Spacetime Interval


by schwarzschild
Tags: spacetime interval
schwarzschild
schwarzschild is offline
#1
Apr6-10, 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?
Phys.Org News Partner Science news on Phys.org
Cougars' diverse diet helped them survive the Pleistocene mass extinction
Cyber risks can cause disruption on scale of 2008 crisis, study says
Mantis shrimp stronger than airplanes
dx
dx is offline
#2
Apr7-10, 12:25 AM
HW Helper
PF Gold
dx's Avatar
P: 1,962
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.
schwarzschild
schwarzschild is offline
#3
Apr7-10, 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.

dx
dx is offline
#4
Apr7-10, 10:24 AM
HW Helper
PF Gold
dx's Avatar
P: 1,962

Spacetime Interval


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.
schwarzschild
schwarzschild is offline
#5
Apr7-10, 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?
dx
dx is offline
#6
Apr7-10, 10:35 AM
HW Helper
PF Gold
dx's Avatar
P: 1,962
I think it would be ok to post them here.
bcrowell
bcrowell is offline
#7
Apr7-10, 11:43 AM
Emeritus
Sci Advisor
PF Gold
bcrowell's Avatar
P: 5,500
Quote Quote by dx View Post
I think it would be ok to post them here.
I would suggest making a new thread if it's a new topic.


Register to reply

Related Discussions
Spacetime Interval Introductory Physics Homework 1
Spacetime interval? Introductory Physics Homework 1
Invariant Spacetime Interval Special & General Relativity 17
spacetime interval of zero Special & General Relativity 15
What is the dimension of the spacetime interval? Special & General Relativity 10