What is the problem with two time dimensions?

Question

I am reading a book "General relativity: The theoretical minimum" by Leonard Suskind.

In page 168-169, the author explains the reason why we don't consider the case with two time dimensions in the metric as follow:

It(metric with two (-) signs and two (+) signs) doesn't mean anything in physics.There are never two time dimensions. There is always one time and three space dimensions. Can you imagine a world with two times? Personally I cannot imagine what it would mean to have two different time dimensions. So we will simply take the view that it is not an option.

I do agree that it is hard to imagine two different time axis. But, I don't think it is very scientifically logical to cancle a case just because it is hard to imagine.

So I want to know what is the logical/mathematical/physical problem of two time dimensions.

Answer 1

This is one of the many death-by-paper-cuts reasons why I really do not like Susskind.

There is a far better experimental argument as to why we do not consider more-than-one time dimensions until some other experiments comes by to force us to consider them.

We have experimentally observed that there are certain purify-able materials that undergo particularly simple types of radioactive decay. Those that are basically at the start of a chain of radioactive decays, they fit the random exponential decay curve extremely well, with one fixed mean lifetime $\tau$, or half-life $T_{\frac12}=\frac\tau{\ln2}$, whatever you prefer to use to characterise this. We also know mathematically that exponential decay in time is linked to a Poisson distribution.

There is, as of yet, no known experimental variable we can do to change $\tau$. It is a constant for a particular material and we might even have good theoretical reasons (in QFT) to believe that they should be alterable by some means when we consider the really extreme conditions. We just have yet to be able to create these extreme conditions in the lab or see them in telescopes, elsewhere in the universe.

The only experimental variable that consistently changes these, is Einsteinian time dilation.

The issue with 2 time directions, is that if you do have them, then either these decays should depend upon the "radial direction" combination of both, or only depend upon one of them. Then we should observe a difference compared to our own time-keeping devices; it would make no sense if we have 2 time directions and we always only measure one certain direction everywhere in the universe. If the universe conspires to only ever allow one certain time direction to be measured, then there is no point in calling that a 2-time-direction theory. But then if both time directions are active, then we always ought to see a deviation from the strict radioactive decay fitting. But we have never been able to get anything more than Einsteinian time dilation.

So this is an experimental argument as to why we should ignore 2-times until future experiments force us to consider them.