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If I have a number of particles interacting with one another locally, then the center of mass of the system moves along a geodesic. Taking this further with the particles interacting via an EM field, I can also say that the center of energy of the electromagnetic system moves also along a geodesic -- a straight line in Minkowski space.

What analogous physical principle applies to space like intervals?

For a closed EM system as an example, I suspect it's something to do with: The total four-momentum leaving a 3-d surface in Minkowski space is stationary, but I'm not exactly sure.

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  • $\begingroup$ There is no restriction on geoesics to be only time-like. Geodesic is simply the "minimal" possible path between any 2 points in space (or sapcetime). The space perpendicular to worldline tangent, which is all spanned by space-like geodesics can be thought as linking to spacetime events that are "simultanious" to that particular point of that particular worldline. $\endgroup$ – Alexander May 27 '15 at 19:40
  • $\begingroup$ Note that this is only true if the particles are small enough that their gravitational interaction and the curvature they cause is small relative to the background. $\endgroup$ – Jerry Schirmer May 27 '15 at 19:51
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I don't think there is an analogous statement for spacelike intervals. The statement "The centre of mass follows a geodesic" is an immediate consequence of global momentum conservation. This is in turn follows from local momentum conservation. \begin{equation} \partial_\mu T^{\mu \alpha} = 0 \end{equation} where $T^{\alpha \beta}$ is the stress-energy tensor. By analogy to the derivation of global four-momentum conservation fix $x^3$ and integrate this equation over some 3d volume with 2 spacelike and 1 timelike dimension, $V$, to get\begin{align} \int_V\mathrm{d}x^0\mathrm{d}x^1\mathrm{d}x^2\; \partial_3T^{3\alpha} &= -\int_V\mathrm{d}x^0\mathrm{d}x^1\mathrm{d}x^2 \left(\partial_0T^{0\alpha}+\partial_1T^{1\alpha}+\partial_2T^{2\alpha}\right)\\ &=-\int_V\mathrm{d}x^0\mathrm{d}x^1\mathrm{d}x^2\; \nabla\cdot T\\ &=-\int_{\partial V}\mathrm{d}\vec{s}\cdot T \end{align} where the integral in the last line over the boundary of $V$. Now in the conventional derivation of a global conservation law from a local one we let the size of $V$ tend to infinity and, since we require all fields to vanish at infinity in space, the left hand side gives us that the time derivative of some quantity is 0. In our case, where we are looking for the spatial analogue to a conservation law, we hit two problems. Firstly our 'conserved' quantity \begin{equation} P^\alpha = \int\mathrm{d}x^0\mathrm{d}x^1\mathrm{d}x^2\; T^{3\alpha} \end{equation} depends on the value of $T$ at all times, which we generally do not know. Secondly in the limit that $t\rightarrow \pm \infty$ the conservation of energy means $T$ cannot in general vanish and since we are interested in $T$ for some fixed $x^3$ the flux of $T$ in $+\infty$ and $-\infty$ limits need not cancel. This means our 'conserved quantity' is not, in fact, 'conserved' but can vary as we vary $x^3$.

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