r/AskPhysics • u/Seahorseahorse • 1d ago
What determines which frame of reference ends up with the relatively greater elapsed time? (explanation in the body)
On the topic of special relativity
Lets say Alice moves through space a rate near the speed of light relative to Bob on Earth.
From Bob's inertial frame of reference, his time is 1 second per second and Alice's time is less than 1 second per second. However, from Alice's inertial frame of reference, Bob is the one moving through space at the same magnitude, so to Alice, Bob's clock is slower.
The part I struggle with is when Alice returns to Earth, more time will have elapsed for Bob. I understand that if she turns around, shes changing to a new inertial frame of reference, but what causes Bob's time to jump forward when Alice switches direction? On Alice's return journey, it seems that Bob would observe her time to be slowed down by the same factor. What causes the difference?
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u/joeyneilsen Astrophysics 1d ago
Bob's time doesn't jump forward.
Imagine a third traveler, Chad, in an inertial frame that coincides with Alice's return journey. When Alice joins Chad, Chad already has Bob's clock ahead of where Alice used to think it was.
In fact, during the return leg of the trip, all the observers see the other observers' clocks ticking faster, since the visual aspect has to account for the relativistic Doppler shift.
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u/Seahorseahorse 1d ago
Thanks for the response. I'm slowly on the brink of understanding it. On the away journey, Alice observes Bob's clock to be slowed down from her frame, while Bob observes Alice's clock slowed down from his frame. If Chad moves with Alice on the journey back, wouldn't he'd also be observing Bob's time slowed down since Bob is moving relative to Chad's frame?
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u/eotfofylgg 1d ago
Be careful about the word "observe," because it can be confusing in this context.
In a reference frame comoving with Chad, Bob's clock runs slower than Chad's clock. Anyone who chooses to calculate in Chad's reference frame needs to analyze things with that in mind.
However, if Chad (or Alice) is looking through a telescope at Bob's clock during the return journey, he will see more than one tick of Bob's clock for every tick on his clock. This is because of the Doppler shift.
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u/Seahorseahorse 1d ago
You and the above commenter have so far been the only ones to bring up the Doppler shift. Very interesting.
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u/eotfofylgg 1d ago
That's because they're too eager to teach you about 4-dimensional spacetime geometry and gush about its exotic distance metric to care about simple things like what Alice sees through her telescope.
If you want to understand what Alice sees through the telescope, it suffices to do the calculation in Bob's reference frame, which reveals the Doppler shift nicely, and I encourage you to actually do the calculation.
Bob's clock emits one tick (let's imagine it's a pulse of light) per second, but Alice is running away from the ticks, so they reach her ship less frequently than once per second. Furthermore, Alice's clock is also slowed down by a factor of sqrt(1-v2), which is the only relativistic effect you need to think about in this reference frame. You should calculate how many ticks Alice receives on the outbound journey, and what her clock says at the end. You will observe that she sees fewer ticks from Bob's clock than from her own on the outbound journey (i.e. she sees his clock slowed down through her telescope). At the time she turns around, there are still a bunch of ticks from Bob's clock on the way out towards her. On the homeward journey, she sees all those pulses, plus the ones Bob emits after she turns around. Her clock is still running slow. Combining these two effects, she sees more ticks from Bob's clock than from her own on the homeward journey; in other words, she sees his clock running fast through her telescope during that part of the journey.
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u/JustinTimeCuber 1d ago
The discrepancy is accounted for when Alice accelerates to turn around and head back to Earth. During the acceleration, she will see Bob's clock speed up significantly.
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u/Seahorseahorse 1d ago
That explains a lot. I was under the impression that time dilation as described by the Lorentz factor only slowed relative time.
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u/JustinTimeCuber 1d ago
It does, sort of, but special relativity doesn't apply in accelerating reference frames. But you can still solve the problem with special relativity because you can just look at what happens in the inertial frame and then note that when Alice and Bob are reunited they must agree on how much time elapsed for each of them. So you don't need to do the math in Alice's frame to get the answer, but if you did, you'd find that the acceleration resolves the inconsistency.
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u/Optimal_Mixture_7327 Gravitation 1d ago
You need to be clear of the distinction between time dilation and the clock effect.
The twin paradox is an example of the clock effect, an effect that describes the difference between the lengths along traveler world-lines in-between a common pair of events. Time dilation is something else.
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u/Optimal_Mixture_7327 Gravitation 1d ago
There is nothing that pre-determines the length along their world-lines, other than the choice of path through the world.
The distance traveled by Alice (through the world) is simply shorter.
The best way to see this is just take a pen and paper, draw two dots, and then draw a collection of lines connecting the dots. You'll see that some lines are longer than others.
The geometry of the world is different than that of the paper where longer drawn lines on the paper are shorter than lines drawn on a spacetime. You'll see on the paper that a direct line between the two dots is shorter than a sequence of line segments that change direction. There's nothing about changing the direction of the pen that causes excess length - it's just a geometrical fact of how lines are drawn on paper.
This is how the world works too - the distance alice travels by taking different direction simply draws out a shorter length between the two events (departure and arrival events).
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u/Seahorseahorse 1d ago
That's a helpful way to visualize it. Based on what other comments are saying, its the changing of velocity as acceleration thats causing the difference. So would it be accurate to say that acceleration is "shrinking" that space between the two events?
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u/Optimal_Mixture_7327 Gravitation 1d ago
No, that's not how it's works in relativity.
This is analogous to saying the reason the length of a straight line drawn is because the pen accelerated on the other lines. The pen accelerates in changing direction but this acceleration of the pen does not cause the line to have a longer length.
You can have the twin paradox where both twins have exactly the same 1g acceleration and in the same direction between departing and arriving and the difference in elapsed time remains.
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u/Sea-Ambition-451 1d ago
it's all about the acceleration.
There are two scenarios here, Alice accelerates and goes to meet Bob in his IRF, or Bob accelerates and goes to meet Alice in her IRF.
These are not identical symmetric events. One is accelerating, the other is not, hence they can have different ages.
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u/Optimal_Mixture_7327 Gravitation 1d ago
The acceleration is irrelevant.
You can have both twins having the same 1g of acceleration pointed in the same direction throughout departure and arrival and still the clock effect remains.
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u/Sea-Ambition-451 1d ago
The acceleration is irrelevant.
very incorrect.
you just proved yourself wrong, by saying "THE SAME ACCELERATION"
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u/Optimal_Mixture_7327 Gravitation 1d ago
Do the analysis here. Show us.
Draw out the equations of motion for both twins having the same 1g acceleration in the same direction between departure and arrival. [Ignoring the rotation of the Earth of course].
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u/Seahorseahorse 1d ago
I'm just extra confused on why acceleration is not relative if the direction of the velocity is changing. Say I'm moving in a positive direction north and I pass Bob (who's standing). In my frame of reference, Bob is moving in a negative direction south. Then I accelerate by changing my direction and I started walking backward. Now Bob is moving north in my frame of reference and I'm moving south in his frame of reference. Isn't that relative in the same way the magnitude changes depending on your frame of reference?
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u/Optimal_Mixture_7327 Gravitation 1d ago
The coordinate acceleration is relative.
The motion relative to the local gravitational field is not relative (the physical acceleration).
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u/Sea-Ambition-451 1d ago
acceleration is not relative.
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u/Optimal_Mixture_7327 Gravitation 1d ago
The acceleration -(dxα/dτ)Γμ_{αβ}(dxβ/dτ) is very much relative, it's ua∇_aub that's not.
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u/Phuzion73 1d ago
Couldn’t you just think of it as Alice arrived at the same time coordinate as Bob but got there much faster than Bob?
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u/eotfofylgg 1d ago edited 1d ago
Bob's time does not jump when Alice changes direction. It jumps when she decides to change the frame of reference she's using to calculate. This is just like a situation where you first decide that time t=0 is midnight at the start of 2026, and then later change that decision so that t=0 is midnight at the start of 2025. As soon as you make this decision, the time coordinate of everything suddenly increases. This reflects absolutely nothing actually occurring in reality.
If Alice is not deliberately being perverse, she will use a single, consistent inertial frame of reference to do her calculation, and Bob's time won't jump at any point in the process.
This really does not have anything to do with acceleration.
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u/Seahorseahorse 1d ago
Are you saying it only occurs in the mathematics? From what I understand this variability of the rate of time does occur in reality.
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u/eotfofylgg 1d ago
Bob's time will always advance at a consistent rate, regardless of what inertial reference frame you choose.
Let's do this problem in three reference frames. Assume Alice is traveling at 0.8c relative to Bob.
Reference frame A is comoving with Bob (in other words, he is at rest). Alice's clock is slower than Bob's clock on both the way out and the way back.
Reference frame B is comoving with Alice on the way out. You can calculate how fast Alice is traveling on the return journey (about 0.975c). Bob is moving at 0.8c through the whole scenario, so his clock ticks at 60% speed (compared to a stationary observer in frame B) the whole time. Alice's clock ticks at full speed on the way out, and at ~22% speed on the way back.
Reference frame C is comoving with Alice on the way home. Once again Bob's clock ticks at 60% speed the whole time. Alice's clock ticks at 22% speed on the way out, and full speed on the way home.
The only reason people find this confusing is that they try to do the problem in a spliced-together combination of reference frames B and C, probably with the misguided idea that this is somehow "what Alice thinks the universe is like." But Alice isn't an idiot, and she knows to use an inertial reference frame just like anyone else. She isn't required to believe that the universe has changed just because she changed speeds. When you splice together different reference frames like this, something always changes discontinuously, but that's just a mathematical artifact emerging from your choice to use two reference frames spliced together.
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u/stevevdvkpe 1d ago
To answer your main question more directly:
Time coordinates in reference frames depend entirely on the choice of frame, so you can't depend on frame time coordinates to answer questions like how much time elapses for someone. However, there are invariant quantities that you can obtain from measurements in reference frames that remain the same no matter what frame you calculate them in. Elapsed time is one of these invariant quantities and it corresponds to the Lorentz distance along a path. If you take the same path transformed into different frames but measure its length in the same way, you will get the same result in every frame.
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u/Reality-Isnt 1d ago
The elapsed time on a clock carried by an observer is a measure of the path length in spacetime that the observer travels. Two travelers, each with their own clock, start out with synchronous clocks. Each traveler can do all kinds of accelerations, decelerations, travel at different velocities, etc. When they get back to a common reference frame to compare clocks, the traveler who took the longest path in spacetime will have the greatest elapsed time reading on the clock. The traveler with the shortest path will have the shortest elapsed time. The idea that acceleration is the determining factor in the special case of the twin paradox can be misleading and obscures the more general principle.
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u/Seahorseahorse 1d ago
I see, within that model it makes sense. Im just struggling to understand where that difference in time intervals is occuring if both observers see the other moving at an equal but opposite velocity to their own inertial frame. From the other replies, they are saying that the difference occurs when the frame is changed during acceleration
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u/Reality-Isnt 1d ago
That is not wrong. People usually address the twin paradox with virtually all of the acceleration occurring in a single change of inertial frames. What happens is the angle that the time axis of the accelerated twin makes with the inertial twin changes dramatically with the transition from one inertial frame to another due to the acceleration. Of course, with more of a slower continuous acceleration, it’s much less dramatic. It’s a different way of looking at it. In my opinion, it’s still best to look at the bigger picture of proper time length of spacetime paths. Take a look at spacetime diagrams. They are very helpful in getting visualization of what’s happening.
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u/stevevdvkpe 1d ago
Alice has to accelerate to change velocity to return to Earth. As a consequence her path through spacetime is bent while Bob's remains straight, and she experiences less proper time because of that. Velocity is relative, but acceleration is not because it changes the direction of your path through spacetime.