Panda"Understand Relationship between Stress-Energy Tensor and Interval

[HeavyMetal]

Hello all,

I have a homework question that I am almost 100% 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.

1.2: $\qquad \Delta \overline{s}^{\,2} = \sum_{\alpha=0}^3 \sum_{\beta=0}^3 M_{\alpha \beta} (\Delta x^\alpha) (\Delta x^\beta)$
1.3: $\qquad \Delta \overline{s}^{\,2} = M_{00} (\Delta r)^2 + 2(\sum_{i=1}^3 M_{0i} \Delta x^i) \Delta r + \sum_{i=1}^3 \sum_{j=1}^3 M_{ij} \Delta x^i \Delta x^j$
where $\qquad \Delta r = \sqrt{(\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2} \qquad$ and $\qquad \Delta r = \Delta t$

My answer described this matrix:

$$\bеgin{vmatrix} t^2 & tx & ty & tz\\ xt & x^2 & xy & xz\\ yt & yx & y^2 & yz\\ zt & zx & zy & z^2 \end{vmatrix}$$

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

$$\bеgin{vmatrix} (t^2) & (tx & ty & tz)\\ \\ \begin{pmatrix} xt\\ yt\\ zt\\ \end{pmatrix} \begin{pmatrix} x^2 & xy & xz\\ yx & y^2 & yz\\ zx & zy & z^2 \end{pmatrix} \end{vmatrix}$$

Entry 00 is represented by $M_{00} (\Delta r)^2$; 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 $2(\sum_{i=1}^3 M_{0i} \Delta x^i) \Delta r$; because these terms can be factored into t(xyz), there is a delta r for each. The remaining entries are represented by $\sum_{i=1}^3 \sum_{j=1}^3 M_{ij} \Delta x^i \Delta x^j$; 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

 

[HeavyMetal]

I'm not quite sure why my matrices weren't showing up properly in the above post.To reiterate, my answer described the matrix below:
\begin{pmatrix}
t^2 & tx & ty & tz\\
xt & x^2 & xy & xz\\
yt & yx & y^2 & yz\\
zt & zx & zy & z^2
\end{pmatrix}I explained it in three terms, each explaining separate parts of the matrix.Entry 00: \begin{pmatrix}
t^2
\end{pmatrix} which is represented by $M_{00} (\Delta r)^2$; because there are two time entries, we observe two delta r terms.Entries 01, 02 and 03: \begin{pmatrix}
tx & ty & tz\\
\end{pmatrix}and entries 10, 20 and 30: \begin{pmatrix}
xt\\
yt\\
zt\\
\end{pmatrix}are the transpose of each other, and equivalent due to the fact that $M_{\alpha \beta} = M_{\beta \alpha}$, and so they were just repeated with a two multiplying out front of that entire term. This was represented by $2(\sum_{i=1}^3 M_{0i} \Delta x^i) \Delta r$; because these terms can be factored into t(xyz), there is a delta r for each.The remaining entries: \begin{pmatrix}
x^2 & xy & xz\\
yx & y^2 & yz\\
zx & zy & z^2
\end{pmatrix} are represented by $\sum_{i=1}^3 \sum_{j=1}^3 M_{ij} \Delta x^i \Delta x^j$; because none of these entries are multiplied by time, the delta r term is absent from this part.I hope no confusion was caused by my formatting issues.

I would also like to repeat that this thread is NOT about the interval, but rather the apparent similarity between this representation and the organization of the stress-energy tensor. The Wikipedia article states:

If Cartesian coordinates in SI units are used, then the components of the position four-vector are given by: x⁰ = t, x¹ = x, x² = y, and x³ = z, where t is time in seconds, and x, y, and z are distances in meters.

I'm not sure if this is a coincidence, or just a result of the $4 \times 4$ nature of this matrix, along with the fact that they are both displayed in spacetime coordinates.

 

[wabbit]

If I understand your question correctly, the formula is general in the sense that it results from the 1+3 block decomposition of any 4x4 symmetric matrix, and as such applies to any quadratic form in 4D (and generalizes to any dimension). The stress-energy tensor being rank-4 symmetric in GR, it applies there too (but not in theories where that tensor isn't symmetric). There may be a deeper connection but I don't know about that.

 

[HeavyMetal]

If I understand your question correctly, the formula is general in the sense that it results from the 1+3 block decomposition of any 4x4 symmetric matrix, and as such applies to any quadratic form in 4D (and generalizes to any dimension). The stress-energy tensor being rank-4 symmetric in GR, it applies there too (but not in theories where that tensor isn't symmetric). There may be a deeper connection but I don't know about that.

Oh yeah, you understood my question quite perfectly! I'm currently studying linear algebra from the Wikibook articles, and I haven't learned about block decomposition yet. I've been using it recently to answer physics questions, but I guess I just didn't know what the name for it was .