Is There More to Special Relativity?

[arydberg]

Is special relativity finished? It seems to me that there is more to special relativity.
If we define C as equal to 1. (time = nanoseconds and length = feet) then we do a special relativity experiment an interval is defined. This interval is given by the square root of X squared minus T squared. The interval is the same regardless of the motion of a observer. That same result is obtained if we use the Pythagorean rule and precede the time with a “i” making time a imaginary number This fact hints that there is something more going on. Also in the physical world there are 3 spatial dimensions and one time dimension. In the mathematical world there are many dimensions that seem to be analogous to spatial dimensions but only one imaginary dimension. It seems to me that the equations of special relativity should include a “i”.

 

[PeroK]

Is special relativity finished? It seems to me that there is more to special relativity.
If we define C as equal to 1. (time = nanoseconds and length = feet) then we do a special relativity experiment an interval is defined. This interval is given by the square root of X squared minus T squared. The interval is the same regardless of the motion of a observer. That same result is obtained if we use the Pythagorean rule and precede the time with a “i” making time a imaginary number This fact hints that there is something more going on. Also in the physical world there are 3 spatial dimensions and one time dimension. In the mathematical world there are many dimensions that seem to be analogous to spatial dimensions but only one imaginary dimension. It seems to me that the equations of special relativity should include a “i”.

The spacetime interval between two events can be defined as: $\Delta s^2 = \Delta x ^2 + \Delta y ^2 + \Delta z ^2 - c^2\Delta t^2$

Clearly, therefore, you could model this using (x, y, z, ict) but that doesn't seem to be standard. At the end of the day, you use whatever mathematics is most useful and practical to model your situation and it seems that SR gets along without using complex numbers.

It's also possible to define a spacetime interval as: $\Delta s^2 = -\Delta x ^2 - \Delta y ^2 - \Delta z ^2 + c^2\Delta t^2$

Then, you could model this as (ix, iy, iz, ct), with time being "real" and your spatial components being "imaginary".

An old professor of mine used to say: "you pays your money and you takes your choice".

Regarding mathematics, you should look up Quaternions. That is a 4D system that has one "real" and three "imaginary" dimensions. Quaternions turned out not to be so useful as complex numbers - so aren't used as much.

Note also, that the word "imaginary" has no significance. It's just a label given to the complex numbers. Complex numbers are no more imaginary than matrices, say.

 

[sweet springs]

There scarce be more in SR, I think. Quantum mechanics and SR were already united. More spatial dimensions than three would not change the essence of SR.

 

[Dale]

That same result is obtained if we use the Pythagorean rule and precede the time with a “i” making time a imaginary number

Yes, that was recognized and proposed more than a century ago. It began to fall out of favor with the development of general relativity because it did not generalize as well to curved manifolds. But you do still see it in older papers and texts.

 

[robphy]

In my opinion, there are still some interesting problems to work out in special relativity.
How would you properly formulate [insert PHY 101 or PHY 102 problem] in relativity? Not merely first-order in "relativistic corrections".
What are the similarities and differences with the how its done in PHY 101 or 102...presumably related by some classical limit?
How would you express that in a tensorial-way (the spacetime viewpoint),
then apply dot-products with observer-4-velocities to obtain measurements made by that observer?

What would those measurements be for uniformly-accelerating observers? rotating-observers? arbitrarily-moving observers?

What are the foundational structures in Special Relativity, which persist when other structures taken for granted are weakened?
For example, what suffers and what doesn't if spacetime is somehow discretized?...sometimes considered when looking for a quantum theory of gravity while recovering classical spacetime in some limit.

Are there other interesting algebraic, geometrical, topological structures not yet fully realized or exploited?

Are there other (possibly better) ways to teach, understand, calculate in relativity?

 

[bcrowell]

Basically it's 110 years old, mature, well understood, and thoroughly tested to incredibly high precision. The remaining unknowns are really in GR and quantum gravity.

But...

Doubly-special relativity https://en.wikipedia.org/wiki/Doubly_special_relativity might be an example of a recent development, although AFAIK it didn't turn out to be successful.

On the experimental side, it's conceivable that tachyons will be discovered as real particles, although the CERN OPERA debacle has probably made most physicists not want to touch the topic, and I think theorists see them now more as pathological things in QFT. But, e.g., people have searched for tachyons in recent years in cosmic ray showers.

People are still doing high-precision tests of Lorentz invariance, e.g., at UW. As they keep adding more decimal places, it's conceivable they'll actually get a non-null result one of these days.

Pedagogically, people have started doing innovative things in the last 10-20 years that IMO are much, much better than the traditional pedagogy. Examples are the books by Takeuchi, Mermin, and Laurent, and Morin's mechanics book.

 

[pervect]

Is special relativity finished? It seems to me that there is more to special relativity.
If we define C as equal to 1. (time = nanoseconds and length = feet) then we do a special relativity experiment an interval is defined. This interval is given by the square root of X squared minus T squared. The interval is the same regardless of the motion of a observer. That same result is obtained if we use the Pythagorean rule and precede the time with a “i” making time a imaginary number This fact hints that there is something more going on. Also in the physical world there are 3 spatial dimensions and one time dimension. In the mathematical world there are many dimensions that seem to be analogous to spatial dimensions but only one imaginary dimension. It seems to me that the equations of special relativity should include a “i”.

That's a good summary, and it can be summarized even more concisely (with a bit of jargon) saying that special relativity is geometry - but because of the fact that you modify the pythagorean theorem by incorporatin a minus sign, it's a Lorentzian geometry rather than a Eucliden geometry.

I'd say it's a mature theory, but I wouldn't use the word "finished". The approach you're using is essentially analytic geometry. There are several other approaches to geometry, consider the difference between the analytic geometry approach and Euclid's axiomatic approach. They both cover the same theory (geometry), but via different techniques. One can also re-visit the analytic approach in the context of using non-Cartesian coordinates, this leads to tensors and differential geometry, both of which can be used for a more advanced understanding of special relativity, with important applications to general relativity.

Aside from the differing theoretical approaches, there is the question of experimental evidence (of which we have a lot, but occasionally more precise tests are done as the technology advances), and exploring the consequences of the theory, such as Thomas precession.

Then there is the occasional discussion of how to solve particular problems, such as how to do relativity in rotating frames, or how to analyze a relativistic sliding block (just to pick two examples that we've seen a lot of discussion of on PF).

Perhaps the biggest thing missing from your description is how to take the kinematic part of the theory (i.e the Lorentz transform) and use it to get equations of motion, i.e. relativistic dynamics. This is also well known (see for instance Goldstein, classical mechanics), but it's something not covered directly by understanding the geometry, though that's an essential first step.