What are these Gaussian coordinates?

[ash64449]

[Mentor's note: This post was moved from another thread as it raised a new question, off-topic in the originating thread]

You don't need a frame of reference. General relativity doesn't even have frames of reference except locally

Albert Einstein in his book Relativity wrote " It is impossible to build up a system(reference body) from rigid bodies and clocks,which shall be of such a nature that measuring rods and clocks,arranged rigidly with respect to one another,shall indicate position and time directly". he was referring to the condition when the gravitational field exist.

Moreover in subsequent paragraphs,he explained how the introduction of Gaussian coordinates overcome the need for the reference body to measure the timing of events.

I cannot understand them.. Can you please help me with that.

 

[pervect]

[Mentor's note: This post was moved from another thread as it raised a new question, off-topic in the originating thread]Albert Einstein in his book Relativity wrote " It is impossible to build up a system(reference body) from rigid bodies and clocks,which shall be of such a nature that measuring rods and clocks,arranged rigidly with respect to one another,shall indicate position and time directly". he was referring to the condition when the gravitational field exist.

Moreover in subsequent paragraphs,he explained how the introduction of Gaussian coordinates overcome the need for the reference body to measure the timing of events.

I cannot understand them.. Can you please help me with that.

Do you mean as described in http://www.bartleby.com/173/25.html ?

I'm not sure what's hard to understand. If you imagine drawing a square grid, you can number the horizontal lines (rows), and vertical lines (columns), and specify an intersection point by giving the number of the horizontal line as one coordinate and the number of the vertical line as the other.

You may have to do a certain amount of interpolation, perhaps this is the issue?

Gaussian coordinates in 2d can just be thought of as replacing the square grid with arbitrary squiggly curves, and using the same row / column technique o describe the coordinates.

 

[ash64449]

No, pervect.. I didn't meant that. I did understand what are Gaussian co-ordinates. Perhaps my post made everyone think that i am asking them. Sorry!

I understand the part of the book(the chapter) that you linked. I have no problem with that chapter.

The trouble that i experienced understanding is this chapter: http://www.bartleby.com/173/27.html

I didn't understand after the paragraph 4 and also the sentence of the paragraph 4 that "The only statements having regard to these points which can claim a physical existence are in reality the statements about their encounters." and "After mature consideration the reader will doubtless admit that in reality such encounters constitute the only actual evidence of a time-space nature with which we meet in physical statements."

I think that after the paragraph 4,he tried to explain how the introduction of Gaussian Co-ordinates overcame the difficulty of assigning 'time' of a given event in the presence of the gravitational field. This part is what i cannot understand. In SR, we have a reference body,which can say 'time' at which events happen in that reference frame.

But there is a difficulty in assigning a reference body, since the 'common time' notion gets complicated due to gravitational field. field varies at each point and clocks at different points go at different rates and hence synchronization cannot be done and hence time at which event happen according to that reference body cannot be determined.

But 'time' at which event happens can be determined in the presence of gravitational field. It was possible after the introduction of Gaussian Co-ordinates. But i didn't understand how because i didn't understand after the paragraph 4 of this link:http://www.bartleby.com/173/27.html

i want help on that.

 

[ash64449]

I thought in the original thread(the thread from which my post was moved from), question was raised about how measuring 'time' possible in the presence of gravitational field. that's why i posted there.

 

[PeterDonis]

after the paragraph 4,he tried to explain how the introduction of Gaussian Co-ordinates overcame the difficulty of assigning 'time' of a given event in the presence of the gravitational field. This part is what i cannot understand.

His point is that Gaussian coordinates just assign four numbers to every event; there is no requirement that anyone of these numbers corresponds to a "time" that is actually measured by anybody. So, for example, suppose I consider two events that mark two "times" shown by a clock (for example, two successive events at which the hand of a stopwatch sweeps past the mark on the top of the stopwatch's face). I can assign Gaussian coordinates to each of these events, but even if I call one of those coordinates "t", that does not necessarily mean that the difference in "t" between the two events corresponds to the time the stopwatch actually measures between those two events.

To obtain a prediction for that actual measured value, I have to know the metric, i.e., the matrix $g_{\mu \nu}$ that appears in the formula $ds^2 = g_{\mu \nu} dx^{\mu} dx^{\nu}$ that I use to calculate actual physical intervals, such as the time measured by the stopwatch. The metric will contain all the information about gravitational fields and how they affect the elapsed time measured by clocks. So the information about actual physical intervals is not contained in the coordinates by themselves; it's only the coordinates plus the metric that contain that information.

 

[PeterDonis]

"The only statements having regard to these points which can claim a physical existence are in reality the statements about their encounters."

This is just a way of emphasizing that coordinates in themselves have no physical meaning, but the fact that two worldlines happen to share an event with a particular set of coordinate values--for example, the worldline of a stopwatch hand and the worldline of the mark at the top of the stopwatch's face--does have physical meaning. In other words, if two worldlines have a particular set of coordinate values in common, that tells you something meaningful, even if the coordinate values themselves don't.

 

[ash64449]

actual physical intervals is not contained in the coordinates by themselves; it's only the coordinates plus the metric that contain that information.

I haven't studied anything about matrix $g_{\mu \nu}$ because i cannot find anything that teaches about it in that book.

So now in Gaussian Co-ordinates, we do not have any meaning of co-ordinates assigned. So, it means there is no meaning to Co-ordinate time that corresponds to the time when an event happen according to that reference body's coordinate system. Is that the reason why we don't have reference body in GR?

 

[ash64449]

if two worldlines have a particular set of coordinate values in common, that tells you something meaningful,

What is that?

 

[ash64449]

So in GR, we cannot assign any time to any given event as coordinate time doesn't make any sense but we can only extract intervals between successive events from that matrix that you were talking?

 

[ash64449]

Does the Gaussian Co-ordinates transform themselves into the Cartesian Co-ordinates when we try to discuss a problem in SR using the former? If so,how? As the co-ordinates themselves don't have meaning in former how does is transform into the latter when we discuss the problem in SR?

 

[PeterDonis]

in GR, we cannot assign any time to any given event as coordinate time doesn't make any sense

No; in GR, we cannot assume, as we do in SR, that coordinate "time" has physical meaning, so if we want to assign a "time" to any given event, we have to use some other method than just looking at the coordinate time. One such method is to look at what the metric tells you the arc length is along timelike intervals; we call this "proper time" and it let's us assign a meaningful "time" to events along a given timelike worldline (i.e., the worldline of any observer).

Does the Gaussian Co-ordinates transform themselves into the Cartesian Co-ordinates when we try to discuss a problem in SR using the former?

Cartesian coordinates (or more precisely, Minkowski coordinates, since we're talking about spacetime) are just a special case of Gaussian coordinates that work when spacetime is flat (i.e., when SR applies).

As the co-ordinates themselves don't have meaning in former how does is transform into the latter when we discuss the problem in SR?

When spacetime is flat, all the problems that prevent coordinates from having physical meaning in general in GR aren't present; so you can in fact choose coordinates that have direct physical meaning. That's what Minkowski coordinates on flat spacetime, the coordinates normally used in SR, are. Note, however, that even in SR, there is no requirement to use Minkowski coordinates; you can use curvilinear coordinates (i.e., more general Gaussian coordinates) even in SR, and in such cases, the coordinates may not have direct physical meaning.

Another way of looking at it is to remember that, as I said before, it's not coordinates themselves that have direct physical meaning, but coordinates plus the metric. Minkowski coordinates look like they have direct physical meaning because the metric that goes with them is very simple: it's just a diagonal matrix with unit elements.

 

[pervect]

No, pervect.. I didn't meant that. I did understand what are Gaussian co-ordinates. Perhaps my post made everyone think that i am asking them. Sorry!

I understand the part of the book(the chapter) that you linked. I have no problem with that chapter.

The trouble that i experienced understanding is this chapter: http://www.bartleby.com/173/27.html

I'm not sure I totally understand that section myself, as it appears to be mostly on philosophy.

But I think I can over-simplify the situation in a way that may be helpful.

Consider a space-time manifold with only two dimensions - one spatial, and one of time - what we usually draw as a "space-time diagram" on a flat 2 dimensional sheet of paper.

Then we can operationally define time coordinate on such a 2d manifold by following Einstein's prescription of "Gaussian coordinates" exactly as he wrote it in chapter 25. You define lines of proper time, that define simultaneous events, and you give them all such lines numbers or labels to identify them.

The process of applying the technique to higher dimensions isn't explained by Einstein. I was going to go into that, but I think it would be a digression, so I will wait to see if there is interest.

I think for the purposes of understanding what Einstein was trying to say, applying it to a simple 2-d space-time manifold and using the prescriptions of chapter 25 is the best way to proceed.