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

[Faiq]

In explaining time dilation we usually say, if velocity of A is greater than velocity of B, then time is slower for A as compared to B. However, if v_A> v_B, using the equation for time dilation, t_A>t_B. So if 60 seconds passed for B, 100 seconds passed for A. How does that imply that time slowed down for A? Time, in above example, increased for A. So doesn't it means that time becomes faster for A?

 

[Doc Al]

In explaining time dilation we usually say, if velocity of A is greater than velocity of B, then time is slower for A as compared to B.

No. We say more like: If B is moving with respect to A, then A will 'measure' B's clocks to run slow (compared to A's). And vice versa, of course.

 

[Ibix]

Also, I think you might have your equation back to front. If you, A and B agree at some point that t=0 then if (according to you) $v_A>v_B$ then (again according to you) A's clock will thereafter always have a lower reading than B's - A's clock ticks slower.

 

[Andrew Mason]

In explaining time dilation we usually say, if velocity of A is greater than velocity of B, then time is slower for A as compared to B. However, if v_A> v_B, using the equation for time dilation, t_A>t_B. So if 60 seconds passed for B, 100 seconds passed for A. How does that imply that time slowed down for A? Time, in above example, increased for A. So doesn't it means that time becomes faster for A?

You must use the Lorentz transformations and keep the concepts of proper time and time dilation straight. You have to be clear on what times you are referring to when you speak of t_A and t_B.

Proper time is the time measured by an observer on a clock that is at rest in the observer's frame of reference. This is necessarily a time between two events that are separated by time only, ie. not spatially separated, in the observer's frame.

The time intervals that observers in other inertial reference frames measure between two events are determined by the Lorentz transformations. The variable t represents time of an event in the "stationary" frame and t' represents time co-ordinate of an event measured in the frame moving at velocity v relative to the first.

Consider an observer in inertial reference frame A and an observer in inertial frame B whose origin is moving at velocity v in the direction of the x-axis relative to the origin of A. The proper time interval that observer in A measures on its own clock between event E1(0,0,0,t1)and event E2(0,0,0,t2) is t2-t1. An observer in B observes the same two events at E1(γ(x-vt1),y, z, γ(t1-vx/c²) and E2(γ(x-vt2),y, z, γ(t2-vx/c²) where x=y=z=0. So the time interval that B measures on its own clock between these events is $\gamma (t2-t1)$ which is greater than the time measured on A's clock. The result is that from B's perspective the time between events measured at the same location in A's frame of reference (A's clock at A's origin), which is moving at velocity -v relative to B, appear to be moving more slowly than B's.

If two events were to occur at the same location in B, A would observe the events to occur at different locations in A and at a different time interval as measured by A on A's clocks. So it is perfectly symmetrical between the two inertial frames. It is just that what one observer sees as a pure interval of time the other sees as an interval of distance and a different interval of time.

Time dilation may be best demonstrated by the muon decay phenomenon. The proper lifetime of the muon is measured in the muon frame of reference. This lifetime can be determined in the laboratory from observing slow moving muons. Muons created in the upper atmosphere by cosmic ray bombardment create muons moving at relativistic speeds relative to the earth. We observe them reach the Earth surface. In order to do that they must survive for a time as measured on Earth that is significantly longer than their proper lifetimes. Based on our time measurements in the Earth frame of reference we say that the muon's clock appears to run slow. That is to say that the time between two events (muon creation to muon decay) is measured to be longer on Earth than it is in the frame of reference of the moving body. So, it appears to us that the fast moving muon's clock is running "slow". That is time dilation.

AM

 

[Faiq]

You must use the Lorentz transformations and keep the concepts of proper time and time dilation straight. You have to be clear on what times you are referring to when you speak of t_A and t_B.

Proper time is the time measured by an observer using a clock that is at rest in the observer's frame of reference.

The proper time intervals that observers measure between two events are determined by the Lorentz transformations. The variable t represents proper time of an event in the "stationary" frame and t' represents proper time of an event in the "moving" frame.

Consider an observer in inertial reference frame A and an observer in inertial frame B whose origin is moving at velocity v in the direction of the x-axis relative to the origin of A. The proper time interval that the observer in A measures on its own clock between event E1(x,y,z,t1)and event E2(x,y,z,t2) is t2-t1. An observer in B observes the same two events at E1(γ(x-vt1),y, z, γ(t1-vx/c²) and E2(γ(x-vt2),y, z, γ(t2-vx/c²). So the proper time interval that B measures on its own clock between these events is $\gamma (t2-t1)$. But the B observer also measures a space interval between the events.

Since γ>1, the time interval measured by B is shorter than the time interval measured by A. Thus an observer in A will conclude that B's clocks are moving slower than A's. That is the concept of time dilation. But this is only because A observes the two events occurring at the same location in A while B observes them occurring at different locations in B. If two events were to occur at the same location in B, A would observe the events to occur at different locations in A and at a shorter time interval as measured by A on A's clocks. So it is perfectly symmetrical between the two inertial frames. It is just that what one observer sees as a pure interval of time the other sees as an interval of distance and a shorter interval of time.

AM

But then why do we say time slows down at higher speeds if you yourself said "the time interval measured by B is shorter than the time interval measured by A. " when A is moving at a greater speed than B ?

 

[Ibix]

But then why do we say time slows down at higher speeds if you yourself said "the time interval measured by B is shorter than the time interval measured by A. " when A is moving at a greater speed than B ?

According to each observer, they themselves are stationary and everybody else is moving. So whose clock is ticking slowest depends on who is doing the measuring. "Time slows down at higher speeds" is something of an over-simplification, I'm afraid.

 

[pervect]

In explaining time dilation we usually say, if velocity of A is greater than velocity of B, then time is slower for A as compared to B. However, if v_A> v_B, using the equation for time dilation, t_A>t_B. So if 60 seconds passed for B, 100 seconds passed for A. How does that imply that time slowed down for A? Time, in above example, increased for A. So doesn't it means that time becomes faster for A?

Saying that the velocity of A is greater than the velocity of B is a frame dependent statement, not an absolute statement. When you say the velocity of A is greater than the velocity of B, you are implicitly creating a frame "C" , in which this statement is true. This is important because time is a frame dependent quantity, and so is time dilation.

The fundamental issue is that saying "time slows down" isn't quite right, it usually indicates a bad model of time, one based on what is commonly called "absolute time". The problem that I have responding to such statements is that frequently people who use absolute time often don't know what I mean when I say the words "aboslute time". The meaning of the phrase doesn't get through :(. Unfortunately, using a lot of words doesn't seem to work well either - the key points seem difficult to convey. Perhaps they seem irrelevant, or perhaps it's a case of tl;dr. One approach I take is to refer readers to Scherr's paper, "The Challenge of Overcoming deeply-held Student Beliefs about the Relativity of SImultaneity" in the hopes that they'll actually read it. I don't get a lot of feedback from people saying that the paper actually helped them, or people who give feedback saying that they read the paper and had questions. So I suspect it often doesn't get read often. Still, it's worth a mention - and perhaps I'm being too pessimistic, I can hope that mentioning the issue could plant a seed that will, at some later date, flower.

 

[Faiq]

Okay then consider my problem this way. Using the equation of time dilation, the answer is always greater than t₀ value ( if v>0). What does this increase in time indicate?

 

[robphy]

This old post and the desmos-visualization might be helpful
https://www.physicsforums.com/threads/time-dilation-speed-relative-to-what.804008/#post-5048164

 

[Faiq]

This old post and the desmos-visualization might be helpful
https://www.physicsforums.com/threads/time-dilation-speed-relative-to-what.804008/#post-5048164

Thank you for the reference but I am pretty clear with the theoritical 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?

 

[Ibix]

Thank you for the reference but I am pretty clear with the theoritical 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?

It isn't.

 

[phinds]

But then why do we say time slows down at higher speeds ...

We DON'T say it. Pop-science treatments say it and it is not correct. Time dilation is an observational phenomenon, as has already been pointed out in this thread, not something that actually happens to anyone. Doc Al explained it perfectly well in post #2.

 

[Faiq]

So time doesn't slows down at higher speeds?

 

[Ibix]

Are you reading the replies you are getting? #3, #6 and #7 all address this. You have the maths wrong and your description is over-simplified.

 

[Nugatory]

Thank you for the reference but I am pretty clear with the theoritical 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?

Listen carefully to what @pervect is saying, that "time slows down" is a bad way of thinking about it; almost certainly you're thinking that there is a correct, not-slowed-down, "absolute" or "real" time that we're comparing with.

Forget the time dilation formula for a moment, as it is tempting you to think about "time slowing down". Instead, work on understanding the relativity of simultaneity (Google for "simultaneity Einstein train" - no math required), as both time dilation and length contraction are consequences of relativity of simultaneity. Don't move on until you clearly understand that the statement "X and Y happened at the same time" is frame dependent; if two things happened at the same time according to Alice, and Bob is moving relative to Alice, then the two things did not happen at the same time according to Bob; and the things that did happen at the same time according to Bob did not happen at the same time according to Alice.

Once you understand that, you are in a position to understand what's really going on with time dilation, and it's not "time slows down". Say Alice and Bob are moving relative to one another, but at some point they meet - either Alice passes Bob or Bob passes Alice, but either way they're at the same place vat the same time at the moment of passing. As they do, they both set their clocks to 1200 noon.

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.

However, relativity of simultaneity means that although "Alice's clock reads 1300" and "Bob's clock reads 1230" happened at the same time according to Alice, these two events did not happen at the same time according to Bob. According to Bob, who is also watching Alice's clock through his telescope, the event "Bob's clock reads 1230" happened at the same time as "Alice's clock reads 1215", and it is Alice's clock that is running slow.

If you are really serious about understanding this... The best thing you can do is to try to forget everything you think you know about relativity and start over with Taylor and Wheeler's book "Spacetime Physics" or equivalent.

 

[Nugatory]

So time doesn't slows down at higher speeds?

It does not, at least not the way you're understanding it.

 

[Faiq]

Wait a minute so both times 1230 and 1330 are correct? If no then, what is the absolute time at the moment Alice looks sees 1300 at her clock?

 

[Ibix]

As pervect said in post #7, there is no such thing as absolute time. So your question is meaningless.

 

[Faiq]

This is some little vague thinking I made after analyzing above posts. Please tell me if it's close to correct

Let's for some simplicity consider time "a line" which is 1 cm long. So 1 cm will be 1 second and so on. This line is different for stationary objects and objects moving.
So let's say v_a >v_b. According to the time dilation equation t_a>t_b but the conventional books/youtube videos tell us that time slows down for moving objects. So here's my idea behind it
The time that books are mentioning is in my opinion the "line" idea provided above. If objects are moving, the length of their "time line" will decrease so in their case 1 sec will be like 0.9 cm, 0.8 cm.
The time that equation refers to is the number of "time lines" one experiences in a time interval.

Now I will refer to an example to be a little clear.
In a 10 second interval measured in a inertial frame
Suppose B is at rest and experiences 10 second.
We can say his length of time-line is 1cm. So he experiences 10 time-lines.

A is moving and by using the time dilation equation we see that he experiences 20 seconds.
We know that the "correct" (I know this is the wrong word here but just for sake of simplicity) time is 10 seconds.
We can arrive at this conclusion if we consider his time-line to be 0.5 seconds. In this way he will experience two times more time-lines in 10 seconds and may measure his time to be 20 seconds.

I know this is too much over-simplified but contradiction between the posts here and the SAT books are driving me crazy.

 

[Doc Al]

So let's say va >vb.

Will you please stop saying this without some context? As has been pointed out several times, such a statement is frame dependent.

 

[Ibix]

So let's say v_a >v_b. According to the time dilation equation t_a>t_b

For the third time, no! If, in some frame, $v_A>v_B$ then the time elapsed on A's clock over some time interval (measured in that frame) will be less than that on B's.

 

[Faiq]

A spaceship of alien sports enthusiasts passes by the Earth at a speed of 0.8c, watching thefinal minute of a basketball game as they zoom by. Though the clock on Earth measures aminute left of play, how long do the aliens think the game lasts?

This is a question taken from a SAT book. The answer is 100 seconds. Can you tell me what does 100 seconds and 60 seconds mean if they are referring to the same event? And how does that relates to the slowing down of time which is again written in the start of the same book?

 

[Ibix]

According to the alien the stadium clock ticks every 1.667s because, to it, the Earth is moving at 0.8c. So it is the Earth's clocks that tick slowly from the alien's perspective.

Edit: ...as I said in #6.

 

[phinds]

but the conventional books/youtube videos tell us that time slows down for moving objects

AND THEY ARE WRONG, as has been explained several times already. FORGET what they say.

 

[Faiq]

Okay, thanks for spending time on me. It was just those books which got me all messed up.

 

[phinds]

A spaceship of alien sports enthusiasts passes by the Earth at a speed of 0.8c, watching thefinal minute of a basketball game as they zoom by. Though the clock on Earth measures aminute left of play, how long do the aliens think the game lasts?

This is a question taken from a SAT book. The answer is 100 seconds. Can you tell me what does 100 seconds and 60 seconds mean if they are referring to the same event? And how does that relates to the slowing down of time which is again written in the start of the same book?

As has been pointed out to you several times already, THERE IS NO ABSOLUTE TIME! Time is frame dependent. It is perfectly reasonable for an observer in one frame of reference to see an event taking a different amount of time than the amount seen by an observer in a different frame.

 

[phinds]

Okay, thanks for spending time on me. It was just those books which got me all messed up.

You are not alone. LOTS of people come here with misconceptions caused by pop-sci books and TV shows.

 

[PeterDonis]

Can you tell me what does 100 seconds and 60 seconds mean if they are referring to the same event?

They aren't referring to "the same event". They refer to two different spacetime intervals, which are intervals between pairs of events. (An "event" is a point in spacetime.)

The 60 seconds refers to the spacetime interval between two particular events on the worldline of the Earth's clock--the event where the clock reads "1 minute left to play", and the event where the clock reads "0 left to play, game just ended".

The 100 seconds refers to the spacetime interval between a different pair of events--a pair of events on the worldline of the alien spaceship. The first event is the event on the alien spaceship that, according to the spaceship's frame, is simultaneous with the event on Earth where the Earth's clock reads "1 minute left to play". The second event is the event on the alien spaceship that, according to the spaceship's frame, is simultaneous with the event on Earth where the Earth clock reads "0 left to play, game just ended".

The general principle being illustrated here is that, whenever you are trying to analyze a scenario with time dilation (or length contraction), you need to also take into account relativity of simultaneity in order to fully understand what is going on. Notice that I was careful to specify simultaneity "according to the spaceship's frame" for the second pair of events. This is not the same as "simultaneous according to the Earth's frame".

 

[phinds]

As has been pointed out to you several times already, THERE IS NO ABSOLUTE TIME! Time is frame dependent. It is perfectly reasonable for an observer in one frame of reference to see an event taking a different amount of time than the amount seen by an observer in a different frame.

EDIT: As Peter clarified, I should not be talking about an "event" taking a certain amount of time. An event is a single point in space-time. I should have said that it is reasonable for observers in different frames to see different durations (or "intervals" as Peter more properly puts it).

 

[Faiq]

thanks everyone

 

[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 theoritical 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".