Understanding space-like hyperplanes in Minkowski space

Question

Here I use: $$ \eta = \begin{bmatrix}-1&0\\0&I_3\end{bmatrix} $$ Two points $X$ and $Y$ are said to be space-like when $(X-Y)^T\eta(X-Y)>0$. That is, light cannot reach from one to the other, they are spatially separated.

A space-like hyperplane is a 3D subspace of Minkowski space, where all points are space-like separated from each other. A time-like hyperplane is a 3D subspace of Minkowski space, where at least two points are time-like separated, $(X-Y)^T\eta(X-Y)<0$.

I see that a snapshot of 3D space at a particular time satisfies the definition of space-like. I think that any Lorentz transform of that hyperplane is also a space-like hyperplane, since $L^T \eta L = \eta$, where $L$ is a Lorentz transform. Are these all the space like hyperplanes? If so, it is very easy to show that: $$ d(X,Y) = \sqrt{|(X-Y)\eta(X-Y)|} $$ Is a metric on any such space-like hyper plane, you just need to transform the plane into the usual 3D space all at a particular time (preferably t=0) and this becomes the usual metric on euclidean space. If I was wrong and and not all space-like hyperplanes are described as above, then how would one show that the distance function defined is indeed a metric on any space like hyperplane? In particular, the triangle inequality must hold. And how would one go about showing that the triangle inequality is reversed in a time-like hyperplane?

I am looking for some intuition about these hyperplanes, the definitions above are all I have from my book.

Answer 1

I will use a bit of the more-standard index notation so please understand $x^\mu$ as your column vector $X$ and $x_\mu$ as the row vector $X^T~\eta.$ Since $\eta~\eta^T = 1$ one can also reconstruct the vector from a covector, $\eta~(X^T \eta)^T.$ The upper/lower index just identifies the vectors distinct from the covectors,¹ but it is helpful to nevertheless think at some level about "components" in various directions or along various axes.

So the 4-positions $r^\mu$ on a hyperplane satisfy $q_\mu~r^\mu = C$ for some 4-covector $q_\mu$ and some $C;$ the requirement that this hyperplane be spacelike means that if two points $r^\mu_{0,1}$ are on this hyperplane then they are spacelike-separated, which is a statement about $\Delta r^\mu = r^\mu_1 - r^\mu_0.$ We can quickly see that $q_\mu~\Delta r^\mu = 0,$ but if $q^\mu$ is not timelike then there exists some Lorentz boost which brings its $ct$-component equal to $0,$ and in the resulting reference frame the timelike 4-displacements $\Delta r^\bullet = (ct, \vec 0)$ solve that equation, so by contradiction: a spacelike hyperplane has a timelike normal vector, $q_\mu~q^\mu < 0$ in the $(-~+~+~+)$ metric.

Since $q^\mu$ is timelike there exists some Lorentz transform which makes it point entirely along the $ct$-axis, and indeed within this special reference frame the invariant intervals $(r_0)_\mu ~ r_1^\mu$ reduce to the normal 3-dimensional dot product $x_1~x_2 + y_1~y_2+z_1~z_2.$ According to this reference frame this is 3D space, "frozen in a moment of time."

Light cones and spacelike hyperplanes

We can also reason in the reverse direction: if you imagine that in your reference frame, at $t = x = y = z = 0,$ some event like a supernova happens: it releases a sphere of light which announces its existence to the world, uniformly expanding at speed $c$. Every 4-position at a later time which is inside this sphere is timelike-separated from the origin. This expanding-bubble is known formally as the "future light cone" of the origin. There is also a "past light cone" of all the places in spacetime whose light has reached the origin, a negative-time expanding-bubble expanding at the same speed $c$. Spacelike hyperplanes through the origin, corresponding to these timelike normal vectors, can tilt in any way such that they're still trapped between the two cones.

Now another event happens somewhere in spacetime: ignoring the degenerate case there are exactly three places it can happen: in the future light cone, in the past light cone, or in the space between the two. Lorentz transforms can rescale some bubbles relative to other bubbles, but it always maps light-cones to light-cones and it always preserves this partition of points within/without those light cones. And you can topologically see the difference: in, say, the future-case, the expanding bubble of the one is entirely contained in the expanding bubble of the other (except at times when it doesn't exist); in the present-case the two bubbles eventually meet on an expanding circle. Now, in the future-case: there exists a reference frame which traveled from one event's location to the other, and it measures a special time between the events called the "proper time", which everyone else measures as this "spacetime interval" between the events, $~(X - Y)^T~\eta~(X - Y).$ It is the actual time for these reference frames at which both events happened at the same location. And we can say definitively: "objectively time-separated implies not objectively space-separated." These reference frames think that both happened at the same place; other reference frames think they might have happened at different places. Other reference frames also must apply a "time dilation" factor $\gamma$ to the proper time between these depending on how fast $|v|$ they think those "proper" reference frames are moving past them.

You can probably see the converse side, but let's go through it: if the two events are objectively space-separated then there exists a spacelike hyperplane through them, we Lorentz-transform this to be perpendicular to the time axis, and the distance between the events becomes a "proper distance"; they are objectively space-separated but not objectively time-separated; some hyperplanes through the origin pass above this other event and think it's in their past, some pass beneath it and think it's in their future. Objectively space-separated implies not objectively time-separated, and the reference frames which sees the two events as happening at the same time measure a "proper distance," which everyone else must "length-dilate" by a factor $\gamma$ according as they see the proper reference frame moving past them, but they also of course see the events happen at different times ($\gamma\beta/c$ times the proper distance).

[RECORD-SCRATCH-SOUND] Length-expand? What about length-contraction?!

That's a little more involved and it comes down to a difference in what "proper" means sometimes when it's used. Two parallel timelike world-lines $s \mapsto r_{0,1}^\mu + \tau^\mu~s$ will have this common tangent vector $\tau^\mu$ which is timelike, so we can boost into the frame where that vector points along the time axis; the resulting "rest frame" for the two objects can also be said to measure a "proper distance" between these two world-lines, and that contracts like $1/\gamma,$ but of course we're measuring the distance between the objects at "different times" compared with the frame where they are at rest. The difference is that the former looks at two events that are spacelike-separated, the latter looks at two objects that maintain a spacelike separation for all times for all viewers.

1. In this "abstract index" notation these upper/lower indices do not actually label individual components; they're just labels for "this is a vector" (upper) and "this is a covector" (lower) and multiple indices refer to an outer tensor product like $x^\mu~y^\nu \leftrightarrow X\otimes Y,$ keeping track of which is which. A repeated upper-and-lower Greek letter indicates "apply this covector to that vector"; if that smells a little mathematically fishy it's because you do need an extra axiom that the tensors living in $T^{\mu\nu}$ can always be decomposed as some sum $\sum_i u_i^\mu~v_i^\nu$ and so on for tensors of other valences: then this action of "applying covectors to vectors" can be thought of as a linear map $T^{\mu\mathcal X}_{\nu\mathcal Y}\to T^{\mathcal X}_\mathcal Y$ which we symbolize by repeating some index above and below, forming a sort of "trace". Anyway we would write $x_\mu = \eta_{\mu\nu}~x^\nu$ according to this sort of notation.