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Or is the question in the title fundamentally wrong? We label each point in space-time with four coordinate values, one of which typically is suggestively called $t$ for time. This made me think that I could just fix, say, $t=0$ and look at the reduced-dimension manifold.

Yet a coordinate system is somewhat arbitrary and can, for example, be rotated such that nearly all points are relabeled. I assume that what, after rotation, is then called suggestively the $t'$ for "time" axis may point in a different direction and we don't have $t=0 \Leftrightarrow t'=0$, meaning that what was previously a "pure space point", i.e. one with $\text{time}=0$ need not be one after rotation.

So let me try a question: Can we relatively freely rotate our 4 dimensional coordinate system for the universe's spacetime such that what was space before (time fixed, say at zero) is rotated "into" the time axis to get varying values of the new time coordinate. Or is there some either mathematical or physical obstacle against completely arbitrary rotations that allows us to talk about space alone in some sense?

This answer actually helped me formulate my question and my hunch is that what is called "foliation" there is roughly my setting the time coordinate to zero in differently rotated coordinate systems.

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Let's suppose you are an observer and you have a clock to measure time and rulers to measure distance. You construct a coordinate system by placing yourself at the origin and using your clock and rulers to measure out your axes.

When you do this there is no ambiguity about what your time axis is because it's simply the time measured by your clock. You can of course choose any coordinate system you want, but that won't change what your clock is measuring. You are free to construct a coordinate system with some different time coordinate $T$, but then the $T$ coordinate won't be what your clock will be measuring.

The point of relativity (both special and general) is that I can also construct a coordinate system with myself at the origin using my clock and rulers, and in general my time axis will not be the same as your time axis. So what you regard as a displacement in time, or a curvature in the time coordinate, would for me be a displacement or curvature in both the time and spatial coordinates.

So as an observer you are free to talk about curvature being only in the spatial coordinates, but that is an observer specific statement and other observers will generally disagree with you.

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A few facts:

  1. In a 4-dimensional manifold such as spacetime you can pick any timelike direction and call it time in the vicinity of any given event. Directions orthogonal to this will then make up 'space'. To extend the definitions, you make a 'threading', that is, many timelike likes smoothly displaced from one another (not intersecting) and you have a notion of time for a continuous region of spacetime. The direction orthogonal to this time can be called space.

  2. In the cosmos at large there is a natural choice to make for the timelike lines, owing to the way the matter is moving. You pick the worldlines of freely-falling matter at the largest scales. This is the standard choice made in cosmology, but for the purposes of your question you do not have to make this choice.

  3. If the 4-dimensional manifold is flat, it is always possible to pick the part called 'space' such that it is curved. (This would be an unusual choice, but it is available).

  4. If the 4-dimensional manifold is curved, it may be possible to find spacelike sections which are flat, or more generally submanifolds which are flat. I think this is a less common situation, and maybe not guaranteed to be possible; I'm not sure. But around a Schwarzschild black hole you can find, remarkably enough, a sequence of spacelike regions with Euclidean metric! Check out Gullstrand-Painlev'e coordinates.

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"Can we relatively freely rotate our 4 dimensional coordinate system for the universe's spacetime such that what was space before (time fixed, say at zero) is rotated "into" the time axis to get varying values of the new time coordinate. Or is there some either mathematical or physical obstacle against completely arbitrary rotations that allows us to talk about space alone in some sense?"

It depends what you mean by "rotate". One way to interpret it is as a transformation that fixes the surfaces a constant distance from the origin. In normal Euclidean space, these are spheres $x^2+y^2+z^2=k$, where $k$ is a positive constant, and we get the rotation we are familiar with where any coordinate axis can be rotated into any other coordinate axis. But in spacetime the surfaces are (4-dimensional versions of) hyperboloids $x^2+y^2+z^2-t^2=k$, with $k$ a positive or negative constant, which are split into two families separated by the lightcone. If $k>0$ then the surface is a hyperboloid of one sheet and the distance to the origin is space-like. If $k=0$ then the surface is the cone representing light rays passing through the origin. And if $k<0$ the surface is a hyperboloid of two sheets in the past and future of the origin. Space and time are distinguished, depending on the sign of $k$.

Thus, rotations that preserve the lengths of the coordinate axes ('length' here being the spacetime interval, not just the space coordinate) cannot transform a time axis (with $k<0$) into a spatial axis ($k>0$) because their 'lengths' are different (the squared lengths have different signs). The time axis can point to anywhere on the part of the two-sheet hyperboloid in the future light cone, but it can never cross the cone. Similarly, the spatial axes can point anywhere on the one-sheet hyperboloid 'outside' the light cone, but never cross it.

That's the situation in special relativity, and I think answers the question you meant to ask. But there is a sense in which the very different question in your title can be interpreted so as to (sort-of) give a different answer in general relativity.

Close to a black hole, gravity has the effect of tilting the light cones towards the black hole. (See 2nd picture here for example.) At the event horizon, the entire future light cone is pointing into the black hole, so it is impossible to escape without going faster than light. In the region inside the black hole the radial direction towards the centre is now time-like, and the direction of time flowing outside the black hole is space-like inside. Space and time are still distinct, separated by a light cone, but the cone itself has twisted round so that what was previously a spatial direction (the direction towards the centre of the black hole) is now pointing future-ward.

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