Why (-,+,+,+) and not (+,+,+,+)?

[kmarinas86]

If I take the Lorentz factor:

$$\gamma \equiv \frac{c}{\sqrt{c^2 - v^2}} = \frac{1}{\sqrt{1 - \beta^2}} = \frac{\mathrm{d}t}{\mathrm{d}\tau}$$

I can derive:

$$\left(\frac{\mathrm{d}\tau}{\mathrm{d}t}\right)^2 + \left(\frac{v}{c}\right)^2 = 1$$

The term $\left(\frac{v}{c}\right)^2$ can be expanded to three terms:

$$\left(\frac{\mathrm{d}\tau}{\mathrm{d}t}\right)^2 + \left(\frac{v_x}{c}\right)^2 + \left(\frac{v_y}{c}\right)^2 + \left(\frac{v_z}{c}\right)^2 = 1$$

$$\mathrm{d}\tau^2 +\left(\frac{\mathrm{d}x}{c}\right)^2 + \left(\frac{\mathrm{d}y}{c}\right)^2 + \left(\frac{\mathrm{d}z}{c}\right)^2 = \mathrm{d} t$$

$$\left(c\mathrm{d}\tau\right)^2 +\left(\mathrm{d}x\right)^2 + \left(\mathrm{d}y\right)^2 + \left(\mathrm{d}z\right)^2 = \left(c\mathrm{d}t\right)^2$$

Using slightly different notation, we have:

$$\left(c \Delta\tau \right)^2 +\left(\Delta x\right)^2 + \left(\Delta y\right)^2 + \left(\Delta z\right)^2 = \left(c \Delta t \right)^2$$

This would give us a metric signature of (+,+,+,+). All components are space-like, with no time-like components.

Mathematically, the spacetime interval would be nothing other than:

$$s^2 = - \left(c \Delta \tau\right)^2$$

Which would mean that either $s$ or $\Delta \tau$ is imaginary.

Is the facility of hyperbolic trigonometric functions the primary motivation of formulating relativity on the basis of the metric signatures (-,+,+,+) and (+,-,-,-)?

It would seem that if relativity were formulated on the basis of non-hyperbolic trigonometric functions, the metric signature (+,+,+,+) would work as well.

 

[Sam Gralla]

I didn't check your equations carefully, but it looks from the end like you're talking about making the time coordinate imaginary. This type of approach is called a "Euclidean approach" (see also "wick rotation"). It works for special relativity but not for general relativity. There's a section in MTW about this ("goodbye to imaginary time" or something like that).

 

[robphy]

Here is an old post of mine which may be useful
https://www.physicsforums.com/showthread.php?p=702497#post702497

 

[Hurkyl]

$$\left(c \Delta\tau \right)^2 +\left(\Delta x\right)^2 + \left(\Delta y\right)^2 + \left(\Delta z\right)^2 = \left(c \Delta t \right)^2$$

This would give us a metric signature of (+,+,+,+). All components are space-like, with no time-like components.

No it wouldn't. In your manipulation, you've forgotten what the variables mean. $(t,x,y,z)$ are the space-time coordinates, and $\tau$ is the variable that parametrizes the world-line implicit in your calculations. (and is only well-defined up to a choice of additive constant)

One could try to revise Special Relativity to this approach, and have even seen people try it. But I don't think it to be a very fruitful approach -- the fact that $\tau$ has absolutely no bearing on whether particles interact appears to be an insurmountable obstacle to any attempt to define a space-time with $\tau$ as a coordinate. Another incredible obstacle would be extending the idea to fields (or even just to objects that are extended rather than point-like).

 

[Dale]

Using slightly different notation, we have:

$$\left(c \Delta\tau \right)^2 +\left(\Delta x\right)^2 + \left(\Delta y\right)^2 + \left(\Delta z\right)^2 = \left(c \Delta t \right)^2$$

This would give us a metric signature of (+,+,+,+). All components are space-like, with no time-like components.

This quantity is not a tensor (it doesn't transform right). So it cannot represent a metric in a Riemannian space.

 

[A.T.]

It would seem that if relativity were formulated on the basis of non-hyperbolic trigonometric functions, the metric signature (+,+,+,+) would work as well.

It just a question how you interpret relativity geometrically. Is coordinate time a dimension, and proper time the path integral or is it the other way around. Both interpreations have good and bad sides for visualization. See my comments here:
https://www.physicsforums.com/showthread.php?p=3276634

 

[Dale]

It just a question how you interpret relativity geometrically. Is coordinate time a dimension, and proper time the path integral or is it the other way around.

Proper time is a dimension in what space? I don't think that it is just a trivial preference or interpretation issue as you suggest here. You simply cannot make a coordinate chart on a manifold using proper time.

 

[A.T.]

Proper time is a dimension in what space?

In any space which is defined using proper time as a dimension. In particular in 4D space-propertime.

I don't think that it is just a trivial preference or interpretation issue as you suggest here. You simply cannot make a coordinate chart on a manifold using proper time.

Why not? If proper time can be interpreted as a distance "traveled" by individual objects (as is the case in Minkowski space time), then why cannot coordinate time be interpreted as a distance "traveled" by all objects (as is the case in space-propertime)? It is just a matter which coordinate chart is more useful to show something.

 

[Hurkyl]

It is just a matter which coordinate chart is more useful to show something.

$\tau$ isn't even a function of the space-time of ordinary special relativity, so $(\tau, x, y, z)$ most definitely cannot be a coordinate chart.

You could set up an entirely different manifold where you plot $(\tau, x, y, z)$ of point-particles, but it is of questionable utility -- there is no physical meaning to two world-lines intersecting on this manifold, symmetry let's us $\tau$-time translate world-lines individually to get equivalent physical states. (as opposed to the usual case where you have to time translate the whole universe) Nothing useful has Euclidean symmetry. There is no clear way to represent objects that aren't point particles.


I know of no advantage this manifold has in terms of physical interpretation, so the only remaining possible utility is for computation. But I know of no way to use this to simplify a computation either...

 

[atyy]

There is a comment in http://www.physics.ucsb.edu/~mark/ms-qft-DRAFT.pdf whose working out I've never seen - is this related to what A.T. is talking about?

"If time becomes an operator, what do we use as the time parameter in the Schroedinger equation? Happily, in relativistic theories, there is more than one notion of time. We can use the proper time τ of the particle (the time measured by a clock that moves with it) as the time parameter. ... Relativistic quantum mechanics can indeed be developed along these lines, but it is surprisingly complicated to do so. (The many times are the problem; any monotonic function of τ is just as good a candidate as τ itself for the proper time, and this infinite redundancy of descriptions must be understood and accounted for.)"

 

[Dale]

In any space which is defined using proper time as a dimension. In particular in 4D space-propertime.

And what are the properties of this space? What symmetries and isomorphisms does it have? What is its connection to physics? You can certainly take random physical quantities and make a "space" out of them (e.g. charge-viscosity-irradiance space), but that space will not in general have any useful properties.

In this case space-propertime is not diffeomorphic to the manifold and the space-propertime (++++) "metric" does not transform as a tensor. You lose the causal structure and all of the geometrical relationships such as intervals and angles. What is left?

It is just a matter which coordinate chart is more useful to show something.

No, space-propertime is not even a coordinate chart on the manifold. It lacks several properties of coordinate charts, specifically it is neither smooth nor invertible. This should be obvious from the fact that the signature is an invariant on the manifold for any valid coordinate chart, and your space-propertime does not yield the correct signature.

 

[PAllen]

I'll add an argument that if start down this path you end up back where you started - with normal Minkowski metric.

As Hurkyl pointed out, proper time is normally only differential or parameter along a world line. So to try to get a coordinate out of it, imagine space filling congruence of world lines, and a spacelike hypersurface. Assign time zero on the hypersurface. For any point, there is a unique worldline of the congruence through that point; its coordinates are the spatial coordinates of its intersection with the chosen hypersurface, and proper time measured from there to the given point (with some arbitrarily chosen positive / negative direction for time).

For simplicity, one would probably want the have coordinate defining world lines be geodesics. Where are we now? We have ended up with standard Minkowski coordinates with the standard Minkowski metric!

 

[A.T.]

You could set up an entirely different manifold where you plot $(\tau, x, y, z)$ of point-particles, but it is of questionable utility -- there is no physical meaning to two world-lines intersecting on this manifold,

Correct. The $\tau$-dimension is on the same footing as the 3 space dimensions. For comparison: In a purely spatial diagram, the intersection of two paths doesn't imply a meeting either.

There is no clear way to represent objects that aren't point particles.

Why? Where is the problem? Any object can be represented as a bunch of "point particles" (See pictures below).

I know of no advantage this manifold has in terms of physical interpretation,

Some pedagogical / visualization related stuff:

- It shows the connection between velocity, time dilation and length contraction in a nice intuitive Euclidean way:

Raum : space
Eigenzeit : proper time

http://www.adamtoons.de/physics/relativity.swf

- The above diagrams could also be used to show geometrically how the clocks in the front & back of the rocked get desynchronized during acceleration

- Visualizing different proper times of two world lines (Twins):

http://www.adamtoons.de/physics/twins.swf

- In GR it shows the connection gravitational time dilation and mass attraction in a nice intuitive geometrical way:

http://www.physics.ucla.edu/demoweb/demomanual/modern_physics/principal_of_equivalence_and_general_relativity/curved_time.gif

http://www.adamtoons.de/physics/gravitation.swf

- The above diagrams could also be used to show geometrically how you reach the event horizon in finite proper time, but infinite coordinate time of the distant observer.

 

[A.T.]

And what are the properties of this space?

I don't know. I could check. But last time you claimed that space-propertime would not be a metric space, it turned out that there is nothing in the definition of metric space that prevents it from being one.

You can certainly take random physical quantities and make a "space" out of them

Yes, you can. And combining space and proper time into one "space", is in no way more "random" than combining space and coordinate time, as Minkowski did.

but that space will not in general have any useful properties.

Depends on what is considered useful. I think that the space-propertime is useful from some things, while Minkowski spacetime is useful for others.

 

[atyy]

It just a question how you interpret relativity geometrically. Is coordinate time a dimension, and proper time the path integral or is it the other way around. Both interpreations have good and bad sides for visualization. See my comments here:
https://www.physicsforums.com/showthread.php?p=3276634

In referring to the path integral, are you thinking of http://arxiv.org/abs/hep-th/0101036, say Eq 1.7 ?

 

[DrGreg]

The attached diagram illustrates what's wrong with "space-propertime".

The left hand diagram is a traditional space-time diagram of the twins paradox. Red observer moves inertially from A to C. Blue observer moves A to B, turns round and moves B to C. C on the diagram represents the event where the two observers meet again.

The right hand diagram is a "space-propertime" diagram of the same scenario. Here point C₁ represents the end of the red journey and point C₂ represents the end of the blue journey. The problem here is that C₁ and C₂ both represent the same event in the real universe, yet they are distinct points in the diagram. That means the diagram isn't much use in practice. Isolated points on the diagram have no meaning, only points on a drawn line, and even then they become ambiguous where two lines meet.

 

[A.T.]

The attached diagram illustrates what's wrong with "space-propertime".

I see nothing "wrong" there. It just shows the difference between space-coordinatetime and space-propertime. I posted an interactive version of that comparison myself:
http://www.adamtoons.de/physics/twins.swf

The problem here is that C₁ and C₂ both represent the same event in the real universe, yet they are distinct points in the diagram.

The concept of "events" as you use it here, is part of the space-coordinatetime concept. Applying it to space-propertime. as you do here, is mixing the two concepts in a nonsensical way.

That means the diagram isn't much use in practice.

"Not usefull" is quite different from "wrong", isn't it? You might not find it useful for some purposes. But equally the space-coordinatetime diagram is quite useless in visualizing the age difference between the twins, while the space-propertime diagram shows is nicely. The space-coordinatetime diagram fails to visualize propertime at all, While space-propertime diagram shows both time quantities directly:
- propertime as the temporal coordiante
- coordinate time as the length of every worldline

 

[PAllen]

The concept of "events" as you use it here, is part of the space-coordinatetime concept. Applying it to space-propertime. as you do here, is mixing the two concepts in a nonsensical way.

No, "event" is a feature of reality, say the collision of the two twins at the end of their journey. So we can say space-proper time cannot describe the set of all possible events as even a topological manifold, let alone a metric space - because it fails to provide a 1-1 mapping between the set of events and R4.

 

[Hurkyl]

Correct. The $\tau$-dimension is on the same footing as the 3 space dimensions.

No it isn't -- the laws of physics distinguish between them.

For comparison: In a purely spatial diagram, the intersection of two paths doesn't imply a meeting either.

So? The real comparison is to a space-time diagram, in which intersection of paths actually means something.

Why? Where is the problem? Any object can be represented as a bunch of "point particles"

This is an assumption that is not required by Special Relativity. Also, you neglect fields.

- It shows the connection between velocity, time dilation and length contraction in a nice intuitive Euclidean way:

Albeit rather divorced from physics. Also incredibly misleading because it suggests physics is preserved by the rotation you use in the diagram.

Also, the key idea in the visualization is the projection onto the axes -- and that is present in ordinary space-time diagrams.


Just for fun, a rough sketch of one particle orbiting another, in $(x, y, \tau)$ coordinates:

 

[A.T.]

No, "event" is a feature of reality,

I was referring the concept of an event as a point in some manifold. This is not part of reality. It is a human invention.

 

[Hurkyl]

I was referring the concept of an event as a point in some manifold. This is not part of reality. It is a human invention.

Special Relativity is a human invention. It has events that are points of some manifold. The corresponding physical theory includes events in its catalog of things that correspond to reality.

That space-propertime cannot talk about events is a huge deficiency of such a formalism.

If you are asserting a theory that doesn't have events at all, then you are not talking about SR, and really shouldn't be hijacking this thread.

 

[A.T.]

No it isn't -- the laws of physics distinguish between them.

Which ones?

So? The real comparison is to a space-time diagram, in which intersection of paths actually means something.

So?

This is an assumption that is not required by Special Relativity.

Which assumption? I just pointed out that there is no problem with non-point-like objects in space proper time, contrary to what you suggested.

Albeit rather divorced from physics.

How so?

Also incredibly misleading because it suggests physics is preserved by the rotation you use in the diagram.

Not sure what you mean here.

Also, the key idea in the visualization is the projection onto the axes -- and that is present in ordinary space-time diagrams.

Not in the simple and intuitive Euclidean way, as it is in space-propertime.

Just for fun, a rough sketch of one particle orbiting another, in $(x, y, \tau)$ coordinates:

You forgot the axes, but it looks wrong anyway. The particle at rest has a worldline along the $\tau$ dimension. The orbiting particle has a helical worldline with the first one as axis. The path lengths of both are the same, so the helix ends at a lower $\tau$ coordinate.

 

[Hurkyl]

You forgot the axes, but it looks wrong anyway. The particle at rest has a worldline along the $\tau$ dimension.

I didn't choose $(x,y)$ from an inertial reference frame where the central particle was at rest.

($\tau$ runs vertically in my diagram)

 

[Hurkyl]

Which ones?

Pretty much all of them. If all four coordinates were on equal footing, then the $\tau$ axis and the $x$ axis would represent physically equivalent configurations.

However, the former is a time-like and the latter is light-like: very different.

Another failed symmetry is that I can take any world-line and translate it in the $\tau$ direction to get an equivalent system. But if I translate it in the x direction, I get an inequivalent system.

Which assumption? I just pointed out that there is no problem with non-point-like objects in space proper time, contrary to what you suggested.

No you didn't. You simply said that you were only going to treat everything as if it was a collection of point-like objects.

 

[A.T.]

Special Relativity is a human invention. It has events that are points of some manifold.

No. Minkowski's geometrical interpretation of SR has events that are points of some manifold.

That space-propertime cannot talk about events is a huge deficiency of such a formalism.

It is usefull for some things, and less usefull for others.

If you are asserting a theory that doesn't have events at all, then you are not talking about SR,

Not what I said. The original SR didn't have events that are points of some manifold.

and really shouldn't be hijacking this thread.

I'm not hijacking anything. The discussion of space-coordiantetime (-,+,+,+) versus space-propertime (+,+,+,+) is exactly what the OP is about.

 

[Hurkyl]

The original SR didn't have events that are points of some manifold.

It most certainly did, I just checked (a translation of) it. It even used the word "event", and had phrases like (x,y,z,t) is a system of values that describe the place and time of any event. The only thing it didn't do is actually use the word "manifold" to refer to the notion.

 

[A.T.]

Pretty much all of them. If all four coordinates were on equal footing, then the $\tau$ axis and the $x$ axis would represent physically equivalent configurations.

However, the former is a time-like and the latter is light-like: very different.

Time-like and light-like are parts of the space-coordinatetime concept.

Another failed symmetry is that I can take any world-line and translate it in the $\tau$ direction to get an equivalent system. But if I translate it in the x direction, I get an inequivalent system.

And what does equivalent system mean here?

No you didn't. You simply said that you were only going to treat everything as if it was a collection of point-like objects.

No, I just said that you can do that, not that you have to.

 

[Hurkyl]

Time-like and light-like are parts of the space-coordinatetime concept.

No, they're actually physical notions. Ones that are rather central to special relativity.

And what does equivalent system mean here?

The things it usually does. The two are physically indistinguishable. The described symmetry preserves all physics. Things like that.

 

[A.T.]

It most certainly did, I just checked (a translation of) it. It even used the word "event", and had phrases like (x,y,z,t) is a system of values that describe the place and time of any event. The only thing it didn't do is actually use the word "manifold" to refer to the notion.

That was my point. The manifold was Minkowski's idea. But even if Einstein would have introduced it himself, it would still be just one possible geometrical interpretation of the math.

 

[WannabeNewton]

That was my point. The manifold was Minkowski's idea. But even if Einstein would have introduced it himself, it would still be just one possible geometrical interpretation of the math.

What are you talking about? You can't characterize space - time in GR without introducing continuously differentiable manifolds.

 

[A.T.]

What are you talking about? You can't characterize space - time in GR without introducing continuously differentiable manifolds.

Yes, but Minkowskis manifold is not the only possible manifold.

 

[Hurkyl]

That was my point. The manifold was Minkowski's idea.

No, the manifold goes back at least to Newton, and I'd imagine much further than that.

 

[A.T.]

No, the manifold goes back at least to Newton, and I'd imagine much further than that.

Back then there was only one absolute time in physics. So you didn't have much choice which time-quantity to use as the time-dimension in your space-time-manifold.

That changed with SR. And Minkowski chose coordinate time, which is fine, but not the only option.

 

[Hurkyl]

Back then there was only one absolute time in physics. So you didn't have much choice which time-quantity to use as the time-dimension in your space-time-manifold.

You've always had the option of "generalized coordinates".

That changed with SR.

SR uses the exact same manifold of events that Newtonian mechanics did.

but not the only option.

Proper time isn't an option to describe events of space-time in SR; the idea simply isn't anything resembling well-defined.

 

[Dale]

I don't know. I could check. But last time you claimed that space-propertime would not be a metric space, it turned out that there is nothing in the definition of metric space that prevents it from being one.

No, I got distracted and we didn't finish the conversation. A metric space is defined as a set which has some notion of distance defined between members of the set. I don't think your space-propertime even constitutes a set. In spacetime the elements of the set are physical events. What are the elements of the in space-propertime set? One space-propertime coordinate can be assigned to multiple events and one physical event may have multiple space-propertime coordinates, so obviously the elements of the set are not physical events. So what are they then? It is certainly not clear to me.

Furthermore, in this thread you have specifically claimed that space-propertime is a coordinate chart. This is a stronger claim than your earlier claim that it is a metric space, and even if the weaker claim that it is a metric space can be justified, the stronger claim that it is a coordinate chart cannot. Coordinate charts are diffeomorphic to the manifold and space-propertime is not. Also, as I mentioned above, the metric signature is an invariant on the manifold regardless of the coordinate chart used, so it is obvious that any convention with a different metric signature can not be a coordinate chart. And again, your "metric" is not a tensor since it does not transform like a tensor.