# SR: which clock was slower?

• I
Gold Member
I don't know much about the math of SR, but this is what's bothering me: if a moving clock B ticks slower than the stationary one I have (A), then from the viewpoint of B, my clock (A) is ticking slower. So if we turn around and meet each other in the middle, which clock was slower than which?

Math is no problem if you want to use it in this thread.

## Answers and Replies

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jedishrfu
Mentor
This is the twin paradox restated.

Here's a more detailed but non-math explanation:

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PeroK
Homework Helper
Gold Member
I don't know much about the math of SR, but this is what's bothering me: if a moving clock B ticks slower than the stationary one I have (A), then from the viewpoint of B, my clock (A) is ticking slower. So if we turn around and meet each other in the middle, which clock was slower than which?

Math is no problem if you want to use it in this thread.
There's no such thing as a "moving" clock or a "stationary" clock. Motion is relative. You can have a clock that is moving inertially: i.e. it is not accelerating. And you can have a clock that is accelerating: i.e. subject to a real force.

If you define an IRF (inertial reference frame), then you can have a clock that is stationary wrt that frame and a clock that is moving wrt that frame. But, in any other IRF the state of motion of the two clocks will be different. There is, therefore, no absolute state of motion.

If two clocks are moving inertially relative to each other, then both are measured to run slow in the inertial frame in which the other clock is at rest.

Suppose the clocks are ##A## and ##B## and they start at the same location, and some time later meet up again. Choose any IRF and compute the following quantities (where the clocks meet at ##t = 0## and again at ##t = T##, as measured in that IRF):
$$\tau_A = \int_0^T \sqrt{1 - v_A(t)^2/c^2} \ dt, \ \ \ \tau_B = \int_0^T \sqrt{1 - v_B(t)^2/c^2} \ dt$$
Where ##v_A(t), v_B(t)## are the speeds of the clocks as measured in the IRF. This gives the "proper" time of each clock (##\tau_A, \tau_B##) and is the time interval recorded on each clock between the two meetings.

Grasshopper and etotheipi
Gold Member
I am still grapling with this; to put it very simple: in my example, either A aged more than B, or B aged more than A, right? I mean, when they meet. It is not symmetrical, so there must be a variable that makes the difference!

That said, acceleration or jumping reference frames (video) could be such a variable.

PeroK
Homework Helper
Gold Member
I am still grapling with this; to put it very simple: in my example, either A aged more than B, or B aged more than A, right? I mean, when they meet. It is not symmetrical, so there must be a variable that makes the difference!

That said, acceleration or jumping reference frames (video) could be such a variable.
If all you know is that they separated and came back together, then they could be the same or one could show less time than the other. The variable is the speed profile of the two clocks (as measured in any IRF).

Gold Member
If all you know is that they separated and came back together, then they could be the same or one could show less time than the other. The variable is the speed profile of the two clocks (as measured in any IRF).
But if I do calculations with IRF where A is stationary and B is moving, I should get the opposite answer if I use the IRF where B is stationary and A is moving, while both are (can be) the exact same situation, right? In the first case A is older and in the second case B is older.

PeroK
Homework Helper
Gold Member
But if I do calculations with IRF where A is stationary and B is moving, I should get the opposite answer if I use the IRF where B is stationary and A is moving, while both are (can be) the exact same situation, right? In the first case A is older and in the second case B is older.
If they separate and then meet again they can't both be inertial.

Gold Member
If they separate and then meet again they can't both be inertial.
No, suppose only one of the two is moving, we get different results when viewed differently.

Ah, the other frame is not inertial, you mean?

Wait, now I'm really confused!

PeroK
Homework Helper
Gold Member
No, suppose only one of the two is moving, we get different results when viewed differently.

Ah, the other frame is not inertial, you mean?
You must be precise about the motion of both clocks. If you establish that one clock was inertial throughout the experiment, then you could use that clock's rest frame as a convenient IRF. The inertial clock must show more time than the other.

But, if neither is inertial, then you need to specify more precisely what is happening.

They can't both be inertial. If they are both inertial then they can only meet once.

Note: we are talking SR here (flat spacetime).

russ_watters and entropy1
phinds
Gold Member
2019 Award
No, suppose only one of the two is moving,
As has already been pointed out (see post #3), this is a meaningless statement. Motion is relative.

What is NOT relative is acceleration, so any differences in their clocks when they meet back up will be based on which one accelerated.

EDIT: If both accelerated, then it gets slightly more complicated but that's just math.

Janus
Staff Emeritus
Gold Member
But if I do calculations with IRF where A is stationary and B is moving, I should get the opposite answer if I use the IRF where B is stationary and A is moving, while both are (can be) the exact same situation, right? In the first case A is older and in the second case B is older.
In order for A and B to meet up again (so that they can directly compare clocks), one or the other has to have changed velocity. Let's say it's B.
As far as A is concerned, B travels away at some speed, turns around and returns at that same speed. All A needs to know to work out how much time passes for B is what speed B was traveling and for how long. (the fact that B spent some time slowing down and then speeding up again in the reverse direction will have some small effect on the total time, but as far as A is concerned, the rate at which B's clock ticks only depends on B's relative speed with respect to A at any given moment. In other words, other than the change in B's speed, the acceleration B is undergoing adds no additional effect).
For B, the above applies during the two legs of his trip. When he and A are separating at a constant speed or approaching at a constant speed, he would measure A's clock running slow at a rate dependent on their relative speed.*
Where B's observations differ from A's is during that period when B is reversing direction and thus changing his own velocity. This is when B is non-inertial. And measurements made from non-inertial frames are not as simple as those made from inertial ones.
During this period, it is not enough for B to know the relative speed between A and himself to determine how fast A's clock is ticking. He also has to factor in the distance to A and how he is accelerating with respect to A.
By transitioning from going away from A to approaching A, he is accelerating towards A, And this causes him to determine that A's clock runs fast by a factor that depends on the magnitude of the acceleration and the distance between A and himself. In other words, B's acceleration does effect how B measures A's clock.
The result is the B would measure A's clock running slow on the outbound trip, running very fast during B's turn-around phase, and then runs slow during the return leg. The end result after returning to A is that more time has accumulated on A's clock than B's clock.
So, while during different points of the trip, A and B will disagree as to what their respect clocks are doing at any moment, when they meet up again, they agree as to how much time has accumulated on each of their clocks.

* And by "measure", I mean what they would determine after accounting to light propagation delay.

PeterDonis
Mentor
2019 Award
if I do calculations with IRF where A is stationary and B is moving, I should get the opposite answer if I use the IRF where B is stationary and A is moving
If A and B separate and then meet up again, and spacetime is flat (so SR applies), it is impossible for there to be a single IRF in which A is always stationary and B is moving, and also a (different) single IRF in which B is always stationary and A is moving. Only one of them can be stationary in the same single IRF the whole time. And that one will be the one who ages the most.

Gold Member
If A and B separate and then meet up again, and spacetime is flat (so SR applies), it is impossible for there to be a single IRF in which A is always stationary and B is moving, and also a (different) single IRF in which B is always stationary and A is moving. Only one of them can be stationary in the same single IRF the whole time. And that one will be the one who ages the most.
But you can apply both of those examples alternately to a single real-life situation, right? Only from a different vantage point (IRF). In real life A is younger than B or the other way round.

Ibix
But you can apply both of those examples alternately to a single real-life situation, right?
I don't quite understand what you mean. In a real life situation, at least one of A and B is not moving inertially. Thus you can have a situation where A is inertial and B is not - in that case, there's an inertial frame where A is always at rest but no inertial frame in which B is always at rest. Or you can have a different situation where B is inertial but A is not - in that case, there's an inertial frame where B is always at rest but no inertial frame in which A is always at rest.

But these are two different scenarios.

entropy1
PeterDonis
Mentor
2019 Award
you can apply both of those examples alternately to a single real-life situation, right?
I don't understand what you mean. It is impossible to have a single real-life situation where both A and B are stationary in a single IRF the whole time. Only one of them can be.

In real life A is younger than B or the other way round.
Yes, and the one that ages more will be the one who is stationary in a single IRF the whole time. Only one of them can be.

entropy1
Mister T
Gold Member
But you can apply both of those examples alternately to a single real-life situation, right?
No, you cannot. One clock changes direction and one clock doesn't.

To fully understand this you need to grasp the relativity of simultaneity.

entropy1
Gold Member
So one of the frames is not inertial. But isn't that dependent on which of the frames (A/B) is considered inertial?

Staff Emeritus
2019 Award
You don't get to pick who is inertial.

Gold Member
You don't get to pick who is inertial.
So where does that depend on? (I guess no acceleration)

PeroK
Homework Helper
Gold Member
So where does that depend on?
Real forces! Newton's laws.

PeterDonis
Mentor
2019 Award
where does that depend on? (I guess no acceleration)
No proper acceleration--the one who never feels any force (no rocket engine firing) is the one who is inertial the whole time.

entropy1
Gold Member
No proper acceleration--the one who never feels any force (no rocket engine firing) is the one who is inertial the whole time.
But then we get to my point: SR clock differences should be the result of velocity, but actually in this twin paradox, it seems to me dependent (also) on acceleration! (who is accelerating delivers an asymmetry)

PeterDonis
Mentor
2019 Award
SR clock differences should be the result of speed, but actually in this twin paradox, it seems to me dependent (also) on acceleration!
Neither of these is correct.

The result of relative speed is time dilation, but time dilation is not an invariant. It's frame-dependent.

The result of acceleration is that two observers who already met once in flat spacetime can meet again; in flat spacetime that is impossible unless one of them accelerates. But the acceleration itself does not affect the rate at which their clocks tick.

The difference in the elapsed time for the two observers when they meet up again is due to the difference in lengths of their paths through spacetime. In other words, it's geometry. It's not that one clock ticked slower than the other: both clocks tick at one second per second. But the path through spacetime that one clock takes is fewer seconds long.

It's the same as if two cars set out from city #1 to city #2 taking two different routes that are different lengths. The elapsed distance on their odometers will be different when they meet up again at city #2, but that's not because either odometer was "ticking" distance at a different rate. It's because the paths they took have different lengths.

entropy1
PeroK
Homework Helper
Gold Member
But then we get to my point: SR clock differences should be the result of velocity, but actually in this twin paradox, it seems to me dependent (also) on acceleration! (who is accelerating delivers an asymmetry)
Ultimately, if you have a real physical clock, then the speed (relative to a given IRF) is determined by the initial velocity and the subsequent acceleration profile. The calculation simply uses the speed - that is all you need to calculate ##\tau##.

entropy1
Ibix