Space-Time Interval: Exploring Schutz's Explanation

In summary: Otherwise, you can continue posting here if it's related to the original topic. In summary, the conversation discusses the presentation of the space time interval in Schutz's A First Course in General Relativity and how the sum Mab + Mba is the only thing that matters in the expansion. The speaker also explains how this sum can be split equally between Mab and Mba to make M a symmetric matrix. They also discuss whether to make a new thread for further questions.
  • #1
schwarzschild
15
1
I have been working through Schutz's A First Course in General Relativity and was a little confused by how he presents the space time interval:

[tex]\Delta \overline{s}^2 = \sum_{\alpha = 0}^{3} \sum_{\beta = 0}^{3} M_{\alpha \beta} (\Delta x^{\alpha})(\Delta x^{\beta}) [/tex] for some numbers [tex] \left\{M_{\alpha \beta} ; \alpha , \beta = 0,...,3\right\} [/tex] which may be functions of the relative velocity between the frames.

And then says:

Note that we can suppose that
[tex] M_{\alpha \beta} = M_{\beta \alpha} [/tex] for all [tex]\alpha[/tex] and [tex]\beta[/tex], since only the sum [tex] M_{\alpha \beta} + M_{\beta \alpha} [/tex] ever appears when [tex] \alpha \ne \beta [/tex]

Anyways I'm confused about his "note" - why can we suppose that?
 
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  • #2
Since Δx1Δx2 and Δx2Δx1 are the same, the only thing that matters is the sum M12 + M21 :

[tex] M_{12} \Delta x^1 \Delta x^2 + M_{21} \Delta x^2 \Delta x^1 = (M_{12} + M_{21})\Delta x^1 \Delta x^2 [/tex]

If this sum were, say, 6, then the term in the expansion would be 6Δx1Δx2, and we can just write this as 3Δx1Δx2 + 3Δx2Δx1.
 
  • #3
Is the following the correct expansion of:

[tex] \Delta \overline{s}^2 = \sum_{\alpha = 0}^{3} \sum_{\beta = 0}^{3} M_{\alpha \beta} (\Delta x^{\alpha})(\Delta x^{\beta})
= \sum_{\alpha = 0}^{3} (M_{\alpha 0} \Delta x^{\alpha} \Delta x ^{0} + M_{\alpha 1} \Delta x^{\alpha} \Delta x ^{1} + M_{\alpha 2} \Delta x^{\alpha} \Delta x ^{2} M_{\alpha 3} \Delta x^{\alpha} \Delta x ^{3}) [/tex]
[tex] = M_{0 0} \Delta x^{0} \Delta x ^{1} + M_{01} \Delta x^{1} \Delta x^{0} + M_{02} \Delta x^{2} \Delta x^{0} + M_{03} \Delta x^{3} \Delta x^{0} + M_{10} \Delta x^{1} \Delta x^{0} + M_{11} \Delta x^{1} \Delta x^{1} + \cdot \cdot \cdot [/tex] [tex]+ M_{13} \Delta x^{1} \Delta x^{3} + M_{20} \Delta x^{2} \Delta x^{0} + \cdot \cdot \cdot + M_{23} \Delta x^{2} \Delta x^{3} + M_{30} \Delta x^{3} \Delta x^{0} + \cdot \cdot \cdot + M_{33} \Delta x^{3} \Delta x^{3} [/tex]

Sorry, but I'm having a little trouble understanding what exactly the summation is.
 
  • #4
Yes, that's correct. (I think you made a typo in the 00 term.)

Notice that the Mab term and the Mba term can always be combined into a single term, and the coefficent of ΔxaΔxb will be Mab + Mba, i.e. only this sum matters. We can always split it up equally between Mab and Mba, and make M a symmetric matrix.
 
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  • #5
Okay, thanks, I'm pretty sure I understand this now. However, I'm probably going to have more questions as I continue through Schutz's treatment of the spacetime interval. Should I post them here, or make a new thread?
 
  • #6
I think it would be ok to post them here.
 
  • #7
dx said:
I think it would be ok to post them here.

I would suggest making a new thread if it's a new topic.
 

1. What is the concept of space-time interval?

The space-time interval is a measure of the distance between two events in space and time. It takes into account both the spatial and temporal components of the events, and is a fundamental concept in Einstein's theory of relativity.

2. How is space-time interval related to Schutz's explanation?

Schutz's explanation of space-time interval is based on the concept of spacetime curvature, which is a fundamental concept in Einstein's theory of relativity. He explains how this curvature affects the space-time interval and how it is different from the traditional concept of distance in Euclidean geometry.

3. What is the importance of understanding space-time interval?

Understanding space-time interval is crucial for understanding the nature of space and time in the universe. It is also essential for understanding the effects of gravity and how it affects the motion of objects in the universe.

4. How is space-time interval measured?

Space-time interval is measured using units of time and distance, such as seconds and meters. It can also be measured using mathematical equations, such as the Minkowski metric in relativity.

5. Can space-time interval be altered?

According to Einstein's theory of relativity, space-time interval is an absolute quantity and cannot be altered. However, it can appear to change depending on the observer's frame of reference due to the effects of gravity and the speed of light.

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