Proper (and coordinate) times re the Twin paradox

[Dale]

Yes, that is the length of the “hypotenuse”

 

[vanhees71]

I'd not use "Euclidean jargon" in connection with the Minkowski space. It's hard, but one must hammer ourselves the fact into the brain that a Minkowski diagram must not be read in Euclidean terms! The lengths defined by the fundamental form of Minkowski space are not what we are used to in the Euclidean plane. The Minkowski plane is rather a hyperbolic space. The indefiniteness of the fundamental form is the key element making it a spacetime manifold, allowing for a causality structure. This cannot be achieved with a proper scalar product of a Euclidean (affine) space.

 

[Dale]

Fair enough, but there is no Minkowski term for the “hypotenuse” so the scare quotes is the best I can do.

 

[Mister T]

Calculate the length of what?

The spacetime interval between the two events. One event is the emission of the light flash and the other is the reception. Note you correctly calculated the value to be zero. The events have a lightlike separation. You seem to be confusing the spacetime interval with the proper length. The value of the spacetime interval equals the proper length only for events with a spacelike separation.

Edit: And the length of a worldline is the proper time. The spacetime interval for events with a timelike separation.

 

[Grimble]

I am finding this intriguing, you are opening up a whole new world to me.
I have never understood, nor seen any indication of, hyperbolic geometry in relation to Minkowski Diagrams.
Yes, I have seen the hyperbola in Minkowski's diagrams in his Space and Time Lecture but had no idea that that led to a completely different geometry...
So perhaps you can understand why I have been so reluctant to let go of the Euclidean perspective.
But are we not still dealing with the same equations?
Isn't the Minkowski hyperbolic treatment just one way of depicting them, because the hyperbola seems to be associated with losing the uniformity of scale on the diagonal lines, which in turn gives rise to what I see as anomalies: planes of simultaneity and jumps on the time axis of twin paradox diagrams.

Don't get me wrong I am not saying there is anything wrong with what you are saying, only that it is intriguing how far I have gone down the wrong road because I have failed to appreciate that hyperbolic geometry is integral with Minkowski diagrams.

For example this from the introduction to Minkowski Diagrams

Minkowski diagrams are two-dimensional graphs that depict events as happening in a universe consisting of one space dimension and one time dimension. Unlike a regular distance-time graph, the distance is displayed on the horizontal axis and time on the vertical axis. Additionally, the time and space units of measurement are chosen in such a way that an object moving at the speed of light is depicted as following a 45° angle to the diagram's axes.

Nowhere can I find any suggestion that it involves a different metric (if that is the correct term?)

 

[Dale]

But are we not still dealing with the same equations?

Clearly not:
$ds^2=dx^2+dy^2+dz^2$
is not the same equation as
$ds^2=-dt^2+dx^2+dy^2+dz^2$

 

[Grimble]

YEs, but one is the aggregate length in 3 dimensions and the other in 4, so does ds represent the same quantity in each case?

 

[Dale]

YEs, but one is the aggregate length in 3 dimensions and the other in 4, so does ds represent the same quantity in each case?

It is not the same quantity, but there are some similarities. In both cases it is an invariant measure of distance in the space. The differences are that in space it is called distance and there is only one kind of distance, while in spacetime it is called the spacetime interval and there are three different kinds of spacetime intervals (space like, time like, and null).

 

[Grimble]

Well if it is not the same quantity then both equations are true... but cannot be equated - or even compared if ds represents different quantities? Or am I missing something here?

The terms in these equations represent specific properties; I agree they are different things, but choosing a type of diagram changes how they are represented on that diagram, not what they represent.
It is still the same Spacetime, consisting of three Cartesian dimensions of space plus another dimension for time
Surely a²+b²+c² is still the aggregate length in Minkowski Spacetime; while -ct²+a²+b²+c² is the Spacetime interval and both are invariant intervals.
The Mathematics doesn't depend on how they are drawn.
It just seems difficult to be sure what the terms mean when the same term ds² means two different things...
It can be very confusing

 

[jbriggs444]

Surely a²+b²+c² is still the aggregate length in Minkowski Spacetime; while -ct²+a²+b²+c² is the Spacetime interval and both are invariant intervals.

It makes little sense to use the term "invariant" to refer to $a^2+b^2+c^2$ in the context of Minkowski Spacetime.

Presumably the formula is shorthand for "the square of the difference in the x coordinates plus the square of the difference in y coordinates plus the square of the difference in z coordinates in some coordinate system".

The obvious question is: The x, y and z coordinates of what?

Without a t coordinate, the only rational answer is "between two lines".

And without an agreed upon coordinate system, the next question is "between which two lines".

Which brings us to the stated conclusion: It makes little sense to use the term "invariant" for this.

 

[PeterDonis]

if it is not the same quantity then both equations are true

True in their respective geometries, yes. But they are different geometries. Euclidean 3-space is not the same geometry as 4-D Minkowski spacetime. Each one has its own formula for $ds^2$. It makes no sense to say the formula for $ds^2$ in Euclidean 3-space is "true" in Minkowski spacetime.

It is still the same Spacetime

Euclidean 3-space is not spacetime. It's Euclidean 3-space.

Surely a2+b2+c2 is still the aggregate length in Minkowski Spacetime; while -ct2+a2+b2+c2 is the Spacetime interval and both are invariant intervals.

No, this is not correct. $dx^2 + dy^2 + dz^2$ is the invariant length in Euclidean 3-space. $- c^2 dt^2 + dx^2 + dy^2 + dz^2$ is the invariant length in 4-D Minkowski spacetime. This length is called a "spacetime interval" in the Minkowski spacetime, but that's just nomenclature.

 

[PeterDonis]

The Mathematics doesn't depend on how they are drawn.

Yes, but it does depend on which geometry you are in.

It just seems difficult to be sure what the terms mean when the same term ds2 means two different things...
It can be very confusing

Welcome to math and physics. Terminology will sometimes be confusing; you just have to learn how to figure out what is intended from context, and ask questions when something is not clear. What you should not do is assume that the same symbol must mean the same thing in a different context.

 

[Grimble]

The main difference in the two geometries is the number of dimensions, is it not?
In Euclidean 3-space, a²+b²+c² is invariant because there is no time component.
So in Minkowski Spacetime the equivalent would be that a²+b²+c² would be invariant at any single specific time.
Similarly, in classical mechanics using euclidean geometry, ds²=(ct')²+a²+b²+c² where ct' is the time axis for a moving body

 

[PeterDonis]

The main difference in the two geometries is the number of dimensions, is it not?

No. That's one difference, but not the main one. The main difference is that Euclidean 3-space is Riemannian (the metric has +++ signature) while 4-D Minkowski spacetime is pseudo-Riemannian (the metric has -+++ signature). That means, as @Dale said, that while in Euclidean 3-space there is only one type of interval/length, in Minkowski spacetime there are three: spacelike, null, and timelike.

In Euclidean 3-space, a2+b2+c2 is invariant because there is no time component.

No, it's because there is no such thing as a "time component" in Euclidean 3-space. There are only three dimensions, and they're all spacelike because the metric has +++ signature.

in Minkowski Spacetime the equivalent would be that a2+b2+c2 would be invariant at any single specific time.

There is no such thing as "any single specific time" because there is no preferred inertial frame in Minkowski spacetime. An interval that has zero $dt$ in one frame will have nonzero $dt'$ in any other frame. And "invariant" means the equation has to hold in every frame, not just one, so your claim is wrong.

in classical mechanics using euclidean geometry, ds2=(ct')2+a2+b2+c2 where ct' is the time axis for a moving body

Wrong. There is no such spacetime interval in Newtonian mechanics.

 

[PeterDonis]

Thread closed for moderation.

 

[PeterDonis]

The thread topic has been sufficiently discussed. The thread will remain closed.