Space-Time Interval: Exploring Schutz's Explanation

[schwarzschild]

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:

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

And then says:

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

Anyways I'm confused about his "note" - why can we suppose that?

 

[dx]

Since Δx¹Δx² and Δx²Δx¹ are the same, the only thing that matters is the sum M₁₂ + M₂₁ :

$$M_{12} \Delta x^1 \Delta x^2 + M_{21} \Delta x^2 \Delta x^1 = (M_{12} + M_{21})\Delta x^1 \Delta x^2$$

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]

Is the following the correct expansion of:

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

Sorry, but I'm having a little trouble understanding what exactly the summation is.

 

[dx]

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^aΔx^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.

 

[schwarzschild]

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]

I think it would be ok to post them here.

 

[bcrowell]

I think it would be ok to post them here.

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