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

[kmarinas86]

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.

It would seem that in a Minkowski diagram, intersecting lines represent simultaneity, whereas in an Epstein diagram, intersecting lines do not represent simultaneity. So I can accept that space-propertime does not transform like a tensor.

The underlying assumption that was initially presented here seems to be Euclidean space, and in such "fixed" geometry, there is no role for anything, be it proper time, or coordinate time, to be somehow seamlessly connected with it. So here, there is no discussion of any workable "space-propertime" manifold. This answers my question, "Why (-,+,+,+) and not (+,+,+,+)?"

As for "elements of set", here is my take on it: The propertime derivative would probably exist as some scalar field that is derived from the norm of a vector field based on relativistic length contractions of observables inside Euclidean space itself. Such does not lead to any "space-propertime" manifold. But one may consider that Euclidean space would be the coordinate space itself, carrying values which are varying with coordinate time $t$. So there would be a set of coordinate times $t_i$, and each element of that set would itself be a set carrying the vectors of each point in the space at some $t_i$.

 

[A.T.]

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

I guess by "SR" you mean Minkowski's interpretation of SR, that uses the space-coordiante time manifold? How do you know that Newtonian mechanics uses the exact same manifold? Newtonian mechanics doesn't differentiate between coordinate time and proper time, there is just time.

Proper time isn't an option to describe events of space-time in SR;

If I want to define events based on coordinate time and if I want to have them represented as points on some manifold, then I obviously should not use space-propertime. But who says that I always want this?

 

[Hurkyl]

I guess by "SR" you mean Minkowski's interpretation of SR, that uses the space-coordiante time manifold? How do you know that Newtonian mechanics uses the exact same manifold?

Er, because I actually learned about them?

Newtonian mechanics doesn't differentiate between coordinate time and proper time, there is just time.

Notions like "coordinate time" or "proper time" or "time" have nothing to do with the manifold. The manifold of events is just (homeomorphic to) the plain old R⁴. And I do mean manifold -- there is no coordinate chart involved, no choice of metric or pseudo-metric or anything else -- just the points and the topology. Well, I suppose it's fine to use the same differentiable structure too. (so, I suppose I mean "differentiable manifold")


If a rocket blasts off from Earth, travels through space, and eventually returns from Earth, then the path of the rocket through space-time is some set of events. These events can be separated into three (potentially overlapping) connected subsets:

• The first set contains what we would describe as "the rocket blasting off", and doesn't contain any of what we would describe as "the rocket landing"
• The third set contains what we would describe as "the rocket blasting landing", and doesn't contain any of what we would describe as "the rocket blasting off"
• The second set contains all of the events that don't fit into the other two categories. (And possibly some that do)

This sort of thing is what people use the term "event" to talk about. The manifold of events is an assertion that events can be arranged into a topological structure of a certain type, in particular encoding the empirical fact that, locally, we can parametrize events (in many different ways) with a system of 4 real parameters such that each quadruple of values describes a unique event, and every event is described by at most one quadruple.


Newtonian mechanics and Special Relativity (but not General Relativity) further asserts there exist differentiable coordinate charts that cover the entire manifold (so every event is described by exactly one quadruple).


Newtonian mechanics asserts some laws of physics of objects living on this manifold. And these laws happen to have a rather simple form in a few special coordinate charts. So much so that the most expedient way to learn the subject is to learn how the laws look in these special coordinate charts, rather than how they look in in general coordinate charts or in a coordinate-free manner. The same is true for Special Relativity.

(General Relativity, of course, does not, preferring to state laws in a way that works for all coordinate charts, or even in a coordinate-free when possible)

But who says that I always want this?

If you want to talk about a manifold whose points aren't events, then you shouldn't use the word "event" to refer to its points.

 

[Hurkyl]

I can put it another way.

We have, in real life, coordinate charts for locating events in (a small region of) space-time -- the four-tuple of latitude, longitude, altitude, and GPST, as reported by a GPS receiver.

(Within the tolerance of the devices) events on two GPS receivers' worldlines have the same reading if and only if they are coincident with each other.

The same cannot be said for the four-tuple of latitude, longitude, altitude, wristwatch time.