Undergrad Minus sign in Minkovsky´s metric

[martinbn]

I hope this isn't too far off-topic, but does anyone know of an application of Riemannian geometry that uses multiple timelike dimensions? That is, that uses a metric which when diagonalized has multiple +1 entries and multiple -1 entries? For that matter, are metrics with both positive and negative entries when in diagonal form used anywhere outside of special/general relativity?

I don't know, but my uneducated guess is that it is very unlikely. The reason I think so is that it would be a lot harder. In the Riemannian (definite metric) case geodesics have minimizing properties that are used in many geometric applications, and the PDE's that typically appear are elliptic or parabolic. In the Lorentzian (one sign different from the others), in some cases geodesics have maximizing properties, which is used in a similar manner as the minimizing in the Riemann case, for example the singularity theorems. The PDE's that appear are usually hyperbolic. In the case of more than one minus and more than one plus sign in the signature the geodesics will have no optimizing properties and the PDE's will be a lot harder to analyze.

About the very last question, Lorentzian geometry is used in the analysis of hyperbolic equations, whether they are GR related or not.

In the same line of questioning it would be interesting to know if there are any applications (to physics) where the metric is more general and not quadratic as in Finsler geometry.

 

[bobob]

That is more due to the fact that there is only one dimension of time rather than two or more instead of the minus sign. If you had two dimensions of time, both with the minus sign, then you could turn around in time too.

I question the validity of that statement. In particular, if you have two time components (and two or more space components), you have no spacelike or timelike hypersurfaces upon which to specify initial data and the spacetime would not make physical sense. (At this point I still see nothing physical in Itzhak Bars' two time theory, so I don't see that as a counterexample). Three time dimensions and one space dimensions describes a spacetime containing only tachyons, which is unphysical.

 

[Dale]

the spacetime would not make physical sense

Agreed, but that doesn’t invalidate anything I said. Purely geometrically two timelike dimensions easily leads to closed timelike curves in flat spacetime.

 

[stevendaryl]

Three time dimensions and one space dimensions describes a spacetime containing only tachyons, which is unphysical.

Actually, no. There is no physical difference between (1) 3 time axes and 1 space axis and (2) 3 space axes and 1 time axis.

 

[PeterDonis]

There is no physical difference between (1) 3 time axes and 1 space axis and (2) 3 space axes and 1 time axis.

I would not put it this way. I would say that there is no physical difference between a metric with a (3, 1) signature and a metric with a (1, 3) signature. That's because either way you can physically interpret the "1" dimension in the signature as the "time" dimension, which means, heuristically, that arc lengths along that dimension are measured with clocks instead of rulers, while arc lengths along the other 3 dimensions are measured with rulers. So there's no physical difference between the two, just a different choice of signature.

But saying that you have 3 time axes and 1 space axis is very different physically: it's saying that you have three orthogonal "directions" along which you measure arc lengths with clocks, and only one along which you measure arc lengths with rulers. That's a physical difference--and again, you could (if experiments supported it) adopt such an interpretation for either a (3, 1) or a (1, 3) signature.

 

[stevendaryl]

I would not put it this way. I would say that there is no physical difference between a metric with a (3, 1) signature and a metric with a (1, 3) signature. That's because either way you can physically interpret the "1" dimension in the signature as the "time" dimension, which means, heuristically, that arc lengths along that dimension are measured with clocks instead of rulers, while arc lengths along the other 3 dimensions are measured with rulers. So there's no physical difference between the two, just a different choice of signature.

But saying that you have 3 time axes and 1 space axis is very different physically: it's saying that you have three orthogonal "directions" along which you measure arc lengths with clocks, and only one along which you measure arc lengths with rulers. That's a physical difference--and again, you could (if experiments supported it) adopt such an interpretation for either a (3, 1) or a (1, 3) signature.

Well, I don’t have the mathematical ability to argue against that, but I don’t think it’s true. The fundamental laws don’t distinguish time from the other dimensions except for the signature. You say that time is something measured by clocks, but clocks work the way do because of the laws of physics.

If there were three time dimensions and one space dimension, then space would work like time, in that specifying conditions for all time at a particular location in space would determine conditions at all other locations in space. I think it’s the fact that there is only one time dimension that makes it work like time intuitively works.

The laws of physics written in terms of three time dimensions and one space dimension would look exactly like the laws of physics for 3 space dimensions and one time dimension, except with the opposite sign convention.

 

[PeterDonis]

You say that time is something measured by clocks, but clocks work the way do because of the laws of physics.

As far as relativity is concerned, this is not true: relativity does not explain why timelike intervals are measured by clocks while spacelike intervals are measured by rulers. It just declares by fiat that this is the case; or, if you like, this is part of the rules for the physical interpretation of relativity, which are not explained, they are taken as axioms of the theory.

There might be ways to get a deeper explanation of how clocks work vs. rulers from quantum theory, but that would probably take us too far afield for this thread; such a discussion should be in a separate thread in the quantum physics forum.

If there were three time dimensions and one space dimension, then space would work like time, in that specifying conditions for all time at a particular location in space would determine conditions at all other locations in space.

This is just quibbling over terminology. Switching the labels "space" and "time" in ordinary language descriptions doesn't change the physics.

I think it’s the fact that there is only one time dimension that makes it work like time intuitively works.

In other words, the reason time works the way it does is basically what @Dale said in post #18? I think that's a reasonable position, at least as far as relativity is concerned; but note that taking this position makes it meaningless to talk about "more than one dimension of time", because a signature such as (2, 2), for example, would describe a manifold that has nothing in it at all that works the way time works in our universe.

 

[stevendaryl]

As far as relativity is concerned, this is not true: relativity does not explain why timelike intervals are measured by clocks while spacelike intervals are measured by rulers. It just declares by fiat that this is the case; or, if you like, this is part of the rules for the physical interpretation of relativity, which are not explained, they are taken as axioms of the theory.

Well, relativity in the operational sense is not capable of answering the question: "What if there were more than one time axis?"

My point is that light travels according to Maxwell's equations, which don't distinguish between space and time, except through the metric signature. Similarly for the Klein-Gordon equation and Dirac Equation and Einstein field equations. So if you try to make variants of those theories with 3 time dimensions and 1 space dimension, the only change would be to switch from the +++- convention to the ---+ convention. Or that's the way it seems to me.

 

[vanhees71]

Well, all of physics very clearly distinguishes between space and time to begin with. You cannot formulate Maxwell's equations without first establishing a space-time model. Historically even a "wrong" space-time model was sufficient to formulate Maxwell's equations, but that's only because the symmetry group of Galilean spacetime is not too different from Minkowski spacetime in the sense that for any inertial observer space is always a Euclidean affine manifold as is the case in Galilean spacetime (where this holds true even for non-inertial observers, but that's pretty irrelevant for the argument since Maxwell's theory was tacitly formulated in an inertial frame to begin with). That's why, even with the "wrong" Galilean space-time model the mathematical objects needed to formulate Maxwell's equations were there, namely vector fields of Euclidean space. Also time is just a parameter labelling the causal order of events in both Newtonian physics (where this holds true for any observer) and Minkowskian space-time (where this for sure holds true for an inertial observer).

This becomes the more clear, if you think in terms of quantum theory (or especially for this discussion simply in terms of QED), where time plays just this role, namely to be a parameter labeling the order of events. It's not an observable in the formalism of quantum theory to begin with, while position is an observable for all massive particles (not for photons, but that doesn't matter for this discussion, because we define the observables in terms of measurement devices which are macroscopic systems and thus massive anyway).

So a clock cannot "show time" in the strict sense since time is not even an observable in theory. What a clock shows is something referring back to time. In the most simple case take a pendulum on Earth with a certain length. What's observed are the positions of the pendulum, and you may count the number of times its motion has gone through full periods, and the period can be defined as time unit.

In principle that's even how the measure of time is defined in the SI, namely the base unit second (and it will stay such defined also in May, when the SI is redefined to the precise standards of 21st century physics). Instead, of course you don't use a simple pendulum, which is way too inaccurate, but a cesium atom's hyperfine transition line, defining its frequency):

The second, symbol s, is the SI unit of time. It is defined by taking the fixed numerical value of the caesium frequency $\Delta \nu_{\text{Cs}}$, the unperturbed ground-state hyperfine transition frequency of the caesium-133 atom, to be 9192631770 when expressed in the unit Hz, which is equal to $\text{s}^{−1}$.

Quoted from

https://en.wikipedia.org/wiki/Second

 

[stevendaryl]

This becomes the more clear, if you think in terms of quantum theory (or especially for this discussion simply in terms of QED), where time plays just this role, namely to be a parameter labeling the order of events.

Well, I don't see how time could play that special role if there were multiple time dimensions.

 

[Dale]

Well, relativity in the operational sense is not capable of answering the question: "What if there were more than one time axis?"

I agree. Counterfactual questions like these can be written down and discussed mathematically, but not operationally. When talking about different signatures it should be understood that it is purely a geometrical discussion, not a scientific discussion.

That said, I tend to agree that three dimensions of time and one dimension of space is exactly equivalent to three of space and one of time. Geometrically the equivalence is clear, and operationally I think it is also equivalent as follows:

Time is what a clock measures and space is what a ruler measures. But to make that non circular you have to include some blueprints or examples of good clocks. The blueprints for 3D clocks would look a lot like ordinary rulers, and the blueprints for 1D rulers would look a lot like ordinary clocks. Thus physically they are still equivalent, the only difference is semantic. What types of devices are labeled “clock” etc.

 

[PeterDonis]

relativity in the operational sense is not capable of answering the question: "What if there were more than one time axis?"

Of course not, since it's an axiom of the theory that there is only one time axis and three space axes. Or, for a statement that doesn't depend on how you assign the labels "time" and "space", see below.

if you try to make variants of those theories with 3 time dimensions and 1 space dimension, the only change would be to switch from the +++- convention to the ---+ convention.

Again, this is just quibbling over terminology: you've decided to call the thingie that appears three times in the signature "time" and the thingie that appears once in the signature "space". But that doesn't change the physics, just the labeling. The physics is that the thingie that appears once in the signature is measured with clocks, and the thingie that appears three times is measured with rulers. That's the rule for interpretation of relativity as we actually use it. If you want to talk about a model where the thingie that appears three times in the signature is measured with clocks and the thingie that appears once is measured with rulers, that is a different physical theory, because the rules for how to translate the math into experimental predictions are different. But just swapping labels is not doing that.

 

[PeterDonis]

The blueprints for 3D clocks would look a lot like ordinary rulers, and the blueprints for 1D rulers would look a lot like ordinary clocks.

I don't see how you could possibly know this, since there are no such things as "3D clocks" and "1D rulers" in our actual universe. The 3D things are rulers, and the 1D things are clocks. That's how our actual universe is, and that's how the rules of interpretation in relativity are defined.

 

[Dale]

you've decided to call the thingie that appears three times in the signature "time" and the thingie that appears once in the signature "space". But that doesn't change the physics, just the labeling.

Precisely. That is why (-+++) is equivalent to (+---), (---+), and (+++-), regardless of how you interpret the - and the +.

It is just a matter of terminology and words used to describe the math. As long as you stick with 3+1D spacetime you can just say the 3D is space and the 1D is time. In that case you know (-+++) is equivalent to (+---)

As soon as you start looking at the geometry (not physics) for other combinations of dimensions you need a general way of labeling space and time that works. So you can just say any + is space and any - is time with the understanding that for dimensions other than 3+1D it is just geometry. But then you have that (+---) is 1D space with 3D time, and you physically considered this to be equivalent to (-+++). So just consider it to be a semantic labeling and the physical equivalence is restored.

 

[vanhees71]

Well, I don't see how time could play that special role if there were multiple time dimensions.

The joke is that there are no multiple time dimensions. It simply doesn't make sense to introduce such a idea.

 

[stevendaryl]

The joke is that there are no multiple time dimensions. It simply doesn't make sense to introduce such a idea.

Well, some of us have curiosity about "what happen if ..."

In my opinion, playful curiosity is what leads people to science, even if, once you become a scientist, you learn to squash it.

 

[vanhees71]

Yes, sure, but there should be some foundation in observational facts. As the history of physics shows that even the greatest geniuses like Einstein couldn't create succuessful theories out of the blue, but they had to be founded in observational facts about nature. This was indeed the case with the younger Einstein (say, until about the 1920ies): He made his great discoveries in an amazingly broad variety of fundamental questions of his time:

(a) The incompatibility of Maxwell's theory of electromagnetism with Galileo symmetry. The ingenious creative act of Einstein's in this case was that he recognized, other than his contemporaries like Lorentz, FitzGerald, Heaviside, and Poincare, the key issue with the problematic interpretation of the missing Galileo symmetry as being due to the existence of a preferred inertial reference frame, which was interpreted as the presence of the socalled aether. The very point was that Einstein recognized that this introduced ideas not based on the observed facts but was just an unjustified assumption about the nature of the electromagnetic field. Famously that lead to his reinterpretation of the math by Lorentz, FitzGerald, Heaviside and Poincare in terms of a new space-time model, modifying the fundamental laws of mechanics rather than Maxwell's theory of electromagnetism, leading finally to Special Relativity and Minkowski spacetime (1905-1908).

(b) The other great insight in his "annus mirabilis" was about the conclusions to be drawn from the hypothesis of the existence of atoms/molecules/corpuscles (or however you want to call the then present ideas about the "atomistic nature" of matter) and the ideas of statistical physics a la Maxwell, Boltzmann or (unknown to him at the time as far as I know) Gibbs: Namely that there must be fluctuations around the mean values described by kinetic theory. Then he realized that Brownian motion of little suspended particles like pollen clearly visible under a microscope is exactly such a phenomenon. This was the beginning of a lot of applications of this idea with the fluctuations and statistical nature of macroscopic objects, e.g., the famous idea how to determine Avogadro's number from the blueness of the sky or the theory of critical opalescence.

(c) Finally there's also his idea of "light corpuscles" as a "heuristic argument". This was based on the insight that Plancks radiation formula for the black-body radiation had both particle and wavelike features, combined with the notion of a fundamental measure for "action" (Planck's constant $h=2 \pi \hbar$). Although the only of his theories which hasn't survived the "quantum revolution", even providing wrong pictures about photons, it was well based on observational facts, particularly on the photoelectric effect (although today we know, it's not due to the quantum nature of the electromagnetic field but of bound quantized electrons and transitions of them due to interaction with an electromagnetic field that can be still described classically in this context [1]). His later derivation of the Planck spectrum from kinetic-theory considerations in 1917, introducing the idea of spontaneous emission, was already very close to the modern formulation in terms of QED, which however was not really possible before Jordan's (1926) and Dirac's (1927) formulation in terms of the quantized photon field. Indeed, today the only consistent explanation for the fact that there is spontaneous emission is due to field quantization, and this indeed is the most simple physical argument (again based on clear observational facts and not unfounded speculations) for the necessity to quantize the em. field to begin with. Later, of course, it was proven in very many other ways too, e.g., the discovery of the Lamb shift in the hydrogen spectra, the anomalous magnetic moment, quantum beats, the violation of the Bell inequality etc.etc.

 

[robphy]

In this recent discussion of multidimensional time, it would be helpful (especially to novices who may be reading) to specify conditions behind blanket statements of “being physical”...and possibly identify where the notion of time arises in the formalism.

 

[stevendaryl]

Yes, sure, but there should be some foundation in observational facts.

I don't know. It seems to me that you can ask, as a purely mathematical question, what do solutions of Maxwell's equations or the Einstein field equations look like when generalized to spacetimes with different signatures than 3+1. Such mathematical playing around sometimes leads to ideas that suggest theories that do have observable consequences. For example, Kaluza-Klein models could possibly describe our own world, if the extra dimensions are little loops. That's an extension to the number of spatial dimensions. I don't know of any interesting consequences of extending the number of time dimensions, but I think saying that you shouldn't think about such things until there is observational evidence for them is overly rigid, it seems to me.

 

[Dale]

Such mathematical playing around sometimes leads to ideas that suggest theories that do have observable consequences

And in any case such mathematical playing around is found in the literature.