Rotate an object about the time axis

Question

Is there a notion of rotating an object about its time axis? I'm not sure if this question totally makes sense, but it seems intuitive to me that an object with dimensions in the three spatial directions and the time dimension (so an object paired with an interval of time) could be rotated about the time axis, just as it could be rotated about its "$x$-axis", "$y$-axis", or "$z$-axis". Is this intuition misguided? If it is possible, what would it look like to perform such a rotation?

Answer 1

This a great question, and leads to some interesting ideas.

Firstly, the notion of a "rotation axis" is restricted to three dimensions. In more than three dimensions the rotation axis concept has to be replaced by a specification of the plane in which the rotation occurs. In 3d a rotation about the $z$ axis is the same thing as a rotation in the $x$-$y$ plane. The $z$ axis is chosen as "the axis" because it is the unique direction perpendicular to the $x$-$y$ plane. In higher dimensions there is more than one perpendicular direction and so the "axis" notion is no longer useful, but the choice of plane still works.

Secondly, once we have four dimensions, one of which is time, there is another complication: the new 4d space is not Euclidean but rather is "Minkowskian". A Minkowski-space "rotation" in the $t$-$x$ plane is same thing as a Lorentz boost in the $x$ direction. It's not a normal rotation because of the minus sign in the Minkowski metric. This sign changes the $\sin \theta$ and $\cos \theta$ of the usual rotation into $\pm\sinh s$ and $\cosh s$ where $s$, called the rapidity, is the analogue of the rotation angle $\theta$.

Answer 2

Is there a notion of rotating an object about its time axis? I'm not sure if this question totally makes sense, but it seems intuitive to me that an object with dimensions in the three spatial directions and the time direction (so an object paired with an interval of time) could be rotated about the time axis, just as it could be rotated about its "x-axis", "y-axis", or "z-axis". Is this intuition misguided?

The idea of "rotating" implies that at one time the object has some orientation with respect to the spatial coordinate axes and at another time the object has some different orientation with respect to the spatial coordinate axes. So there is a before rotation and an after rotation.

Spacetime has all time, all future and all past. So there cannot be a before rotation or an after rotation in spacetime because there is no before or after, all time is already included in spacetime. An object is simply one unified 4D shape in spacetime embedded in a 4D manifold.

Instead, in spacetime a spatial rotation is a "twist" along the length of the object. The orientation of the cross-section with respect to the spatial axes is different at different points along the time dimension. This is similar to the sculpture below, where the vertical direction represents the time axis.

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So similarly, the orientation of the cross-section with respect to the time axis could be different at different points along the time dimension. This would be a bend in the spacetime shape and would correspond to an acceleration of the object.

Answer 3

There can be such a notion, and it ties in with the notion of curvature of spacetime. Curvature of spacetime is measured by parallel transport of a 4 vector around a small square. If spacetime is curved, the square doesn't come back to the same point and the 4 vector changes direction and/or magnitude.

You can see the same thing in a curved 2D space, such as the surface of the Earth. Consider a square with sides 1 m long oriented parallel to meridian and longitude lines. Start with a 2D vector at the southeast corner pointing north. Move it north, then west, then south, then east back to the start. The change is too small to see. The surface of the earth is very nearly flat.

Try again with a smaller sphere, where the distance from the equator to the north pole is 1 m. The square now looks different. It goes from the equator north to the north pole. It makes a 90$^o$ turn and goes south to the equator, then east along the equator, then north back to the north pole. If you follow the direction of the vector, you see that it has rotated 90$^o$ in its journey.

You can make similar loops in spacetime. Since you cannot travel backward in time, it is more convenient to go halfway around the square in opposite directions. The square can have sides parallel to the 3 space axes and/or the time axis. The sides $x$ and $t$ are of equal length if $x = ct$.

General Relativity tells us that mass curves spacetime. Time travels more slowly in a gravitational well than in space far from a planet. See Why can't I do this to get infinite energy?.

Let us choose a square that starts above a black hole. One side is spatial, toward the center. The next is time - you just wait a bit. For one half-trip, you wait $t$ and then find a point $x$ below you. For the other, you find the point $x$ below you, and wait there. Since time travels more slowly at the lower point, you arrive at two different events.

To see how a vector changes, you define a vector by placing a rock at the start and seeing how far it moves in a short time. The rock both the time and space component of the vector are different if you wait then move vs move then wait.

See this for more on visualizing curvature of spacetime - A new way to visualize General Relativity

Answer 4

Your intuition is interesting, but rotating an object about the time axis isn't quite analogous to rotating an object in spatial dimensions. In special relativity, however, there is a concept that somewhat resembles your idea called Lorentz transformation. Lorentz transformations can be seen as "rotations" in a four-dimensional spacetime, which includes three spatial dimensions and one time dimension.

In Minkowski spacetime, the time and spatial coordinates are combined into a four-vector called the spacetime interval. A Lorentz transformation relates the coordinates of an event in one inertial frame to the coordinates of the same event in another inertial frame, which is moving with a constant velocity relative to the first frame. This transformation mixes the time and spatial coordinates, similar to how rotations mix the spatial coordinates in three-dimensional Euclidean space.

For a Lorentz transformation along the $x$-axis with a relative velocity $v$, the transformation equations are:

\begin{align} t' &= \gamma \left( t - \frac{v x}{c^2} \right) \\ x' &= \gamma \left( x - vt \right) \\ y' &= y \\ z' &= z \end{align}

where $\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$ is the Lorentz factor, $c$ is the speed of light, and $(t, x, y, z)$ and $(t', x', y', z')$ are the spacetime coordinates in the unprimed and primed frames, respectively.

Although Lorentz transformations can be seen as a kind of rotation in spacetime, they don't have the same geometric interpretation as spatial rotations. They involve hyperbolic functions, rather than the trigonometric functions used in spatial rotations, and their effects are not visually intuitive in the same way.