Rotation in the x-t plane

I am currently studying special relativity using tensors. My lecture notes (which happen to be publicly accessible, see top of page 99) say that the standard configuration can be viewed as a rotation in the x-t plane. Can anyone explain this a bit? Is there a good way to visualize it?

The Standard configuration; I would prefer an intuitive (not rigorous) explanation over a maths one, but if you have a maths one then for completeness it would be good to post it.

All other notes from these lectures are available here.

You probably already understand that merely rotating a coordinate system does not change the physical system you're modeling. Given one set of axes, any rotation of those axes just represents some freedom of choice to do math how you see fit.

So with that in mind, if you were pointing northeast (and chose a pair of coordinate axes to point northeast and northwest) and had a compass (pointing north), you'd have the freedom to rotate your coordinate axes to point north instead, simplifying the description of the compass direction.

Similarly, an object traveling with nonzero velocity traces out a diagonal line on the $tx$-plane, but there is a single, well-defined group of operations that could align the $t$-axis with such a diagonal line. This is analogous to a rotation.

I say analogous it's not exactly a rotation the way we typically think of it. We think of rotations as tracing out circles. That's because rotations preserve distance, and a circle is a uniform distance from its center.

But in spacetime, on the $tx$-plane, we have to think of "distance" as being the spacetime interval, and a curve of constant interval isn't a circle--it's a hyperbola. And in this case, we're talking about a hyperbola with diagonal asymptotes. Those asymptotes are the lines that light follow, unreachable by any "rotation" on the $tx$-plane.

This operation is every bit a "rotation" as our usual 3d rotations are; perpendicular vectors stay perpendicular under it, and so on. It's just the equivalent of a rotation for a different kind of underlying space(time) than regular old 3d space.

And so, it is convenient that we don't typically call this a "rotation" in so many words, but we give this operation its own name: the Lorentz transformation.

Using the Lorentz transformation to put some align a four-velocity with the $t$-direction is no different from aligning one's $xyz$-directions with east, north, and up. It's an arbitrary choice, one that you don't have to abide by if it's inconvenient, but just as there always exists a rotation to align one's $xyz$-directions that way, there always exists a Lorentz transformation to align the four-velocity of a massive object with the $t$-direction.

• This preserved length, is it the Minkowski distance? The invariant hyperbola you describe seems like the hyperbola drawn on pg 93 (www2.ph.ed.ac.uk/~rhorsley/SII14-15_socm/lec18.pdf). It that it? – Jekowl May 1 '15 at 7:25
• Yes, the Minkowski distance. - The hyperbola drawn there is appropriate for spacelike objects. Imagine the $tx$-plane cut into quarters by the diagonals (the lightlike lines). Each quarter corresponds to a distinct region: the quarter containing $+ct$ is the causal future, for instance--the set of points that can be influenced by an object at the origin. A point on the hyperbola in the causal future will stay on that hyperbola piece no matter what Lorentz transformations are applied, and cannot reach the lightlike lines on either side (since the hyperbola never intersects them). – Muphrid May 1 '15 at 7:33
• Thank-you, this is a very good conceptual explanation and ties in with understanding of other elements of minkowski diagrams. – Jekowl May 1 '15 at 8:01

Bernhard Schutz discusses this reasonably well in his book A First Course in General Relativity.

Consider sending a light beam horizontally along an $x$ axis and then receiving it back again. A space time plot of this would look like Here's an example of the rotation you are describing And this is how everything becomes distorted when you create such an $xt$ plane rotation: Note some major properties

1. Simultaneity broken
2. It shrinks moving objects (length contraction)
3. Slows down moving clocks (ie time dilation)

But it works and allows $$v_{light} = c$$

See this video for a visual (it's really cool!):

http://youtu.be/C2VMO7pcWhg

And buy/rent/borrow Schutz's book. He nicely walks you through this part. The SR section is actually my favorite part in the whole book.

In matrix form,

$$\begin{bmatrix} ct'\\ x'\\ \end{bmatrix}= \begin{bmatrix} \cosh \theta & -\sinh \theta\\ -\sinh \theta & \cosh \theta \\ \end{bmatrix}\begin{bmatrix} ct\\ x\\ \end{bmatrix}$$

Video Explaining Rotation Matrix:

http://youtu.be/lRnGpBJQluQ

Note:

The OP mentioned tensors. Someone might like to add a bit about the role of tensors in Lorentz transformations.

• Thank-you for your answer, it is a great compliment to the other answer here. If I ever have enough rep here I will come back to upvote both answers. – Jekowl May 1 '15 at 8:03 ... Another way to look at it is to use geometry which includes the vertical time axis, the horizontal space axis, and the stacking of both motion vectors( c, v ) and length scalars. Rotation determines the direction of travel in space-time, thus determines the velocity across space and the rate of the ticking of time.

Here, twin spaceship A is a rest in space, but is in motion across time at c, and twin spaceship B is moving at a velocity of 260,000 km/s. From this simple geometry we determine the Length Contraction equation, the Time Dilation equation, and we can also quickly derive the Velocity Addition equation, along with the Lorentz Transformation equations, all in a matter of minutes.