Timelike v. spacelike, is it arbitrary?

[BruceW]

well, we both agree that when there is no matter around, such a model works fine, right? So to introduce matter, I guess I would need to show that tachyons are at least a consistent model. They might have negative mass (whatever that means), but apart from that, I'd guess they mostly work the same as normal matter? But anyway, this is still related to the fact that normal matter all travel along the same kind of curve (which we call timelike curves). If you call that an assumption of relativity, then fine, there is a real difference between timelike and spacelike curves. But if not, then I still do not see any difference.

 

[ghwellsjr]

Maybe start with the simple case of special relativity. ds^2 = -dt^2+dx^2+dy^2+dz^2 ...

relativity puts space and time on an equal footing. The component we choose as 'time' (i.e. to get the negative sign) is an arbitrary choice. There is no non-arbitrary choice of 'space' and 'time', there is only spacetime.

To put it another way, when we write down a line element with one sign being different from the other three, the physical difference that we are reflecting in the math is that we measure one dimension using a clock, and the other three using a ruler.

The spacetime interval between two arbitrary events falls into one of three categories. They are all physically different from each other. We can use devices (clocks and rulers) to physically measure spacetime intervals. In only one of those categories can the spacetime interval be measured with a single device.

We can measure time-like spacetime intervals using a single inertial clock that passes through both events. This does not require any synchronization convention since there is only one clock and it does not require any ruler.

However, for space-like spacetime intervals, even though we are measuring it with a single ruler, it is not sufficient that two points on that ruler pass through both events, they must pass through at the same time as determined by two synchronized clocks located at the two points of measurement.

This is a non-arbitrary choice between 'space' and 'time'. (Unless you want to claim that there is no non-arbitrary choice between a ruler and a clock.) Spacetime intervals are not called that because there is only spacetime. All spacetime intervals are either exclusively 'space' or exclusively 'time' or 'neither' ('null').

 

[yuiop]

... when the whole point of the term "timelike" is that it labels the "odd one out" dimension, with a different sign of its squared length from the other three.

Besides the sign, I have always imagined that one of the special properties of the timelike dimension that makes it unique, is that you can only move forward in this dimension while you can move forwards or backwards in the spacelike dimensions. Inside a black hole the radial spatial coordinate is said to become timelike and we find that indeed we can only move in one direction radially, which is inwards and the time coordinate becomes spacelike, because inside the black hole it is possible to move forwards or backwards through coordinate time.

 

[Dale]

For example, in a universe with radiation but no matter, then surely there is no way to discern which curves are 'timelike' and which curves are 'spacelike'.

In a universe with radiation but not matter timelike curves and spacelike curves can still be identified. Timelike curves are curves inside the light cone at each point, and spacelike curves are outside the light cone at each point. The difference is physical, not coordinates.

 

[Dale]

I don't see why not. Let's say you have some curve in spacetime. You could define the timelike dimension to lie along that curve (in which case you'd label the curve as 'timelike'). Or you could define one of the spacelike dimensions to lie along the curve, in which case you'd label that curve as spacelike.

Not without changing the physics. Once you have specified if the curve lies inside or outside the light cone then the physics is set and the timelike or spacelike character of the line is fixed.

 

[Dale]

OK, you can call it 'geometry', but it is still arbitrarily defined geometry. It does not correspond to anything physical.

It corresponds to physical light cones.

 

[BruceW]

We can measure time-like spacetime intervals using a single inertial clock that passes through both events. This does not require any synchronization convention since there is only one clock and it does not require any ruler.

However, for space-like spacetime intervals, even though we are measuring it with a single ruler, it is not sufficient that two points on that ruler pass through both events, they must pass through at the same time as determined by two synchronized clocks located at the two points of measurement.

this is a good example of what I am talking about. right, so let's say we have defined what are the time-like intervals. And 'normal' matter all moves along these time-like intervals. So now, we can use some normal matter to measure any time-like spacetime interval. (for example, a muon beam acts a very good clock, since the muon's mean lifetime is accurately known). i.e. we send a muon beam along a time-like interval, and the fraction remaining tells us the arc length along that path.

But now, suppose we come across some not-normal matter that moves along a space-like interval. Again, suppose it is like a muon, i.e. the amount of muons that decay depends only on the arc length that the beam has traveled. Now, we could use this weird matter to measure the arc length along a space-like interval, simply by observing the fraction of muons that remain, since this tells us the arc length along that path.

So again, the point I am trying to get across is that if we assume (as part of the theory of relativity) that all matter travels along the timelike curves, then yes there is a physical difference between which group of curves we label as 'timelike' and which group of curves we label as 'spacelike'. But if we do not make such an assumption, then the choice is arbitrary.

 

[Dale]

well, we both agree that when there is no matter around, such a model works fine, right?

I disagree.

 

[BruceW]

Not without changing the physics. Once you have specified if the curve lies inside or outside the light cone then the physics is set and the timelike or spacelike character of the line is fixed.

I agree that once you have specified if the curve is inside or outside the light cone, then the timelike or spacelike character of that line is fixed. What I am saying is that to begin with, specifying the curve to lie inside or outside the light cone is an arbitrary choice that we can make.

 

[Dale]

specifying the curve to lie inside or outside the light cone is an arbitrary choice that we can make.

Not without changing the physics.

 

[BruceW]

Not without changing the physics.

when you say 'the physics', if you mean an arbitrarily defined non-physical convention, then I agree. Else, I disagree.

Edit: ah, or if you want to say that we define all matter to move along the timelike curves, then I would agree that in a universe with matter, we have a non-arbitrary way to assign timelike and spacelike curves.

 

[Dale]

when you say 'the physics', if you mean an arbitrarily defined non-physical convention, then I agree. Else, I disagree.

Edit: ah, or if you want to say that we define all matter to move along the timelike curves, then I would agree that in a universe with matter, we have a non-arbitrary way to assign timelike and spacelike curves.

In a universe with only radiation and no matter we can physically distinguish a line which lies within a light cone from a line which lies outside a light cone. There is nothing arbitrary about it. A line inside is timelike; a line outside is spacelike.

 

[BruceW]

we seem unable to understand each other's line of reasoning. I am trying though :) OK, so let's say we have components a,b,c,d (and the order I have listed them does not imply anything about the metric). (And I have avoided the usual t,x,y,z because that would imply which component should be the timelike component). Now, let's say I look at a path that goes completely along the c component. Is this path timelike or spacelike?

There is no way to tell, because I have not told you which component is the timelike component. And further, it doesn't matter which component I choose as the timelike component. The only way it would be important is if we define matter to move along timelike curves, since I would then have to define the timelike component so that matter travels along timelike curves.

 

[WannabeNewton]

There is no need to look at components in a frame. Forget components, because that is not geometry! You don't need to tell me anything about that. All I need to check is if the squared norm of the tangent vector is everywhere negative.

 

[Dale]

The only way it would be important is if we define matter to move along timelike curves, since I would then have to define the timelike component so that matter travels along timelike curves.

No. It would also be important if you have radiation. You would have to define the timelike compnent to be the one inside light cones.

The reason that I cannot tell which is which is because you eliminated both matter and radiation (i.e. your scenario involves no physics). As soon as there is any physics, whether it is matter or radiation, the choice is constrained by the physics. A choice which is constrained by the physics is not arbitrary.

 

[BruceW]

No. It would also be important if you have radiation. You would have to define the timelike compnent to be the one inside light cones.

The reason that I cannot tell which is which is because you eliminated both matter and radiation (i.e. your scenario involves no physics). As soon as there is any physics, whether it is matter or radiation, the choice is constrained by the physics. That makes it not arbitrary.

alright, say there is a beam of light with tangent vector (0,1,1,0) Now is the path with tangent (0,0,1,0) timelike or spacelike?

 

[Dale]

alright, say there is a beam of light with tangent vector (0,1,1,0)

There is no such thing. It doesn't satisfy Maxwell's equations.

Give me any EM radiation field which actually satisfies Maxwell's vacuum equations (since you want to eliminate matter) with physically possible boundary conditions in terms of your a,b,c,d coordinates and I can tell you which is the timelike and which is the spacelike.

 

[BruceW]

what do you mean? I haven't said which component is the timelike component.

 

[ghwellsjr]

We can measure time-like spacetime intervals using a single inertial clock that passes through both events. This does not require any synchronization convention since there is only one clock and it does not require any ruler.

However, for space-like spacetime intervals, even though we are measuring it with a single ruler, it is not sufficient that two points on that ruler pass through both events, they must pass through at the same time as determined by two synchronized clocks located at the two points of measurement.

this is a good example of what I am talking about. right, so let's say we have defined what are the time-like intervals. And 'normal' matter all moves along these time-like intervals. So now, we can use some normal matter to measure any time-like spacetime interval. (for example, a muon beam acts a very good clock, since the muon's mean lifetime is accurately known). i.e. we send a muon beam along a time-like interval, and the fraction remaining tells us the arc length along that path.

But now, suppose we come across some not-normal matter that moves along a space-like interval. Again, suppose it is like a muon, i.e. the amount of muons that decay depends only on the arc length that the beam has traveled. Now, we could use this weird matter to measure the arc length along a space-like interval, simply by observing the fraction of muons that remain, since this tells us the arc length along that path.

Your supposition is speculation, isn't it? It has nothing to do with the universe we live in, the physics we use to describe it, or teaching relativity which is the purpose of this forum.

So again, the point I am trying to get across is that if we assume (as part of the theory of relativity) that all matter travels along the timelike curves, then yes there is a physical difference between which group of curves we label as 'timelike' and which group of curves we label as 'spacelike'. But if we do not make such an assumption, then the choice is arbitrary.

There is no assumption in the theory of Special Relativity that matter travels along timelike curves. There are two assumptions (postulates) and from them we get the three distinctly different categories of spacetime intervals and the concept of Proper Time. Your misunderstanding that Proper Time or a time-like spacetime intervals are no different than Proper Length or space-like spacetime intervals or null spacetime intervals is not part of teaching Special Relativity. You are attempting to promote something different.

 

[Dale]

what do you mean?

I mean that, strictly speaking, a beam of light is a fiction. It does not satisfy Maxwell's equations. There is always some divergence of the beam, and that divergence (required to satisfy Maxwell's equations) identifies the other two spatial dimensions.

Radiation satisfies $\Box A^{\mu} = 0$, and if you provide any physically possible A which satisfies that equation then timelike and spacelike are determined.

 

[BruceW]

Your supposition is speculation, isn't it? It has nothing to do with the universe we live in, the physics we use to describe it, or teaching relativity which is the purpose of this forum.

I was just saying that if we use the definition that all matter travels along timelike curves, then yes, we can say which curves are timelike and which are spacelike. But If we do not use this definition, then we cannot.

Your misunderstanding that Proper Time or a time-like spacetime intervals are no different than Proper Length or space-like spacetime intervals or null spacetime intervals is not part of teaching Special Relativity. You are attempting to promote something different.

I agree that we can choose a set of time-like curves and a set of space-like curves. But I am saying that for a given physical situation, it is our choice for which ones are time-like and which ones are space-like. (Unless we define all matter to travel along timelike curves, in which case the choice is made for us by this definition).

 

[Dale]

BruceW, besides the physical point which I am making there is the geometrical point which WBN and PD are making. The geometry is determined by the metric. Any Lorentzian metric will have three eigenvalues with one sign and one eigenvalue with the opposite sign. The eigenvector corresponding to the one with the opposite sign is timelike. No matter, no physics, pure relativity, pure geometry. The only way that you can make it ambiguous is by NOT specifying the geometry (i.e. by not specifying the metric).

Simply put, there is no way in which the choice is arbitrary. It is constrained by the geometry, it is constrained by the physics, that is as far from arbitrary as is possible. The only possible ambiguity/arbitraryness comes from not specifying either the geometry or the physics.

 

[PeterDonis]

But now, suppose we come across some not-normal matter that moves along a space-like interval. Again, suppose it is like a muon, i.e. the amount of muons that decay depends only on the arc length that the beam has traveled. Now, we could use this weird matter to measure the arc length along a space-like interval, simply by observing the fraction of muons that remain, since this tells us the arc length along that path.

Have you actually tried to construct such a model? If so, please show your work, as I asked before. If not, you are just waving your hands and assuming that such a model would work the way you say and still be consistent. That's why I asked you before to actually do the math, instead of just speculating. (And you should really look into the literature on tachyon models; as I've said a couple of times already, there are subtleties lurking there.)

 

[ghwellsjr]

Have you actually tried to construct such a model? If so, please show your work, as I asked before. If not, you are just waving your hands and assuming that such a model would work the way you say and still be consistent. That's why I asked you before to actually do the math, instead of just speculating. (And you should really look into the literature on tachyon models; as I've said a couple of times already, there are subtleties lurking there.)

Isn't it still speculation, even if he does come up with consistent math?

 

[PeterDonis]

OK, so let's say we have components a,b,c,d (and the order I have listed them does not imply anything about the metric). (And I have avoided the usual t,x,y,z because that would imply which component should be the timelike component). Now, let's say I look at a path that goes completely along the c component. Is this path timelike or spacelike?

There is no way to tell, because I have not told you which component is the timelike component.

No, there's no way to tell because you haven't told us what the metric is. Which means, as DaleSpam said, that you aren't doing physics; you're just throwing letters and numbers around. As soon as you define a metric, you have defined which paths are timelike and which are spacelike. You don't have to make any assumptions about what kinds of objects travel on what kinds of paths.

 

[PeterDonis]

Isn't it still speculation, even if he does come up with consistent math?

"Speculation" is a broad word. PF's prohibition is on "personal" speculation. There is, as I mentioned, plenty of literature on tachyon models, so talking about those models and what kinds of subtleties arise in them would not be personal speculation, even though no tachyons have ever actually been observed.

 

[PeterDonis]

for a given physical situation, it is our choice for which ones are time-like and which ones are space-like.

Incorrect; the "physical situation" includes the metric, which, as I said, determines which curves are timelike and which are spacelike, regardless of what kinds of objects travel on what kinds of paths.

 

[PeterDonis]

we both agree that when there is no matter around, such a model works fine, right?

No, I don't, because you haven't shown me such a model. I can't say anything about such a model until you actually show me one.

 

[BruceW]

I'm trying to think of different ways to explain the point I'm trying to get across... OK, suppose we have some model for the universe, and we have chosen a metric with signature (-1,1,1,1). OK, now instead let's say we choose the metric with signature (1,-1,1,1). Then is our new physical description going to work as well? (apart from the fact that matter only travels along timelike curves).

I would (intuitively) say that our new physical description would also work. As long as our equations are manifestly covariant, then I would be surprised that a (-1,1,1,1) metric would work but a (1,-1,1,1) metric would not. (again, ignoring the fact that matter travels along timelike curves).

edit: to be more specific, when I say "instead choose the metric with signature..." I mean keep things like the stress-energy tensor and the distribution of matter the same. But choose the metric to be different (i.e. choose a different timelike component).

 

[WannabeNewton]

What you described is not a different signature; the number of $+$ and $-$ signs are the same in both.