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Spacetime Intervalby schwarzschild
Tags: spacetime interval 
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#1
Apr610, 10:53 PM

P: 15

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? 


#2
Apr710, 12:25 AM

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PF Gold
P: 1,961

Since Δx^{1}Δx^{2} and Δx^{2}Δx^{1} are the same, the only thing that matters is the sum M_{12} + M_{21} :
[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 6Δx^{1}Δx^{2}, and we can just write this as 3Δx^{1}Δx^{2} + 3Δx^{2}Δx^{1}. 


#3
Apr710, 10:16 AM

P: 15

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
Apr710, 10:24 AM

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PF Gold
P: 1,961

Spacetime Interval
Yes, that's correct. (I think you made a typo in the 00 term.)
Notice that the M_{ab} term and the M_{ba} term can always be combined into a single term, and the coefficent of Δx^{a}Δx^{b} will be M_{ab} + M_{ba}, i.e. only this sum matters. We can always split it up equally between M_{ab} and M_{ba}, and make M a symmetric matrix. 


#5
Apr710, 10:32 AM

P: 15

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?



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