# 4-Space time rotations

1. Aug 22, 2006

### lokofer

2 Weeks ago..i was reading the novel "The time Ships"... by Stephen Baxter (the second part of "the time machine" by H.G Wells) in a paragraph the "time traveler" explained how his machine worked.... inducing a "rotation" in the space time so the "time interval" became an "space interval"..of course according to Relativity this is impossible..but why?..in fact let be the metric:

$$ds^2 = dx^2 + dy^2 + dz^2 +dT^2$$ (imaginary time ict=T )

But this metric above is just an "Eculidean" 4-dimensional metric...if we have that under no potential or force so the metric element is not altered the metric is just invariant under any rotation (4-dimensional) so you couldn't say exactly what is x and what is t :grumpy: :grumpy:

2. Aug 22, 2006

Staff Emeritus
1. Modern relativists avoid the ict notation for preisely the reason that "you can't be just a little bit complex". Stick complex variables into special relativity and all sorts of weird wonders come with it. You can continue around the singularity at v = c for example. Better not open tha can of worms when we have no evidence that kind of stuff occurs.

2. having said that there is a mathematical transformation called Wick rotation that does exactly what you're thinking, but backwards. It assumes we have a real t, with a pseudo-euclidean (aka Minkowskian) metric. Then replace this real t with it wherever it occurs and continue around to replace the positive real axis in the complex plane with the upper imaginary axis. This converts the pseudo-euclidean into a proper euclidean one, and that in turn makes Feynaman path integrals convrge at infinity, whereas they just oscillate at infinity in the pseudo-euclidean metric. After they get the convergent answer, they just continue back to the real t and the pseudo-euclidean metric of relativity.

Last edited: Aug 22, 2006
3. Aug 22, 2006

### pervect

Staff Emeritus
The space-time in a Schwarzschild metric of a black hole is rotated so that the r coordinate becomes time-like.

I think we can also say that any such rotation must be associated with an event horizon, i.e. a trapped null surface, i.e a region that light can't escape. (This just came to me as an insight, it's not a well known result from a textbook like the previous remark).

Consider the contribution of a coordinate dz to ds using the -+++ sign convention.

If the contribution is positive, the coordinate is spacelike.
If the contribution is negative, the coordinte is timelike.

If a coordinate is switching from space-like to time-like smoothly, there must be a region where the contribution of changing the coordinate to the metric is zero,.

Setting that coordinate equal to zero will produce a null geodesic of a "trapped" photon, for any trajectory in which ds=0 is the trajectory of a photon, and this is a photon that is "standing still".

Of course black holes don't necessarily allow time travel, either, in the sense of closed timelike curves.

On the other hand, there are some known metrics in General Relatiavity that do have closed time-like curves. The jury is still out on a conclusive proof that time machines do not exist (or do exist) in our universe according to GR.

http://www.mcs.vuw.ac.nz/~visser/general.shtml#why-wormholes

talks about some of the issues in the context of one particular form of time machine, the wormhole time-machine.