Proper (and coordinate) times re the Twin paradox

In summary, the stay-at-home twin is at rest in her frame and her clock must therefore measure proper time. The traveling twin, carries his clock with him; it is therefore at rest in his frame and must also measure proper time. As each twin is moving relative to the other, they will each measure coordinate time for their twin. Their proper times will be identical. Their coordinate times will be identical. As their relative speeds are the same, their Lorentz transformations will be the same. When the traveling twin slows on his return and comes to rest in his twin's frame they are both once again in the same frame and will have traveled exactly the same each relative to the other. It is only if the traveller continues
  • #316
Yes, that is the length of the “hypotenuse”
 
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  • #317
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.
 
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  • #318
Fair enough, but there is no Minkowski term for the “hypotenuse” so the scare quotes is the best I can do.
 
  • #319
Grimble said:
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.
 
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  • #320
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
Wikipedia said:
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?)
 
  • #321
Grimble said:
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##
 
  • #322
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?
 
  • #323
Grimble said:
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).
 
  • #324
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
 
  • #325
Grimble said:
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.
 
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  • #326
Grimble said:
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.

Grimble said:
It is still the same Spacetime

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

Grimble said:
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.
 
  • #327
Grimble said:
The Mathematics doesn't depend on how they are drawn.

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

Grimble said:
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. :wink: 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.
 
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  • #328
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
 
  • #329
Grimble said:
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.

Grimble said:
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.

Grimble said:
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.

Grimble said:
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.
 
  • #331
The thread topic has been sufficiently discussed. The thread will remain closed.
 
<h2>1. What is the Twin Paradox?</h2><p>The Twin Paradox is a thought experiment in which one twin travels through space at high speeds while the other remains on Earth. When the traveling twin returns, they have aged less than the twin who stayed on Earth. This paradox challenges our understanding of time and relativity.</p><h2>2. How does the concept of proper time relate to the Twin Paradox?</h2><p>Proper time is the time measured by an observer who is in the same frame of reference as the event being measured. In the Twin Paradox, proper time is used to calculate the time dilation experienced by the traveling twin. It is the time experienced by the twin on their journey through space.</p><h2>3. How does the concept of coordinate time come into play in the Twin Paradox?</h2><p>Coordinate time is the time measured by an observer who is in a different frame of reference than the event being measured. In the Twin Paradox, coordinate time is used to calculate the time experienced by the twin who stayed on Earth. It is the time experienced by an observer who is not in the same frame of reference as the event.</p><h2>4. Can the Twin Paradox be explained by the theory of relativity?</h2><p>Yes, the Twin Paradox can be explained by the theory of relativity. According to the theory, time is relative and can be affected by factors such as speed and gravity. In the Twin Paradox, the traveling twin experiences time dilation due to their high speed, causing them to age slower than the twin on Earth.</p><h2>5. Are there any real-life examples of the Twin Paradox?</h2><p>While the Twin Paradox is often used as a thought experiment, there have been real-life examples that demonstrate its principles. For example, astronauts who spend extended periods of time in space experience time dilation and age slower than those on Earth. This has been observed in astronauts who have spent time on the International Space Station.</p>

1. What is the Twin Paradox?

The Twin Paradox is a thought experiment in which one twin travels through space at high speeds while the other remains on Earth. When the traveling twin returns, they have aged less than the twin who stayed on Earth. This paradox challenges our understanding of time and relativity.

2. How does the concept of proper time relate to the Twin Paradox?

Proper time is the time measured by an observer who is in the same frame of reference as the event being measured. In the Twin Paradox, proper time is used to calculate the time dilation experienced by the traveling twin. It is the time experienced by the twin on their journey through space.

3. How does the concept of coordinate time come into play in the Twin Paradox?

Coordinate time is the time measured by an observer who is in a different frame of reference than the event being measured. In the Twin Paradox, coordinate time is used to calculate the time experienced by the twin who stayed on Earth. It is the time experienced by an observer who is not in the same frame of reference as the event.

4. Can the Twin Paradox be explained by the theory of relativity?

Yes, the Twin Paradox can be explained by the theory of relativity. According to the theory, time is relative and can be affected by factors such as speed and gravity. In the Twin Paradox, the traveling twin experiences time dilation due to their high speed, causing them to age slower than the twin on Earth.

5. Are there any real-life examples of the Twin Paradox?

While the Twin Paradox is often used as a thought experiment, there have been real-life examples that demonstrate its principles. For example, astronauts who spend extended periods of time in space experience time dilation and age slower than those on Earth. This has been observed in astronauts who have spent time on the International Space Station.

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