# Sabine Hossenfelder says time dilation is due to acceleration in the twin's paradox. Is this true?

Sabine Hossenfelder says time dilation is due to acceleration in the twin's paradox. Is this true?

At 12 minutes into this video https://www.youtube.com/watch?v=ZdrZf4lQTSg, Hossenfelder states, "This is the real time dilation. It comes from acceleration."

Looking at the equations for time dilation, time dilation comes from velocity, not acceleration.

How can Hossenfelder state, "This is the real time dilation. It comes from acceleration."?

• Can you move somewhere and back without acceleration? No. The point Sabine Hossenfelder makes is completely pointless. The twin paradox is simply Doppler plus signal delay. Commented Apr 22, 2023 at 16:43
• As I said in physics.stackexchange.com/a/562529/123208 Acceleration itself does not cause time dilation, only the velocity which arises due to that acceleration causes time dilation. This is known as the clock postulate Commented Apr 22, 2023 at 17:50
• Sabine is right in this sense: one of the two twins will be the younger one. Which one? The one that accelerated. The twin paradox is not a "real" paradox exactly because acceleration makes the two twins physically different. Related: physics.stackexchange.com/q/503069/226902 physics.stackexchange.com/q/53228/226902 Commented Apr 22, 2023 at 21:41
– hft
Commented Jan 28 at 18:07

This is a matter of terminology. The term time dilation can be used in two different ways and you and Hossenfelder are using it in those two different ways.

If you are moving at velocity $$v$$ relative to me then your clock runs more slowly than my clock, and we can define the time dilation as:

$$\frac{t_{me}}{t_{you}} = \frac{1}{\sqrt{1 - v^2/c^2}}$$

and I would guess this is what you are thinking of. Defined like this you are quite correct the time dilation depends only on the relative velocity $$v$$.

However what Hossenfelder is saying is that if you and I start at the same point and later meet at the same point then we can compare our clocks to see how much elapsed time we measured. Then time dilation means the difference in our clocks.

Now, unless we both just sat stationary in the same place the only way we can start together, separate, then meet up again is if one or both of us have accelerated. And roughly speaking the person who accelerated least measured most time on their clock. This is what Hossenfelder means by saying that acceleration causes time dilation i.e. it is the fact that one of us accelerated that caused our clocks to record different elapsed times.

If you are interested in a more rigorous explanation see What is time dilation really?, and if you're feeling really brave see What is the proper way to explain the twin paradox?

• So are you saying that acceleration causes time dilation in the twin's paradox, and thus one can ignore any time dilation due to velocities? Where is your equation that calculates the time dilation based on acceleration? Commented Apr 22, 2023 at 15:31
• @EpicMythology This is explained in detail in the questions I linked. The elapsed time is proportional to the length of the trajectory on a spacetime diagram, but due to the weird way relativity calculates lengths the line that looks longer to us is actually shorter. Acceleration causes the trajectory on the spacetime diagram to curve, which makes it shorter (even though it looks longer to us). Commented Apr 22, 2023 at 15:35
• I see that this answer was posted two hours before mine, though I somehow hadn't seen it. It pretty much renders mine superfluous. Commented Apr 22, 2023 at 18:40

Sabine is not just wrong, she is spectacularly wrong.

Here are 5 counter examples to Sabine's claim that time dilation and differential aging is caused by proper acceleration.

Counterexample 4 involves neither gravity nor circular motion. Only linear motion in flat spacetime is involved, and this is the final nail in the coffin of the "acceleration causes time dilation" argument.

Counter example 1:

This example comes from Mathpages. One projectile goes straight up, reaches apogee and falls back down to the starting point, while another projectile orbits the gravitational body. Neither projectile is under power. Both projectiles are free-falling during the experiment, so neither projectile experiences proper acceleration, but they experience different elapsed proper times. There is no proper acceleration anywhere in this experiment to explain the differential ageing, falsifying Sabine's claim.

Counter example 2:

One rocket hovers above a black hole using thrust to remain stationary, at a Schwarzschild radius greater than $$3GM/c^2$$. Another rockets orbit with its engine off. They both experience the same gravitational time dilation due to having equal altitude but the orbiting rocket experience additional time dilation due to its orbital velocity. In this case, the orbiting rocket is in free-fall so experiences no proper acceleration, while the hovering rocket does experience proper acceleration due to the thrust required to keep it stationary. In this case, the rocket with proper acceleration experiences less time dilation than the rocket in free-fall, in direct contradiction to Sabine's claims.

Counter example 3:

When particles are put in a centrifuge and spun up to high angular velocities, they experience extreme proper acceleration, yet all the time dilation (As measured in changes in half life) is due to only the tangential speed, not the acceleration. This "... has been verified experimentally up to extraordinarily high accelerations, as much as $$10^{18g}$$ in fact. See the Clock Postulate. by Baez.

Real experiments carried out by scientists confirm that acceleration has no effect on the half life of radioactive elements, in direct contradiction to Sabine's claims.

Counter example 4:

A slightly modified Twin's paradox. Consider the diagram below that was created by DrGreg of Physicsforums.

In this spacetime diagram, (time up, space across) twin A accelerates away from Earth and returns as in the usual Twin's paradox. Twin B also accelerates away from the Earth, but turns around sooner, returns to Earth and waits. Both twins experience exactly the same proper acceleration for exactly the same durations. The slower ageing of twin A that travels the furthest in the Earth frame can not be attributed to a difference in proper acceleration, in direct contradiction to Sabine's claims.

Counter example 5:

Consider 2 rockets following circular paths in space. The path of rocket A has a radius of 1 light-year and has a tangential velocity of 0.4c.
Rocket B's path has a radius of 16 light-years and a tangential velocity of 0.8c.

When their locations coincide, that is the start of the experiment. After rocket B has completed one revolution, it is back at the start and rocket A is also simultaneously back at the start, but has completed 8 revolutions.

The relativistic equation for centripetal acceleration using units such that c=1, is $$a = v^2/r \sqrt{1-v^2}$$ where v is the tangential speed as measured by an inertial observer that remains stationary at the mutual starting coordinate. The proper centripetal acceleration as measured by the rocket pilots is simply $$a' = v^2/r$$, the same as the Newtonian expectation. Rocket A following the smaller circular path experiences a proper centripetal acceleration (as measured by an onboard accelerometer) of $$v^2/r = 0.4^2/1 = 0.16 \ ls/s^2$$ while rocket B with the greater velocity following the larger circular path experiences a proper acceleration of $$v^2/R = 0.8^2/16 = 0.04 \ ls/s^2$$.

Rocket B experiences less proper acceleration but experiences the greater time dilation, in direct contradiction to Sabine's claims.

• These examples were much needed because they cut through all this linguistic/semantic sophistry. Example 4 is the clearest one because it stays within the confines of special relativity (which was the context of Sabine's video), which is what I was looking for on this page. Commented Jan 26 at 22:59
• Example 4 is intriguing but without further expalanation, its hard to take it as a "nail in the coffin" without me doing the math to verify if you're right or not (which I don't have the skills to do).
– B T
Commented Feb 15 at 20:10

You and Sabine are using the term 'time dilation' in different ways. I would say that Sabine is at fault here- for someone who aims to help people understand physics, she should have been more careful with her terminology.

Generally, the time interval between two events in flat space-time is 'path dependent'. If you coast directly from one event to another, the time interval will be longer than if you follow an indirect path, such as a curved or zig-zag path. The difference is a property of the geometry of spacetime. It is somewhat misleading to say that acceleration 'causes' the difference in elapsed time- it is better to say that the elapsed time is a property of the path you follow, and acceleration is just the means by which you follow a particular path.

There is a straightforward analogy with 3-D Euclidian space. The distance between two points in space depends on the path you follow from one to the other. It is shortest if you go in a straight line, and longer if you follow some zig-zag path. To follow a zig-zag path you have to accelerate, so in Sabine's terminology you might say that acceleration 'causes' the difference in path lengths- but it should be obvious to you that the length is a property of the path, and acceleration is just the means of following a particular path- acceleration doesn't 'cause' the path length to differ.

Finally, the term 'time dilation' is most commonly used to mean a special case comparing the time between two events that occur in the same place in one frame with the time between the same two events in some other frame where they occur in different places. By using the term 'time dilation' in a broader sense, Sabine is spreading confusion, in my view, as your question proves.

The answer to your question depends, of course, on what you mean by the phrase "time dilation".

I take it that "time dilation" refers to the fact that the same clock can be ticking at two different rates in two different reference frames, whereas "the twin paradox" refers to the fact that twins, present at the same place at the same time, can have two different ages.

With those definitions, time dilation is due entirely to velocity; acceleration is irrelevant. The twin paradox in special relativity is due to a combination of time dilation and acceleration. The twin paradox in general relativity, as AccidentalTaylorExpansion has explained, can be due to time dilation without acceleration.

I believe the definitions I've used are quite standard and widely used. Of course, there are always people who use language in quirky ways, and if your definition of time dilation differs from mine, then your answer to the question might differ.

While the twins are moving apart there is also time dilation. But, because they are in separate frames, these time dilations cannot be compared with each other. It is like comparing velocities in different reference frames. You think the train is going 120 km/h? Well, I think the train is going zero km/h.

While they are travelling at constant velocity they each see their other twins clock travelling at a slower rate than usual, but it is impossible to tell yet who ages more. It is only when the twins meet up that they can compare their clocks. One of the twins has aged more. How can that be? The problem is symmetric right? Each twin sees the other twin moving away, so you would expect the clocks to be equal. It turns out that one of the twins has accelerated and this has caused that twin to have aged less.

• @EpicMythology : Repeating your question after it's been answered is not a good way to get people to want to help you in the future. Commented Apr 22, 2023 at 15:18
• @EpicMythology Time dilation like in the twin paradox is caused by acceleration. The amount each twin has aged can be calculated using the proper time $c^2\Delta\tau^2=c^2\Delta t^2-\Delta x^2$. For the non-accelerating twin you get $\Delta\tau=\Delta t$. You can approximate the path of the second twin as two straight lines. This will give you $\Delta\tau=2\sqrt{(\Delta t/2)^2-\Delta x^2/c^2}=\sqrt{\Delta t^2-4\Delta x^2/c^2}$. This last quantity is less. There is also time dilation for moving observers. This does not depend on acceleration but it is a very real effect. Commented Apr 22, 2023 at 15:18
• So don't quote me on this, but handwavingly you could say time dilation is caused by velocity, but acceleration causes some of this time dilation to be carried over to other frames. Acceleration causes you to switch frames continuously. Proper time is a nice tool to keep track of this. Commented Apr 22, 2023 at 15:22
• @EpicMythology Not entirely. The time dilation that the twins experience when they get together is definitely due to acceleration, but saying that time dilation is solely caused by acceleration I would not agree with. In general, you can calculate the proper time of a path using $\tau=\int dt\sqrt{1-\dot x^2/c^2}$. Just plug in two paths $x(t)$ and then compare the proper times when they meet. Commented Apr 22, 2023 at 15:43
• @EpicMythology With the formula I gave you you can calculate time dilation in general. I don't know if there is a formula that only includes acceleration. I have a masters degree in theoretical physics. Commented Apr 22, 2023 at 15:55

No, you can still get a "twin paradox" without acceleration.

Although some solutions attribute a crucial role to the acceleration of the travelling twin at the time of the turnaround, others note that the effect also arises if one imagines two separate travellers, one outward-going and one inward-coming, who pass each other and synchronize their clocks at the point corresponding to "turnaround" of a single traveller. In this version, physical acceleration of the travelling clock plays no direct role.

The main reason is a change in inertial reference frame, which of course can arise from an acceleration, but it does not need to.

• @EpicMythology That's not what BioPhysicist is saying. If one (or both) twins accelerates then the difference in their proper times can be attributed to their acceleration. But you can also achieve the same effect without acceleration by changing reference frames. Commented Apr 22, 2023 at 16:33
• How in the world are you coming back to your starting point without acceleration? That is a silly idea. It is still not the acceleration that causes the effect of less aging in the travelling twin. That is caused by Doppler effect and signal delay. Commented Apr 22, 2023 at 16:49
• @EpicMythology Time dilation, as it is commonly known and presented, is due to velocity. The asymmetric time difference between the two twins when one returns to Earth is in part due to acceleration (within the specific thought experiment setup... there are variations that take out acceleration). If there is any disagreement, it is over the wording that Sabine Hossenfelder used. Commented Apr 22, 2023 at 17:44
• @FlatterMann There is still something to be learned from this variation. It shows us that acceleration is sufficient but not necessary. And from there it shows us we can search for something more fundamental that is both necessary and sufficient. Commented Apr 23, 2023 at 2:59
• @FlatterMann Thanks for the input. I just answer the questions though Commented Apr 23, 2023 at 3:52

In special relativity you can't compare the clocks without acceleration, but in general relativity there are examples without acceleration as well, for example on a round trip with constant velocity in a closed universe or between a twin on a circular and one on an elliptic orbit who also both experience no proper acceleration, but have different proper times everytime their paths cross.

By the way, here is a rebuttal to Sabine Hossenfelder's video in question.

• "you can't compare the clocks without acceleration" You can send light signals. Commented Apr 22, 2023 at 15:18
• @Maximal Ideal: sure, but we are talking about a comparison that leads to the same result in all frames of reference, and only the values you compare when you meet each other are invariant. If you just send light signals they won't agree who is younger and who is older, then both will say the other one's clock runs slower. Commented Apr 22, 2023 at 15:20
• The term "time dilation" does not even refer to the twin paradox. It refers to the instantaneous difference between the rate of two clocks in the same place that happen to move against each other with a velocity v. What most people don't realize is that this is not what we are actually observing. We are always observing Doppler, i.e. the visible clock rate will change depending on whether the other clock is coming towards us or moving away from us. Lorentz transformations are great if you want to know how fields behave, but they are the wrong approach for this scenario. Commented Apr 22, 2023 at 16:47
• @FlatterMann: the title of the question is obviously about the twin paradox, so my answer is as well. Commented Apr 22, 2023 at 22:44
• @Qmechanic removed the general relativity tag, so my answer and the previous comment about the time dilation being dt/dτ=√(gᵗᵗ)/√(1-v²/c²) becomes obsolete, but I'll leave it up nevertheless since it already got some upvotes. Commented Apr 23, 2023 at 2:58

Compare it to distances in space.

Suppose we take a journey from New York to Washington D.C. (in the U.S.A.) One route goes in a straight line. Another route goes via San Francisco (on the west coast). The second route is a lot longer. Why? Because it is longer. What else can one say?

But if you want to spot some property of the second route which gives a clue to the fact that it is longer, you can notice that it includes a change of direction (the turn-around at San Francisco). So you might say that it is longer "because" it has this change of direction. That is essentially what is going on when people focus on the acceleration part of the twin paradox.

In the spacetime case the worldline with the turn-around is the "shorter", in the sense of having less proper time, as compared with a straight worldline. The reason it has less proper time is that if you add up all the little bits of proper time along that worldline then the sum total is smaller than along the straight worldline. The straight worldline has the most proper time (between any given pair of timelike-separated events in a spacetime without curvature).

Instead of twins, think of triplets; one goes to Mars, one goes to Jupiter and one stays home. The Mars and Jupiter triplets accelerate exactly the same to the same top speed of 1/2 c, and they turn around at their planet in exactly the same manner. Any explanation that works on the basis of acceleration or of turning around would have to state that the Mars and Jupiter triplets would return and be the same age as each other. But this is simply not true. The Jupiter triplet would be younger than the Mars triplet or the Earth triplet. Special relativity time dilation is only concerned with velocity and the amount of time spent at that velocity. Acceleration or turning around has absolutely no impact on special relativity time dilation and should not be considered at all.

Physics isn't due to mathematics: it happens whether you calculate or not. The math is a story we tell about the physical phenomena. Of course, it's a very credible and effective story because we check it against reality. But in math, there is generally more than one way to tell the story. Unless the ways have different consequences for some experiment, physics can't decide which is the correct telling.

Oh, this is attracting good answers. Yukterez is giving an example from GR. AccidentalTaylorExpansion's answer is the immediately relevant one.

But I want to expand upon ATE's answer. Yes, there is no comparison if you never meet up again to compare. But to meet up, at least one of them must have accelerated (under SR). If it were completely symmetric, there will be no way to know which one of the pair should be aging. Thus acceleration is necessarily the cause of the time dilation.

However, this is still a velocity issue, not an acceleration issue. This is because you can change the importance of the velocity part v.s. the acceleration part simply by changing the wait time during the constant velocity coasting parts. This is particularly clear if you look at the Minkowski diagram---the acceleration part can be reduced to an arbitrarily small contributor, except that it is also the thing that dumps all the constant velocity accrued time dilation onto the poor twin. Which is why it is ok to study time dilation under basic SR and not have to only be considered using GR.

• So then you disagree with Sabine Hossenfelder who says time dilation is due to acceleration in the twin's paradox, stating "This is the real time dilation. It comes from acceleration." Commented Apr 22, 2023 at 15:38
• No, I am agreeing with basically everybody. Time dilation is an SR effect, and the one that determines who gets the time dilation factor thrown on them, is only to be decided by acceleration. Commented Apr 22, 2023 at 15:49
• Time dilation is NOT the lesser aging of a traveling twin in the twin paradox. It's a term reserved for the effect of the Lorentz transformation on clock rates. The Lorentz transformation simply doesn't apply here because the two clocks are not in the same location. The lesser aging in the twin paradox is caused by Doppler effect (red shifted on the way out, blue shifted on the way in) and signal delay (the Doppler signal changes immediately for the traveling twin upon reversal of his velocity but it takes a while for that to propagate to the stationary twin because of signal delay). Commented Apr 22, 2023 at 16:52