Examining Time Dilation and the Effects of Velocity on Perception of Time

[Nugatory]

This line is different for stationary objects and objects moving.

What are these "stationary objects" that you're talking about? What does it mean to say that something is "stationary"?

If you are standing next to a road and a car drives by with its speedometer reading 100 km/hr, does that mean that you are stationary and the car is moving at 100 km/hr? Is it not just as reasonable to consider the car to be stationary while you and the road are moving backwards at 100 km/hr?

Before you answer those questions, consider that the surface of the Earth is moving at several thousand km/hr because of the Earth's rotation, the earth,, the road, and the car are all moving around the sun at several km/sec, the sun itself is orbiting the center of our galaxy, our galaxy is moving towards the Andromeda galaxy at several hundred km/sec, both galaxies are part of a larger galactic cluster that is moving through space, ...

The point here is that it never makes sense to say that something is moving or not moving, or moving at speed V, unless you also say what that speed is relative to.

 

[Herman Trivilino]

So if 60 seconds passed for B, 100 seconds passed for A. How does that imply that time slowed down for A?

If you measure 60 seconds of elapsed time between two events that occur in the same place, but someone's clock measures 100 seconds of elapsed time between those same two events then they would conclude your clock is running slow.

One thing that can help sort this out is the notion of proper time. If the two events occur at the same location, then the time that elapses between them is called a proper time. For an observer moving relative to those two events, imagine a reference frame in which he is at rest. In that moving frame those two events will occur at different locations. Thus the time that elapses between them is not a proper time. And the time he measures will always be greater than the proper time.

 

[Andrew Mason]

Thank you for the reference but I am pretty clear with the theoretical explanation of slowing effect of time. What confuses me is the mathematics. Why is t>t₀ when v>v₀ (as measured from an inertial frame ) if time is slowed down?

If you measure 60 seconds of elapsed time between two events that occur in the same place, but someone's clock measures 100 seconds of elapsed time between those same two events then they would conclude your clock is running slow.

One thing that can help sort this out is the notion of proper time. If the two events occur at the same location, then the time that elapses between them is called a proper time. For an observer moving relative to those two events, imagine a reference frame in which he is at rest. In that moving frame those two events will occur at different locations. Thus the time that elapses between them is not a proper time. And the time he measures will always be greater than the proper time.

Yes. This is the key to resolving the OP's question.

The OP is comparing co-ordinate time between two events in different frames and, because of the $\gamma$ factor notes that the ' frame (moving at speed v relative to the rest frame) measures the time interval between two events to be longer than the rest frame. [Note: This is true for proper time intervals in the unprimed frame but not necessarily true for time intervals between events that are spatially separated in that frame. The time measurement in the ' frame may be longer or shorter depending on the spatial co-ordinates of the events]. The confusion is between co-ordinate time and proper time.

Co-ordinate times of events are measured by inertial clocks that are synchronized in their frame of reference and physically placed at the spatial location of each event. A single inertial clock measures proper time intervals. In asking whether observer A's "clocks run slow" one is asking whether other inertial observers measure a proper time interval in A's inertial reference frame (ie the time between two positions of the hands of one of A's clocks) to be longer than A's measurement. They all do. So A's clock appears to run slow to other inertial observers with whom A is in relative motion. A good illustration of this is the muon decay example (see my amended post #4.)

AM

 

[David Lewis]

The fundamental issue is that saying "time slows down" isn't quite right, it usually indicates a bad model of time,

If we say "length contracts," is it a bad model of length?

 

[Herman Trivilino]

If we say "length contracts," is it a bad model of length?

Fundamentally, saying lengths contract could be just as misleading as saying time slows down. Learners (for example, students) might be less likely to encounter the conceptual disconnect presented by the OP when it comes to considering lengths, but it is nonetheless there. When transforming a length $\Delta x$ from a stationary frame to the $\Delta x'$ of a moving frame we could find, for example when $\Delta t$ is zero, that $\Delta x'$ is larger than $\Delta x$.

 

[Ibix]

If we say "length contracts," is it a bad model of length?

Agree with @Mister T here. When you use a different frame you are using a different definition of length and duration, so "length contraction" is a bit of a misnomer. An analogy would be to draw a unit square on a piece of paper and then rotate it 45°. Now you have a diamond that is $\sqrt 2$ wide and high. Has the square "length dilated"? No - you're measuring something different and calling it a width.

I think length contraction and time dilation are unhelpful in some ways because they imply that something is happening to the object, whereas what is actually happening is that (for one reason or another) a mismatch develops between your "natural" definition of length and duration and the "natural" definition of length and duration for the object.

I'm not sure how that works pedagogically...

 

[stevendaryl]

Agree with @Mister T here. When you use a different frame you are using a different definition of length and duration, so "length contraction" is a bit of a misnomer. An analogy would be to draw a unit square on a piece of paper and then rotate it 45°. Now you have a diamond that is $\sqrt 2$ wide and high. Has the square "length dilated"? No - you're measuring something different and calling it a width.

I think length contraction and time dilation are unhelpful in some ways because they imply that something is happening to the object, whereas what is actually happening is that (for one reason or another) a mismatch develops between your "natural" definition of length and duration and the "natural" definition of length and duration for the object.

I'm not sure how that works pedagogically...

I think it's confusing to newbies, no matter how it's presented. Thinking of time dilation as a physical effect acting on clocks in motion that causes them to run slower, that can't be right, because there is no absolute rest frame, so there is no absolute notion of a clock running slow. On the other hand, thinking of time dilation as a matter of perspective can be misleading in another sense: If you set a clock in motion and then return it to where it started, the elapsed time on the clock (compared to a clock that was "at rest" the whole time) will be exactly what is predicted by the time dilation formula. How can a matter of perspective lead to objective, physically measurable differences in clocks?

I'm not saying that these are unsolved problems for SR---they obviously were figured out 100 years ago. But they are confusing to the beginner.

 

[Herman Trivilino]

I'm not sure how that works pedagogically

I think that the pedagogical issue arises because of the pressure in many situatuations to teach relativity in a short amount of time. Such abbreviated treatments address the relativity of simultaneity only in a qualitative way. Length and time interval "transformations" are done quantitatively, but one of the lengths is always a proper length and one of the time intervals is always a proper time. In those cases the terms length contraction and time dilation are used to describe these "transformations". It's an over-simplification, really.

 

[robphy]

In addition, I think that physics textbooks only highlight Einstein's "transformations" approach
but hardly any of Minkowski's "spacetime geometry" approach [which is often regarded as too difficult].
In my opinion, words and logic are not enough... "a spacetime diagram is worth a thousand words".
What's wrong with using geometry (or at least a sketch of the geometry) as a crutch for a non-intuitive subject?

Imagine doing kinematics with just words and no position-vs-time diagrams.
Imagine doing dynamics with just words and no free-body-diagrams.
[Yes, I know that lots of students try to proceed without such diagrams.]

 

[Quandry]

Later Alice looks at her clock and sees that it reads 1300. At the same time that her clock reads 1300, Bob's clock reads 1230 (How does she know this? Well, suppose their relative speed is .5c, so Bob is now .5 light-hours distant. If she's watching Bob's clock through her telescope, what she sees at 1330 is the image of Bob's clock at the same time her clock read 1300 - it took the light 30 minutes to get to her telescope), so she concludes that Bob's clock is slower than hers.

This, to me, is a very clear explanation which I believe most people can understand without difficulty. It is not that Bob's clock runs at a different speed to Alice's. It is that Alice sees an image of Bob's clock from a past time.

 

[Nugatory]

This, to me, is a very clear explanation which I believe most people can understand without difficulty. It is not that Bob's clock runs at a different speed to Alice's. It is that Alice sees an image of Bob's clock from a past time.

It's both. Even after Alice corrects for the light travel time, Bob's clock is slower than hers. The image she sees at 1330 is the image that left Bob when Alice's clock read 1300 and Bob was 30 light-minutes away. That image shows Bob's clock reading 1230 not 1300, so Bob's clock is slower than hers; at the same time that her clock read 1300 his read 1230.

Time dilation is what's left over after you've allowed for light travel time.

 

[Elnur Hajiyev]

@Faiq , you misunderstood the math behind the time dilation. $t_A$ should be greater than $t_B$ for the observer A, if the observer B is moving relative to the observer A.
Let's say both observers measure the time elapsed between two events—for example, between 12pm and 6pm on the clock of the observer B. Pay attention here: both observer measure the same events! Time dilation formula says if the observer A wants to measure time between events TAKE PLACE on observer B's reference it would be as : $$t_A=\frac{t_B}{\sqrt{1-\frac{v^2}{c^2}}}$$ where $v$ is the speed of the observer B that the observer A measures, $t_B$ is the time interval between the same events relative to the observer B himself. See, as the observer B speeds up relative to A, time interval that A measures between events happened on B increases, and A says, 12 hours passed between 12pm and 6pm on the clock of the B, not 6. However, B says, on A's clock 3 hours passed while I wait 6 hours here. It seems paradoxical, and inconsistent, but both are correct! There is no real time, absolutie time about which you can say it is the correct time.

Remember: the time dilation formula compares time intervals relative to two observer between the same events.

 

[PeterDonis]

the time dilation formula compares time intervals relative to two observer between the same events.

No, it doesn't. It compares two different spacetime intervals, i.e., the intervals between two different pairs of events. See post #28.

The spacetime interval between the same two events is invariant; it's the same for all observers.

 

[Elnur Hajiyev]

No, it doesn't. It compares two different spacetime intervals, i.e., the intervals between two different pairs of events. See post #28.

The spacetime interval between the same two events is invariant; it's the same for all observers.

I mean, time intervals, not spacetime intervals. Are not being 12 pm and 6 pm considered as two different events for both observers? And both observer measures different time and space interval but same spacetime interval between these events?

 

[Ibix]

I mean, time intervals, not spacetime intervals. Are not being 12 pm and 6 pm considered as two different events for both observers? And both observer measures different time and space interval but same spacetime interval between these events?

6pm isn't an event - it's a spacelike slice of spacetime.

The time dilation formula relates the coordinate time difference between two time-like separated events. Unfortunately, much of the confusion in this thread has come from a failure by the OP to distinguish between coordinate time differences and proper time differences, and you didn't specify which you meant in your previous post. In fact you said "time interval", which is unfortunate in that "interval" is the term for $\Delta s^2=c^2\Delta\tau^2$.

 

[robphy]

One way to describe time-dilation is with a Minkowski-right-triangle [with a timelike leg and orthogonal spacelike leg ] with a timelike hypotenuse.
The time-dilation factor is the ratio of the timelike sides: longer-leg/shorter-hypotenuse...
The Minkowski-angle between the timelike leg and timelike hypotenuse is the rapidity and
the time-dilation factor is cosh(rapidity) [so the timelike leg is the side adjacent to the rapidity].
An identity for cosh(rapidity) takes a possibly familiar form: cosh(rapidity)=1/sqrt(1-tanh^2(rapidity)).

 

[Elnur Hajiyev]

6pm isn't an event - it's a spacelike slice of spacetime.

No. I did not mention it as a time. I meant 6 pm as an event. Ok, let's say going to toilet and coming from toilet of B instead of 12 pm and 6 pm. And by saying time interval, I tried to say that, time dilation compares how much time elapsed from the perspective of A and B. In this case: how much time did B spend in toilet for A and how much time did B spend in toilet for B.

Time interval in the perspective of B is equal to the proper time between these two events. However in the perspective of A this interval(how much time passed) is determined by the time dilation formula.

 

[PeterDonis]

I meant 6 pm as an event.

There is no such thing. An event is a point in spacetime. A time by itself is not enough to pick out a particular point in spacetime; you need four coordinates for that, not one.

let's say going to toilet and coming from toilet of B

That will do as a pair of events; but there is only one spacetime interval associated with them. See below.

how much time did B spend in toilet for A and how much time did B spend in toilet for B.

And only one of those (the second) is the spacetime interval between the two events. The other is a coordinate time difference.

Time interval

As has already been pointed out to you, you should not use the word "interval" here; it only causes confusion with the proper usage of that word, which is to denote the spacetime interval between two events.

 

[Andrew Mason]

Fundamentally, saying lengths contract could be just as misleading as saying time slows down. Learners (for example, students) might be less likely to encounter the conceptual disconnect presented by the OP when it comes to considering lengths, but it is nonetheless there.

But proper lengths or displacements in one frame are always measured to be shorter in another frame that is moving in the direction of that displacement. An observer in a rocket ship moving at .9999999999c relative to the Earth in a direction toward the Andromeda Galaxy, which is located at a proper displacement of 1,000,000 light years from earth, measures Andromeda as being only about 14 light years away. And the observer finds that his rocket ship only takes about 14 years to get there.

When transforming a length $\Delta x$ from a stationary frame to the $\Delta x'$ of a moving frame we could find, for example when $\Delta t$ is zero, that $\Delta x'$ is larger than $\Delta x$.

This would not be the case if Δx is a proper length eg. the length of a ruler at rest lying on the x-axis in that inertial frame. All inertial observers moving relative to that ruler in the direction of that x-axis will measure that ruler to be shorter than its proper length. They would also measure the clocks in that frame to be moving slower than the observer's clocks.

These effects are real, as in the muon decay example. Another illustration is the narrow forward-directed cone of light emitted by relativistic charges in a synchrotron.

AM

 

[Ibix]

No. I did not mention it as a time. I meant 6 pm as an event. Ok, let's say going to toilet and coming from toilet of B instead of 12 pm and 6 pm.

I believe you understand the physics here - the problem is that your use of language isn't very precise. You talked about 6pm as an event, but 6pm is not an event. 6pm on someone's worldline is an event - and that was what you meant, but it was not what you wrote.

Similarly, using "time interval" is confusing in this context because "interval" has a precise meaning related to proper times, and you were talking about coordinate times. I recommend "coordinate time differences".

 

[Justintruth]

I think the problem has nothing to do with relativity. Pretend I was just talking about any mechanism to make two clocks disagree.

Dilation means basically expansion. So if you have a clock that is dilated relative to some other clock the time between ticks expands, meaning it is longer. Now if the time between ticks is longer then we usually say the clock has slowed down.

tick - tick - tick

vs

tick ----------- tick -----------tick

Which is faster? I say the first. The second is ticking slower.

Here is what you said: " ... if vA> vB, using the equation for time dilation, tA>tB." Here tA>tB means, for example, that the time between ticks of clock A is longer than the time between ticks in clock B. So clock A is slower.

You then said: "So if 60 seconds passed for B, 100 seconds passed for A." So let's rewrite what you wrote a little: "So for example if 60 seconds pass between successive ticks of Bs clock, 100 seconds pass for between successive ticks of A's clock" Again you get A running slower because it takes 100 sec to tick and B takes 60 seconds to tick. Think about the dial. Let's say the dial moves one second on the hand each tick. A's clock is ticking slower so the dial is moving slower. B's clock is ticking faster so it get's ahead of A's clock. A's clock is therefore running slower.

But not only is As clock running slower but if you saw someone reading As clock in that frame they would be running slower. Everything is a clock in a sense. All physical processes go slower. So we say not only that the clock is running slower but time itself is running slower.

Here you can see where you are confused: You wrote: "Time, in above example, increased for A. So doesn't it means that time becomes faster for A?" You should have wrote: "Time between clock ticks, in above example increased for A. So doesn't that mean that time becomes faster for A?" Then the answer would be a simple. "No! It means that time becomes slower when the time between ticks is longer.

So "time" - meaning the rate of time - did not increase for A. Rather the time between successive ticks of As clock got bigger and so the rate of time got slower. Time dilation does not mean that the rate of ticks gets bigger. It means the time between ticks get's bigger. That means the time rate actually gets smaller.

And all of that is relative. Better to say the time between ticks of As clock when measured by a stationary clock got bigger.

Hard to believe that one clock can be running slower when measured by another but when you measure the other by the first it is also slower? That is what is strange. Normally one clock running slower as measured by another will result in the other running faster if measured by the first! But not so here!

Here is a way to see it. If I had two rulers each claiming to be 1 ft long and one was shorter than the other and I measured the longer with the shorter I would find it was longer than 1 foot but of I measured the shorter with the longer I would find it shorter than a foot! So why doesn't that happen with clocks here!

Well imagine for a moment you have two rulers 1 ft long each. But instead of laying one next to another you separate them by some angle and then "measure" one with the other by dropping a perpendicular line from the tip of the one being measured to the one doing the measurement. If by perpendicular you mean perpendicular to the one being measured the length will be larger. And surprisingly if you do the exact same in reverse it is also longer. So you can have a measuring procedure that has one ruler measured as longer by another and the other measured longer than the first as long as you use the right measurement procedure. If you change the meaning of perpendicular to meaning perpendicular to the one measuring you will get a smaller value but again both rulers will measure the other as shorter using this procedure.

The same thing happens in clocks in relativity. You can take a space-time interval that has only time components - say the interval between two ticks of a stationary clock and "rotate it into space" my starting it moving so that it is not now all time but part space. The interval can stay the same just like the rulers stay the same length when you rotate one and then measure another but now part of the length is in another dimension.

That to me is why they call it space-time. Because you can rotate a temporal interval such that part of it is now in the space direction.

 

[stevendaryl]

I think the problem has nothing to do with relativity. Pretend I was just talking about any mechanism to make two clocks disagree.

Dilation means basically expansion. So if you have a clock that is dilated relative to some other clock the time between ticks expands, meaning it is longer. Now if the time between ticks is longer then we usually say the clock has slowed down.

tick - tick - tick

vs

tick ----------- tick -----------tick

Which is faster? I say the first. The second is ticking slower.

Here is what you said: " ... if vA> vB, using the equation for time dilation, tA>tB." Here tA>tB means, for example, that the time between ticks of clock A is longer than the time between ticks in clock B. So clock A is slower.

You then said: "So if 60 seconds passed for B, 100 seconds passed for A." So let's rewrite what you wrote a little: "So for example if 60 seconds pass between successive ticks of Bs clock, 100 seconds pass for between successive ticks of A's clock"

Sorry for being pedantic, but it's not correct to say that "60 seconds pass between successive ticks of Bs clock, 100 seconds pass for between successive ticks of A's clock". There are two different ways of measuring elapsed time between two events:

1. If you have a single clock that moves inertially from event to the other, then you can use that clock to measure the proper time between the events.
2. If you have a pair of synchronized clocks at rest relative to each other, then you can use one clock to record the time of one event, and the other clock to record the time of the other event. By subtracting the two times, you can compute the time between the events.

Time dilation is a matter of comparing the one way of computing time between events to the other way. The way that uses a single clock always shows less elapsed time than the way that uses two clocks. But it's not a matter of comparing ticks on one clock to ticks on another clock.

 

[Elnur Hajiyev]

@PeterDonis , @Ibix , I think I have some difficulties to express myself. Please read my previous post carefully as I did not give 6pm and 12pm examples as time concepts, but as an event of clock ticking on the specific time on the specific location, because I was talking about specific experiment to make things clear, not about a concept. I noticed that it is causing a confusion here and changed the example to the toilet example in the subsequent post, but I don't understand, why are you insisting on discussing flaws of an example even after that rather than continuing to discuss on the main question. But I agree, I violated physics formalism by using "time interval" phrase, I should use "time difference".
So we changed the example, now I am asking: does not the time dilation formula compare elapsed time for A and B during which B is in the toilet?

Please ignore some tiny formalism errors for the sake of simplicity just a moment if you find. I know, there may be some of those due to my not-perfect English. Please focus on the main issue here. I still cannot understand where is the fundamental mistake in my explanation?

 

[Nugatory]

...does not the time dilation formula compare elapsed time for A and B during which B is in the toilet?
...
Please ignore some tiny formalism errors for the sake of simplicity just a moment if you find. I know, there may be some of those due to my not-perfect English. Please focus on the main issue here. I still cannot understand where is the fundamental mistake in my explanation?

It is in using the term "elapsed time" without having clearly defined what it means. Usually "elapsed time" is understood to mean the proper time along some worldline, while the time dilation formula relates differences in coordinate time using one frame to differences in coordinate time using another frame.

 

[vanhees71]

In addition, I think that physics textbooks only highlight Einstein's "transformations" approach
but hardly any of Minkowski's "spacetime geometry" approach [which is often regarded as too difficult].
In my opinion, words and logic are not enough... "a spacetime diagram is worth a thousand words".
What's wrong with using geometry (or at least a sketch of the geometry) as a crutch for a non-intuitive subject?

Imagine doing kinematics with just words and no position-vs-time diagrams.
Imagine doing dynamics with just words and no free-body-diagrams.
[Yes, I know that lots of students try to proceed without such diagrams.]

Well, I usually have more difficulties with Minkowski diagram than with the formlae. The formulae are usually clean and clear, the diagrams need a lot of forgetting about the Euclidean connotation we have with the plane on which we are drawing it. You must completely forget the Euclidean meaning but think in terms of the somewhat hyperbolic geometry of Minkowski space (it's of course not really hyperbolic geometry in the sense of non-Euclidean geometry; the Minkowski space is still a flat space but not Euclidean but pseudo-Euclidean). That's more difficult than to handle the formulae which are not too complicated.

 

[stevendaryl]

So we changed the example, now I am asking: does not the time dilation formula compare elapsed time for A and B during which B is in the toilet?

As I and several other people have pointed out, time dilation compares two different things:

1. Elapsed time on a single clock in one frame.
2. Computed time differences between a pair of synchronized clocks in another frame.

So it's not a comparison of two different elapsed times.

Here's a drawing showing time dilation from two different frames, illustrating these two different types of time comparisons. We have two identical rockets, one at rest in frame 1 and the other at rest in frame 2. Their relative speed is 0.866c, leading to a time dilation factor and a length contraction factor of 2. In the first rocket, there is clock A on the left end and clock B on the right end. In the second rocket, there is clock C on the left end and clock D on the right end. At event 1, clocks A and D pass each other, and they both show time 12:00. At event 2, clocks A and C pass each other, and clock A shows time 12:30, while clock C shows time 1:00. At event 2, clocks B and D pass each other, and clock B shows time 1:00 while clock D shows time 12:30.

Both frames 1 and 2 agree on the above facts. What they don't agree about is what is going on elsewhere at the same time as the above events. In particular:

1. They disagree about the locations and times of clocks C and B at the time of event 1. According to frame 1, C is only half a rocket length away from clock D, and is not synchronized with clock D, showing time 12:45 instead of 12:00. According to frame 2, B is only half a rocket length away from clock A, and is not synchronized with clock A, showing time 12:45.
2. They disagree about the locations and times of clocks B and D at the time of event 2. According to frame 1, D is only halfway between A and B, and is showing time 12:15. According to frame 2, A is only halfway between C and D and is showing time 12:15.
3. They disagree about the locations and times of clocks A and C at the time of event 3. According to frame 1, clock C is halfway between A and B, and is showing time 1:15. According to frame 2, clock B is halfway between C and D and is showing time 1:15.

Note: the two frames also disagree about whether event 2 happens before or after event 3.

Time dilation compares the 30 minutes between events 1 and 2, as calculated by the single clock, A, with the 1 hour between those events, as measured using the pair of clocks, C and D. (We subtract the time of event 2, according to clock C, from the time of event 1, according to clock D, to get 1 hour). The single clock shows less elapsed time.

Time dilation compares the 30 minutes between events 1 and 3, as calculated by the single clock, D, with the 1 hour between those events, as measured using the pair of clocks, A and B. (We subtract the time of event 3, according to clock B, from the time of event 1, according to clock A.) Again, the single clock shows less elapsed time.

So time dilation is not a comparison of two different elapsed times---it's a comparison of a one-clock time measurement to a two-clock time measurement.

 

[robphy]

Well, I usually have more difficulties with Minkowski diagram than with the formlae. The formulae are usually clean and clear, the diagrams need a lot of forgetting about the Euclidean connotation we have with the plane on which we are drawing it. You must completely forget the Euclidean meaning but think in terms of the somewhat hyperbolic geometry of Minkowski space (it's of course not really hyperbolic geometry in the sense of non-Euclidean geometry; the Minkowski space is still a flat space but not Euclidean but pseudo-Euclidean). That's more difficult than to handle the formulae which are not too complicated.

Not too complicated...? Maybe for you... (so congratulations :) )
However, I think threads like this suggest that formulas are not sufficient
when the terms aren't well understood or when terms aren't described well enough for others to interpret.

Maybe history will support this conclusion.
It appears that increased attention and acceptance of special relativity occurred after Minkowski's spacetime formulation.
http://scottwalter.free.fr/papers/einstd7.pdf "Minkowski, Mathematicians, and the Mathematical Theory of Relativity" by Scott Walter
In my opinion, the geometrical objects and relationships in a spacetime formulation gives something more tangible than "strings of carefully arranged words".
In addition, there appears to be a consistent [though possibly unfamiliar] geometrical system which one could use as a crutch as one hobbles through this non-intuitive terrain. (Without it, students will fall into the same old traps and will need an extraction from the terrain... leaving puzzled, likely never to return.)

(While there is a hyperbola playing the role of the circle, you get "hyperbolic trigonometry"... not to be confused with the curved "hyperbolic geometry", as you point out. [Similarly, "circular trigonometry" isn't to be confused with "elliptic or spherical geometry".] While one can't use much of Euclidean geometry taken absolutely-literally, many general ideas still persist... if you are willing to expand your meaning of "circle" [for example]... or willing to put pseudo- everywhere and introduce minus signs where needed. In other words, you can expand your intuition by geometric analogy.)

(Shameless plug)
I claim that my diamond-studded spacetime diagram like the one below
concisely encodes many of the issues that folks discuss about time-dilation.

 

[Andrew Mason]

So time dilation is not a comparison of two different elapsed times---it's a comparison of a one-clock time measurement to a two-clock time measurement.

Exactly. Each observer is measuring the other's clock speed with a two-clock time measurement using spatially separated clocks that are at rest and synchronized in the observer's rest frame. Each observer measures the other's clock as being slower than his own. But each observer (A) could also say that the other's (B's) measurements of the speed of its (A's) clock using two clocks (in B) are the result of those (B's) clocks not being synchronized.

AM

 

[PeterDonis]

does not the time dilation formula compare elapsed time for A and B during which B is in the toilet?

"Elapsed time for A" is a spacetime interval between a different pair of events than "elapsed time for B". The time dilation formula compares these two intervals. With that clarification, your statement is true, and it might well be what you meant to say. But you have repeatedly used language which does not clearly convey that meaning.

An even better way to make sure you are being clear is to use math instead of ordinary language. For example: there are two events on B's worldline, event B1 where he enters the toilet, and event B2 where he leaves it. The spacetime interval between these two events is the "elapsed time for B". This interval is given by $t_B = \sqrt{\left( t_{B2} - t_{B1} \right)^2 - \left( x_{B2} - x_{B1} \right)^2}$.

There are also two events on A's worldline, event A1 which is simultaneous, according to A's rest frame, with event B1, and event A2 which is simultaneous, according to A's rest frame, with event B2. The spacetime interval between these two events is the "elapsed time for A". This interval is given by $t_A = \sqrt{\left( t_{A2} - t_{A1} \right)^2 - \left( x_{A2} - x_{A1} \right)^2}$.

If we now assume that we are using A's rest frame (so A is always at $x = 0$), and that event A1 has time coordinate $t = 0$, then the elapsed time for A is just $t_A = t_{A2}$. And if we assume that B is moving at a constant speed $v$ in the $x$ direction relative to A (I am using units in which $c = 1$), and that event B1 is the same as event A1 (B and his toilet are just passing by A when B enters the toilet), then we have $t_B = \sqrt{ t_{B2}^2 - x_{B2}^2 } = \sqrt{ t_{B2}^2 - \left( v t_{B2} \right)^2 } = \sqrt{ 1 - v^2 } t_{B2} = \sqrt{ 1 - v^2 } t_{A2} = \sqrt{ 1 - v^2 } t_A$, where we have used the fact that events A2 and B2 are simultaneous in A's rest frame, as noted above. This can be rearranged to give the time dilation formula you gave:

$$
t_A = \frac{t_B}{\sqrt{1 - v^2}}
$$

 

[Andrew Mason]

"Elapsed time for A" is a spacetime interval between a different pair of events than "elapsed time for B". The time dilation formula compares these two intervals.

I don't think that is what Elnur was saying. He was comparing A's measurement of the time between two events that occur at the same location in B's frame to the proper time interval ie. shown by a single clock in B at that location. In this case, the two events are 1. B enters toilet and 2. B exits toilet. The toilet, hopefully, was at rest in B's frame of reference. So they are the same two events and they are measuring the same space-time interval.

AM