Hi, Tiny-tim, thanks for correcting my spelling !
Obviously, if one intends to solve Einstein's equation, he must input the energy-momentum tensor, whose form clearly hinges on which frame of reference has been chosen. A complete frame of reference involves necessarily a clock. So, the question is, what is the relation between the coordinate "t" in Schwarzschild metric and the reading of the clock he uses to record the motion of the point mass (He records the motion so as to obtain the energy-momentum tensor) ?

 

[tiny-tim]

Hello hiyok!

… which frame of reference has been chosen. A complete frame of reference involves necessarily a clock. So, the question is, what is the relation between the coordinate "t" in Schwarzschild metric and the reading of the clock he uses to record the motion of the point mass (He records the motion so as to obtain the energy-momentum tensor) ?

General Relativity doesn't really have frames of reference.

As ZikZak says …

t is the time read out on clocks arbitrarily distant from the gravitating body and at rest with respect to it.

 

[hiyok]

You could check the metric to see under what conditions d(tau) = dt. In your case, t is the time read out on clocks arbitrarily distant from the gravitating body and at rest with respect to it. Verify this by letting dr=dtheta=dphi=0 and then letting r become large.

Hello, ZikZak. Related to your reply, other questions form in my mind: (1)Is that a generic rule to spot clocks? (2)Given a metric, how does one differentiate time coordinate and space coordinates?

 

[ZikZak]

Hello, ZikZak. Related to your reply, other questions form in my mind: (1)Is that a generic rule to spot clocks? (2)Given a metric, how does one differentiate time coordinate and space coordinates?

(1) I don't know what you mean by "generic rule for spotting clocks." Your question was about under what conditions the t coordinate in the Schwarzschild metric was the proper time.

(2) I don't know what a "time coordinate" or a "space coordinate" is. Time and space are relative; only spacetime intervals are invariant. If you wish to check whether a certain short interval is timelike or spacelike, use the metric to determine whether ds^2 is positive or negative for that interval.