Just noodling for fun. This is speculative, not authoritative.
And, on inspection, I clearly have run Blogger's TeX interface way past its limits. I don't know if splitting this up will help--I doubt it. Ah well. Where you see boxes around things, just read the TeX inside, or skim over it. I may have to try to embed gifs, or maybe make a pdf. It's hard to see if I made a typo. And I don't know how to put large parentheses around the matrix--Blogger doesn't handle some of the TeX options.
What does a world look like if it has two dimensions for time instead of just one?
Your intuition undoubtedly says--"That's silly, it would have to look very different from ours, so why bother?"
- Because it is an amusing way to spend time on the bus
- Because some string theorists have come up with modest arguments that we do have two time dimensions. Bars, Vongehr, and Gogberashvili have been looking at systems with 2 time dimensions. I hold no brief for string theory, but hey, it's an excuse.
- In one of his speculative moods, Eddington wrote about interfaces between (3,1), (2,2), and (1,3) spaces, where the numbers in parenthesis are the number of space and time dimensions respectively. The notion fascinated me ever since.
- Because there are serious problems with invisible mass in cosmology, and visibility of matter on other timelines is probably going to be a problem—just as an intuitive guess.
I just want to take a preliminary look right now--not trying to figure out what quantum mechanics would look like, for instance. And by macroscopic time dimensions I mean large enough to use a wall clock or a calendar to measure.
There are a few questions about visibility, the speed of light, and causality that don't have obvious answers. I'll try to keep it logical. Assume that the speed of light always appears constant. I'll also assume that objects on different timelines can interact--at some point they had to, so why not now also? Also, if A measures B's relative timeline angle, it should be the same as B measuring A's.
Wait, what do I mean by a "timeline?"
Assume that the two time directions can be viewed as a Euclidean plane, with time-1 in one direction and time-2 at right angles. As an undisturbed object ages, it will assume time-1 and time-2 ($t_1$ and $t_2) values which lie along a line in the plane.
Here are two examples. In the left the upper line has the object moving more along $t_1$ than $t_2$, and the lower line tilts more along $t_2$. Where zero is is arbitrary, by the way.
The right-hand image shows a complication that we need to keep in mind. Timeline A is kind of banal. Timeline B, relative to A, also seems ordinary. It has positive components of its timeline direction both parallel to A and perpendicular to A. So from A's perspective B will not go backwards in time.
However, when B meets C, it will appear to be going backwards in at least one time component.
Should we allow that in our initial study? We can work non-causality in if we rely on small interaction rates, or demand that it only work on small distances, but that seems ad hoc. Let's pretend it isn't going to happen and plo
The obvious first approach is to modify the Einsteinian formalism. In some coordinate system, denote points by $(t_1, t_2; x, y, z)$, where I separate space and time components with a semicolon. Use the same convention for momentum: $(E_1, E_2; P_x, P_y, P_z)$. For two points $a$ and $b$, assume an analogous invariant to Einstein's: $(t_{1a} - t_{1b})^2 + (t_{2a} - t_{2b})^2 - (x_a - x_b)^2 - (y_a - y_b)^2 - (z_a - z_b)^2$. Assume that a transformation to a different frame of reference will be linear.
To keep things simple, just ignore $y$ and $z$ for now, and use $\delta {t_1}^2 + \delta {t_2}^2 - \delta x^2$ as the separation.
A linear (and symmetric) transform can be parameterized as
| ${x}^'$ | | | | 1 | $\alpha$ | $\beta$ | | $x$ |
| ${t_1}^'$ | = | $A$ | | $\alpha$ | $\lambda$ | $\epsilon$ | | $t_1$ |
| ${t_2}^'$ | | | | $\beta$ | $\epsilon$ | $\tau$ | | $t_2$ |
Invariance requires that ${{t_1}^'}^2 + {{t_2}^'}^2 - {x^'}^2 = {t_1}^2 + {t_2}^2 - x^2$, from which we can derive equations which specify $A$, $\tau$, $\epsilon$, and $\lambda$ in terms of $\alpha$ and $\beta$, where the latter act like the $\beta$ in the usual 1-time dimensional theory, just for the two different time axes. Think of them as $\beta_1$ and $\beta_2$. By looking at the limit $\alpha=0$ we can determine the right sign for the square root.
For ease in reading the result, define:
$\gamma \equiv {1 \over{ \sqrt{1 - \beta_1^2 - \beta_2^2}}}$
$A=\gamma$
$\epsilon = {{\beta_1 \beta_2} \over {\beta_1^2+ \beta_2^2}} (1 - {1 \over \gamma})$
$\lambda = 1 - {{\beta_2^2} \over {\beta_1^2+ \beta_2^2}} (1- {1 \over \gamma})$
$\tau = 1 - {{\beta_1^2} \over {\beta_1^2+ \beta_2^2}} (1- {1 \over \gamma})$
And of course $\beta_1$ and $\beta_2$ are the speeds as a fraction of c along the $t_1$ and $t_2$ axes respectively. Their squared sum will never exceed 1, and so $\gamma$ is always real. Yep, this assumes that nothing exceeds the speed of light in any frame. And if you define a rotation in the $t_1:t_2$ plane--a rotation to a different timeline--you can turn a simple boost with one time direction to one with a mix, and it matches the parameterization here ($R^{-1} B_{1,0} R = B_{\beta_1,\beta_2}$), where $\beta_1$ and $\beta_2$ are the original $\beta^'$ times the sine or cosine.
Recalling that energy is non-negative (skipping QM subtleties), an object with momentum $(E_1, 0, p_x, p_y, p_z)$ will not break into objects with non-negative $E_2$, since there's no existing energy in that "bin" to give them. It could break into $(E_1^', 0, p_x^', p_y^', p_z^') + (E_1-E_1^{'}, 0, p_x-p_x^{'}, p_y-p_y^{'}, p_z-p_z^{'})$, but not something with a positive $E_2$ component.
In practice that means that if you don't have any local $t_2$ activity, you won't get any. The situation is stable.
Is there any way to detect the other time dimension?
Typically you and everything about you is going along the same timeline--what is there to make it change? Your best bet would be something distant, or something from a great distance that comes to pay a visit.
Suppose that you always measure the speed of light as the same, no matter what timeline it is on or came from.
Suppose you have two objects, with the same timeline direction, but one starting from a different $t_2$ time, $T_2$. It is at some nearby position $x$.
Suppose the first object has interacted with something in its past so that it has a certain amount of $E_2$ energy available to emit a photon that can reach, and bounce off, that second object. Without that, you'll never emit anything with $t_2$ component non-zero, so you'll not hit the second object. OTOH, assume that the first object has enough $E_1$ that the total $E_2$ is negligable. That way the bulk of what you measure is along the $t_1$ line.
In this thought experiment, what you would measure is the time between the emission and absorption of the photon along the $t_1$ axis, since by assumption you're not measuring anything in $t_2$. That time will be twice the travel time $\tau$, as projected onto $t_1$ axis, or $2 \sqrt{x^2 - {T_2}^2}$. (Observe that if $T_2 = x$, the photon will only reach it going along $t_2$, without any $t_1$ component. It will go there and back again in $0$ $t_1$ time; not detectable. If larger, the separation is time-like, but in a time direction invisible to you; again it will seem dark.)
Since the speed of light is always measured as the same, you will predict $\tau^2 - {x^'}^2 = 0$; that the $x^'$ will appear to be closer to you.
OK, so far so curious.
If you receive a photon from a different timeline, and only measure the energy component parallel to your own, you will underestimate the photon's energy.
But if you measure a foreshortened distance, and can also measure how many wavelengths away the object is, you will predict the photon's wavelength to be less, and therefore its energy higher. If you know a priori what kind of photon you emitted, and measure the energy of the returning one, the energies won't match.
This little contradiction might offer a way to indirectly detect other timelines--one of the assumptions won't hold.
