# Minkowski space and geometric intuition

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I'm interested in why less time passes on the spaceship than on Earth in the twins paradox of special relativity, and the only explanation I've ever seen is that the spaceship takes a "shorter path through spacetime". By "explanation", I mean a proposal for a physical mechanism that would slow down the time evolution of the spaceship's wave function, or some other theory that provides a framework for understanding the phenomenon. I'm not interested in the quantitative aspects of the problem; I take Lorentz's transformation equations as given. I may be dumb, but I'm not that dumb! đ

The only reason I use the twins paradox as an example is because it eliminates the argument that time dilation is an artifact of the measurement process. If you consider the three-observer variant of the paradox,* the synchronization events define a triangle in spacetime, so the total elapsed times are for paths between the same two points (the Earth fly-byes).

So, what does the word "shorter" mean in in the context of Minkowski space? I know what it means in ordinary Euclidean space: It means "not as long", and length is a measure of how many physical objects can be packed into the space between two points, i.e. how many units of distance are required to span the gap between the points. Even in a curved space, length is reasonably intuitive if we use units that are small enough that the space appears flat locally. No problem.

So what about Minkowski space? The phrase "shorter path" is typically presented without explanation, as though it's intuitively obvious because we all learned what it means when we were children. But what does it mean in Minkowski space?

Technically speaking, there are different ways to formulate Minkowski space. Minkowski originally added ict as the fourth dimension and calculated intervals in the normal way, as a sum of squares, but nowadays we use ct as the fourth axis and a metric tensor with -1 for the time element.

But these conventions make our geometric intuition irrelevent. There's no such thing as packing blocks or rulers or meter sticks into a negative length. In Minkowski space, "length" and "shortness" are purely mathematical concepts with no intuitive physical or geometric meaning. They're not physics.

Anyway, I don't want to start the umpteen millionth flame war about the twins paradox. I'm too old for flame wars. I just want to register my disapproval of the language that physicists and/or physics popularizers use to "explain" time dilation. I think it's a cop-out, I think it's misleading wordplay, I think the only real scientific answer early in the 21st century is "Nobody knows". Which is okay. Maybe someone will figure it out sometime in the future. đ

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* In the three-observer variant of the twins paradox (the triplets paradox?đ€), one spaceship handles the outbound leg of the journey and another handles the return leg. There's no need for acceleration anywhere in the experiment. The three events in the experiment are (1) the outbound ship flies by Earth, (2) the two ships cross paths somewhere out in space, and (3) the inbound ship flies by Earth. Each spaceship can be assumed to travel at constant velocity during its part of the experiment, and the total time elapsed in the spaceships (i.e. along the path 1-2-3) is defined as the time on the first ship between 1 and 2 plus the time on the second ship between 2 and 3.

Dr. Lincoln doesn't actually provide an explanation for anything, but he does describe the triplets version of the paradox.

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Edited by Lorentz Jr
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1 hour ago, Lorentz Jr said:

By "explanation", I mean a proposal for a physical mechanism that would slow down the time evolution of the spaceship's wave function

I think it is best to just keep the situation fully classic, and consider only physical clocks to begin with, rather than wave functions. The question of evolution operators in RQM is complex and very non-trivial, and does little to illuminate this underlying question.

Time dilation is a relationship between reference frames, and not something that physically âhappensâ to a single clock. Asking for a mechanism that âslows downâ some clock is thus meaningless - clocks always tick at the same rate within their own frames. So the correct question would be why inertial frames are related via hyperbolic rotations in spacetime - thatâs a very valid question, but it isnât one that any of our present theories can answer. So to make a long story short, we donât have an explanation of why this happens, only a description of it. Thatâs not the same thing at all.

1 hour ago, Lorentz Jr said:

But what does it mean in Minkowski space?

The length of a world line between given events in Minkowski spacetime is defined to be equivalent to the proper time of a clock travelling between these events that traces out that world line. In other words, itâs simply the total elapsed time thatâs physically measured on a clock that travels along a specific spatial path between events.

1 hour ago, Lorentz Jr said:

But these conventions make our geometric intuition irrelevent.

Intuitiveness is not a necessary condition for a mathematical model to be valid and useful. It just needs to be internally self-consistent, and produce results that can be verified using the scientific method. I think you would agree that SR does this quite well.

Beside, something being intuitive (or not) is a very subjective measure - many things I find intuitive might appear otherwise to you, and vice versa.

1 hour ago, Lorentz Jr said:

I think it's a cop-out, I think it's misleading wordplay, I think the only real scientific answer early in the 21st century is "Nobody knows". Which is okay. Maybe someone will figure it out sometime in the future. đ

I would, by and large, agree with you - though I wouldnât put into such strong terms. I just think many depictions of physical concepts get the differences between what is an explanation and what is a description muddled up, especially within pop-sci publications. We do not yet know the underlying mechanism of why spacetime is what it is, but we do have an excellent description of its features. To fully understand why spacetime gives rise to the phenomenology we see, weâd have to figure out first how spacetime itself comes to be, and if it can be broken down further into more fundamental concepts. Such attempts are under way, but at present they are just ideas and conjectures.

1 hour ago, Lorentz Jr said:

They're not physics.

I disagree. Physics makes models of the world around us, but not all of these models purport to be a fundamental explanation in ontological terms. As such, SR is a very good model that is in excellent agreement with experiment and observation. Itâs just important to not confuse a model with an (ontological) explanation, because they are not the same.

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53 minutes ago, Markus Hanke said:

clocks always tick at the same rate within their own frames.

Who cares? Clocks are also motionless in their own frames. One of the most important lessons of Physics 101 is that you can't directly compare quantities measured in different reference frames. It's apples and oranges.

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So to make a long story short, we donât have an explanation of why this happens, only a description of it.

Right. That's my point.

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SR is a very good model that is in excellent agreement with experiment and observation.

So is Lorentz's "aether" theory, but everybody says that's been "discredited".

Feel free to ask that this thread be moved to the Speculation category if you want, Markus.

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11 minutes ago, bangstrom said:

The twins paradox still includes a time dilation as a part of the explanation.

So time dilation is the explanation of time dilation.

Okay, thanks. đ

Edited by Lorentz Jr
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46 minutes ago, Markus Hanke said:

Time dilation is a relationship between reference frames, and not something that physically âhappensâ to a single clock. Asking for a mechanism that âslows downâ some clock is thus meaningless - clocks always tick at the same rate within their own frames. So the correct question would be why inertial frames are related via hyperbolic rotations in spacetime - thatâs a very valid question, but it isnât one that any of our present theories can answer. So to make a long story short, we donât have an explanation of why this happens, only a description of it. Thatâs not the same thing at all.

2 hours ago, Lorentz Jr said:

But what does it mean in Minkowski space?

The length of a world line between given events in Minkowski spacetime is defined to be equivalent to the proper time of a clock travelling between these events that traces out that world line. In other words, itâs simply the total elapsed time thatâs physically measured on a clock that travels along a specific spatial path between event

I would like to add to the second answer to the question "what does it mean....?"

Minkowski imposes a cartesian coordinate system, which as Eddington pointed out a century ago, actually adds unneccessary mathematical structure.

The same result could be achieved by a set of events and a set of the invariant intervals between them.

This would form a network of linked events, with no cartesian structure superimposed.

The sum of these invariants connecting any pair of eventsÂ  as a mathematical 'graph' (since this thread is partly about geometry, whch includes geometrical graph theory) then includes the shortest path.

Ref

Eddington The Mathematical Theory of Relativity pages 8 - 16, Cambridge 1923Â  (my ed 1954)

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2 hours ago, Lorentz Jr said:

Who cares? Clocks are also motionless in their own frames.

Yes, thatâs precisely my point. It is meaningless to speak of length contraction and time dilation âhappeningâ to rulers or clocks. Itâs always a relationship between two rulers, or two clocks.Â

2 hours ago, Lorentz Jr said:

So is Lorentz's "aether" theory

No. That would be like saying that a topographical map of your local area relies on an âaetherâ just because it uses a coordinate grid. To be sure, you can make that claim without affecting the usefulness of the map itself, if you so wish, but it doesnât add anything to the information contained therein.Â

Spacetime is just the same - itâs quite simply a map of events that allows you to determine separations and angles. There is no implication that we need to reify this into some kind of physical substance.

2 hours ago, Lorentz Jr said:

One of the most important lessons of Physics 101 is that you can't directly compare quantities measured in different reference frames. It's apples and oranges.

This guy disagrees:

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2 hours ago, studiot said:

The same result could be achieved by a set of events and a set of the invariant intervals between them.

This would form a network of linked events, with no cartesian structure superimposed.

The sum of these invariants connecting any pair of eventsÂ  as a mathematical 'graph' (since this thread is partly about geometry, whch includes geometrical graph theory) then includes the shortest path.

Yes! Very important observation +1

This very lucidly demonstrates why a specific choice of coordinate system cannot carry physical relevance, so far as the form of physical laws is concerned.

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4 hours ago, Markus Hanke said:

clocks always tick at the same rate within their own frames.

3 hours ago, Lorentz Jr said:

Who cares? Clocks are also motionless in their own frames.

42 minutes ago, Markus Hanke said:

Yes, thatâs precisely my point. It is meaningless to speak of length contraction and time dilation âhappeningâ to rulers or clocks. Itâs always a relationship between two rulers, or two clocks.

And the relationship between the clock on Earth and the ones on the spaceships is that the latter run more slowly on average than the former. That will be true as measured by any observer in any inertial reference frame, as long as they use the same reference frame for all measurements, i.e. compare apples to apples.

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3 hours ago, Lorentz Jr said:

One of the most important lessons of Physics 101 is that you can't directly compare quantities measured in different reference frames. It's apples and oranges.

42 minutes ago, Markus Hanke said:

This guy disagrees

????????????????????????????

Is that the Relativity Police?

Are they going to arrest me for calling them out on an elementary Physics 101 mistake? đ±

Edited by Lorentz Jr
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1 hour ago, Lorentz Jr said:

And the relationship between the clock on Earth and the ones on the spaceships is that the latter run more slowly on average than the former. That will be true as measured by any observer in any inertial reference frame, as long as they use the same reference frame for all measurements, i.e. compare apples to apples.

But they don't.Â  The travelers in the Triplet scenario exist in two inertial reference frames, while the earth observer exists only in one.Â  The only reference frame an observer "uses" is her own. The point of the fly-by scenario is to remove deceleration from the picture and get to the core of the paradox and resolve it with the multiple IRFs flashing digital readouts to each other.Â  As long as we're in effectively flat Mink spacetime, with no large masses around, that's all that's needed.Â  Maybe I'm not really understanding your problem with that, but you seem to think Don hasn't really explained it.Â  If I skimmed past something my apologies - I was probably too near c.

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11 hours ago, Markus Hanke said:

So the correct question would be why inertial frames are related via hyperbolic rotations in spacetime - thatâs a very valid question, but it isnât one that any of our present theories can answer. So to make a long story short, we donât have an explanation of why this happens, only a description of it. Thatâs not the same thing at all.

It's easy to explain the hyperbolic rotation relation in terms of other things. If you assume that the speed of light is constant, and consider a path of light c*tau long in one frame, then consider it in another frame (as commonly done with a light-clock at rest and moving, in thought-experiments used to explain time dilation), chosen to form a right triangle with a sides x and c*tau and hypotenuse c*t (right? am I messing up the details?). Then by the Pythagorean Theorem, the value of x^2 - (ct)^2 is constant for a given tau. That equation happens to describe a hyperbola. You don't need to explain why it's described by a hyperbolic rotation, if you instead explain why the speed of light is constant and why Pythagoras' theorem holds. "Hyperbolic rotation" is just an equivalent mathematical description of the relationship. But likewise, a constant speed of light and Pythagorean theorem could also be just descriptions of what's happening, equivalent to some other things. Maybe there's some simplest explanation of what's "really" happening, or maybe they're all just equivalent descriptions of what we observe.

I think we have the same point, that you don't need to explain a given description of a phenomenon to explain the phenomenon itself. Likewise, you don't need things to "really" be rotating for a mathematical equation for rotation to be a valid description.

Edited by md65536
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5 hours ago, TheVat said:

The point of the fly-by scenario is to remove deceleration from the picture

I know. That's why I introduced it.

The video is ancillary material in support of a footnote, not the main topic of the thread.

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But they don't.

How is this implied by anything you wrote in the rest of your post? It all seems like a humongous non sequitur.

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If I skimmed past something my apologies

That's okay, I do it all the time. đ

Edited by Lorentz Jr

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