Frame-Independent Set of Events?

[greswd]

In SR, an event is a particular point in space and time.For simplicity, we only use one dimension of space.

We have x'=γ(x-vt) if when x, t = 0 , x', t' = 0Let's say we have a bunch of inertial frames, all moving at different speeds, but all sharing a common origin x, t = 0

From the POV of one of the inertial frames, we have a series of events in spacetime with the coordinates (x1,t1) , (x2, t2) etc.If we switch to another frame, the space-time coordinates will be transformed.

How do I describe this set of events in a manner that is independent of any inertial frame?What about describing a continuous set of events, such as the continuous existence of the entire length of a rod that is floating in space?

 

[mfb]

How do I describe this set of events in a manner that is independent of any inertial frame?

You cannot. No matter how you describe it, a reconstruction in different frames will lead to different results. You need at least a second reference point.
You can make one invariant quantity, the magnitude of the four-vector of the event, t² - x², which is the same in all reference frames, but that's not sufficient for two degrees of freedom.

 

[Matterwave]

How do I describe this set of events in a manner that is independent of any inertial frame?What about describing a continuous set of events, such as the continuous existence of the entire length of a rod that is floating in space?

It depends on what you mean by "describe". You can describe the set of events as "event 1, event 2, event 3, etc." or "event A, event B, event C, etc." or "event photon was emitted, event photon was absorbed, event my foot made contact with his butt, event his head made contact with the ground, etc.) with a unique descriptor for each event, and that will be frame independent (be careful of how you describe the events though, like "event 2 is the event after event 1" is not a good descriptor). But as far as finding a frame-independent coordinate system or something, that is, by definition basically, not possible.

 

[PeterDonis]

How do I describe this set of events in a manner that is independent of any inertial frame?

By finding invariant relationships between them. For example, the spacetime interval between any pair of events is invariant in SR. You can also define the spacetime equivalent of "angles" between the lines connecting events (but that requires more care).

What about describing a continuous set of events, such as the continuous existence of the entire length of a rod that is floating in space?

By picking out a set of worldlines (timelike curves in spacetime) that describes the points on the rod. The "world tube" formed by all these worldlines (which are assumed to entirely fill the "world tube", i.e., the volume of spacetime occupied by the rod) basically is the rod, from an SR point of view. This description of the rod is independent of coordinates.

If the rod's motion satisfies some (fairly stringent) conditions, then you can go further and define an invariant "distance" between any pair of points on the rod, also independent of coordinates. But many motions do not satisfy the conditions, and if the rod's motion does not, then there is no invariant way to define "distance" along the rod.

 

[greswd]

@PeterDonis thanks for the answer.

what is the spacetime equivalent of an angle? Let's use 3 events as a reference, like one uses 3 points in space to draw an angle.

 

[PeterDonis]

what is the spacetime equivalent of an angle?

If the two lines are both timelike, the angle between them is their relative velocity at the point where they meet; more precisely, it's the inverse hyperbolic tangent of the relative velocity. Note that this is a hyperbolic "angle", so when you try to do trigonometry with it, you have to use the hyperbolic functions, $\sinh$, $\cosh$, $\tanh$, etc., instead of the usual circular functions $\sin$, $\cos$, $\tan$, etc.

If the two lines are both spacelike, the angle between them is just an ordinary angle.

For other cases, the "spacetime angle" between lines doesn't have a simple intuitive interpretation like the above two cases; but you can always compute the inner product between any two vectors using the metric, so the angle between any two lines in spacetime can always be defined using the inner product of their tangent vectors at the point where they intersect.

 

[greswd]

The reason I'm doing this is because I think it will help with the issue of the "block universe", the issue of relativistic pre-determinism.

I've come up with something that I think will work, it may not be complete.

Let's start with the Galilean transformation. We have three events, O, A & B. O is always at the origin. This is for one dimension of space only.

Then we use three different numbers: the time interval between A & O, the time interval between B & O, and lastly the distance between A & B.

The point A will be located at the appropriate time interval, and it can have an arbitrary x-position, thus it is independent of any specific frame. The location of point B will depend on the chosen location of A as well its own corresponding time interval.Now we switch to the Lorentz transformation. Same three points, O, A & B.

We use another three numbers, the spacetime intervals between O & A, O & B and A & B.

With these numbers we can describe a set of events independent of any frame.So for the Galilean, the set is a collection of time intervals and distance intervals. For the Lorentzian, it is a collection of spacetime intervals.

Some other parameters might be needed to give a complete picture but I think this is what I was looking for when I started this thread.

 

[PeterDonis]

Now we switch to the Lorentz transformation. Same three points, O, A & B.

Yes, but note that, as you say in your OP, these are points in spacetime, not space. They don't identify three objects; they identify three objects at three instants of their proper times, i.e., three objects that each carry a clock, with given readings on each of the three clocks.

We use another three numbers, the spacetime intervals between O & A, O & B and A & B.

With these numbers we can describe a set of events independent of any frame.

Yes, but again, these are events, not objects. You might want to think about what that means.

 

[greswd]

Yes, but note that, as you say in your OP, these are points in spacetime, not space. They don't identify three objects; they identify three objects at three instants of their proper times, i.e., three objects that each carry a clock, with given readings on each of the three clocks.

haha, I have to walk before I can run. First points, then extended objects, and also the remaining 2 dimensions of space. But I need to make sure this method works first.

 

[PeterDonis]

First points, then extended objects

Extended objects are not the next step after points in spacetime. The next step after points in spacetime is worldlines, i.e., curves that describe the entire spacetime history of a point-like object.

 

[greswd]

Extended objects are not the next step after points in spacetime. The next step after points in spacetime is worldlines, i.e., curves that describe the entire spacetime history of a point-like object.

True. There'll be an infinite number of points to deal with anyway.

Anyway, do you think the method suggested so far (for singular events only) requires any tweaks?

 

[PeterDonis]

do you think the method suggested so far (for singular events only) requires any tweaks?

If all you're trying to describe is the relationships between events, then what you've done is correct: the invariant description of those relationships is the set of spacetime intervals between them. (Note that in the presence of gravity, the spacetime geometry implied by these relationships will be curved, not flat.)

 

[greswd]

If all you're trying to describe is the relationships between events, then what you've done is correct: the invariant description of those relationships is the set of spacetime intervals between them. (Note that in the presence of gravity, the spacetime geometry implied by these relationships will be curved, not flat.)

I'm hoping there are no problems such as getting two solutions and then having to eliminate one, as is in the case of ordinary trilateration.

 

[PeterDonis]

I'm hoping there are no problems such as getting two solutions and then having to eliminate one, as is in the case of ordinary trilateration.

That's only a problem if you don't know the relationships between the events, but are trying to derive them based on other observations. If you already know the invariant spacetime intervals between all the events, you have a unique solution, at least as far as those events are concerned.

 

[greswd]

That's only a problem if you don't know the relationships between the events, but are trying to derive them based on other observations. If you already know the invariant spacetime intervals between all the events, you have a unique solution, at least as far as those events are concerned.

Thank you. I need this for those endless block universe and determinism arguments.