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I have a homework question that I amalmost100% sure that I solved, so I do not believe that this post should go into the "Homework Questions" section. This thread does not have to do with the answer to that homework question anyways, but rather a curiosity about whether or not this formulation is related -- in some way -- to the stress-energy tensor.

I apologize if this is in the wrong section. Please let me know if this is improper.

The question, taken from "A First Course in General Relativity" by Bernard F. Schutz, asks us to derive equation 1.3 from equation 1.2.

My answer described this matrix: 1.2: [itex]\qquad \Delta \overline{s}^{\,2} = \sum_{\alpha=0}^3 \sum_{\beta=0}^3 M_{\alpha \beta} (\Delta x^\alpha) (\Delta x^\beta)[/itex]

1.3: [itex]\qquad \Delta \overline{s}^{\,2} = M_{00} (\Delta r)^2 + 2(\sum_{i=1}^3 M_{0i} \Delta x^i) \Delta r + \sum_{i=1}^3 \sum_{j=1}^3 M_{ij} \Delta x^i \Delta x^j[/itex]

where [itex]\qquad \Delta r = \sqrt{(\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2} \qquad[/itex] and [itex]\qquad \Delta r = \Delta t[/itex]

[tex]

\bеgin{vmatrix}

t^2 & tx & ty & tz\\

xt & x^2 & xy & xz\\

yt & yx & y^2 & yz\\

zt & zx & zy & z^2

\end{vmatrix}

[/tex]

I explained it in three terms, each explaining separate parts of the matrix:

[tex]

\bеgin{vmatrix}

(t^2) & (tx & ty & tz)\\

\\

\begin{pmatrix}

xt\\

yt\\

zt\\

\end{pmatrix}

\begin{pmatrix}

x^2 & xy & xz\\

yx & y^2 & yz\\

zx & zy & z^2

\end{pmatrix}

\end{vmatrix}

[/tex]

Entry 00 is represented by [itex]M_{00} (\Delta r)^2[/itex]; because there are two time entries, we observe two delta r terms. Entries 01, 02 and 03 are the transpose of entries 10, 20 and 30, and so they were just repeated with a two multiplying out front of that entire term. This was represented by [itex]2(\sum_{i=1}^3 M_{0i} \Delta x^i) \Delta r[/itex]; because these terms can be factored into t(xyz), there is a delta r for each. The remaining entries are represented by [itex]\sum_{i=1}^3 \sum_{j=1}^3 M_{ij} \Delta x^i \Delta x^j[/itex]; because none of these entries are multiplied by time, the delta r term is absent from this part.

Please tell me if there is an error in my work.

Anyways, I can see that this matrix is broken up in the very same way as the stress-energy tensor (see image at the top of the Wikipedia page). I know virtually nothing about the stress-energy tensor, though! To learn about it was one of the main motivations for me to read this book. Is there a relationship between the stress-energy tensor and the interval? Or are all second-order matrices broken up in this way?

Thanks,

HeavyMetal

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# Relationship between Stress-Energy Tensor and Interval

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