Pythorean Theorem and General Relativity

[MetricBrian]

Hi have a question. Am I right in thinking that calculating the curvature of space is mathematically treated as the sum of an infinitely small set of flat spaces?

and that the calcutation of the distance in a flat space involves a modified version of the pythagorean theorem where fourth dimension is minus the time squared?

Thanks in advance Brian.

 

[pervect]

Hi have a question. Am I right in thinking that calculating the curvature of space is mathematically treated as the sum of an infinitely small set of flat spaces?

If you want to learn about curvature I'd suggest reading something like http://www.eftaylor.com/pub/chapter2.pdf, it's too involved to explain in a post.

and that the calcutation of the distance in a flat space involves a modified version of the pythagorean theorem where fourth dimension is minus the time squared?

You've got this right, though - this is exactly how SR works in flat space-time.

 

[MetricBrian]

Thanks For Repyling, But really all I wanted to establish is this: is it correct (generally speaking) to say that the modified pythagorean theorem plays in an important role in calculating the distance of a curved spacetime.

 

[belliott4488]

I don't really know what's behind your question, but I do think you might be blending two ideas: the modification of the concept of distance in spacetime, and the modification of distance in curved spaces in general.

In spacetime, even flat Minkowski Space, the concept of distance (and thus the Pythagorean theorem) is in fact modified as you said, by the relative negative sign between the squares of the space components and the square of the time component. In a 4-dimensional space, even a 4-D Euclidean space, curvature is defined in reference to distances between points, which you could think of as integrated applications of the Pythagorean Theorem.

I think these two "modifications" are very different, but both apply in the curved spacetime of GR.

 

[pervect]

Thanks For Repyling, But really all I wanted to establish is this: is it correct (generally speaking) to say that the modified pythagorean theorem plays in an important role in calculating the distance of a curved spacetime.

I may be getting into more detail than what you want, sorry - but I'm not quite sure what you're asking.

In a flat space-time, the Lorentz interval ds is given by what you call the modified Pythagorean theorem:

ds^2 = -dt^2 + dx^2

where I've suppressed dy and dz for simplicity.

ds here is a very fundamental quantity - it's the same for all observers, it is an invariant. In fact, it can be regarded as the fundamental entity, whose description describes space-time. If that's all that you're asking, then you're on the right track. But that may not be all that you were asking.

We can get distances and times out of the lorentz interval as follows.

If ds^2 is positive, you have a spacelike interval, and sqrt(ds^2) is a distance interval. If ds^2 is negative, you have a timelike interval, and sqrt(-ds^2) is a time interval.

Now, how does this change in a general space-time?

In a general space-time, we would write instead a more general expression involving the metric coefficients $g_{ij}$ to find the Lorentz interval, i.e. the modified Pythagorean theorem gets further modified.

$$ds^2 = g_{00}d t^2 + g_{01} (dt dx + dx dt) + g_{11} dx^2$$

Here the metric coefficients $g_{ij}$ are in general a function of (t,x)

If $g_{00} = -1, g_{01} = 0, g_{11}=1$ everywhere, then you have a flat space-time.

 

[MetricBrian]

Hi Pervect, thank you for your helpful reply. Let me try to be clearer. I am writing a paper on the philosophy of science where I am trying to document how new iscientific ideas are connected to past ones, hence my interest in the role that the pythagorean theorem plays in special and general relativity.
It seems to me that from Pythagorus all the way to General Relativity, there is a common theme: an invariant relationship between the distance squared and the sum of the square of the components.

 

[yenchin]

Hi, MetricBrian, I have written a similar article from mathematical point of view. You may access it at the following website:

http://203.81.38.114/databank/courseweb/YENCHIN//Metric.pdf

It might be helpful for you philosophy paper.

 

[pervect]

Hi Pervect, thank you for your helpful reply. Let me try to be clearer. I am writing a paper on the philosophy of science where I am trying to document how new iscientific ideas are connected to past ones, hence my interest in the role that the pythagorean theorem plays in special and general relativity.
It seems to me that from Pythagorus all the way to General Relativity, there is a common theme: an invariant relationship between the distance squared and the sum of the square of the components.

The nitpicky point I was stumbling over is that sometimes distances are given by quadratic forms, rather than the sum of the squares.

This may be "too much information" for your paper, on the otherhand maybe not, I dunno.

See http://mathworld.wolfram.com/QuadraticForm.html for a definition. (The formula I wrote above was an example of a quadratic form in two dimensions and two variables, dx and dt).

The good news is that by a suitable choice of variables and scaling factors, any quadratic form can be diagonalized so that the quadratic form reduces to the pythagorean theorem.

In GR, though, while you can chose coordinates to make the metric such a diagonal quadratic form at anyone given point, you can't chose coordinates to make the metric diagonal everywhere - i.e. any given point, you can define coordinates so that the Pythagorean theorem works unmodified near that point, but you can't define coordinates so that the Pythagorean theorem works unmodified everywhere. To express the idea of distance everywhere in GR, you need a more general formula for it than the Pythagorean formula.

 

[MetricBrian]

O.K.
So is it correct to say that distances in GR can be calculated by a Generalized Pythagorean Theorem or is this statement open to question?

Also is it correct to say that the the integral techinque for calulating the distance for a curved spacetime surface is by divinding the curve into a infinietly number of very small flat spacetime pieces?

 

[pervect]

O.K.
So is it correct to say that distances in GR can be calculated by a Generalized Pythagorean Theorem or is this statement open to question?

Also is it correct to say that the the integral techinque for calulating the distance for a curved spacetime surface is by divinding the curve into a infinietly number of very small flat spacetime pieces?

Both of those statements look OK to me.

 

[MetricBrian]

Thanks for your help!