How can time dilation be the same for both observers?

[Jonathan Scott]

The universe is the same regardless of reference frame, but the way of identifying events in space and time depends on the choice of reference frame.

You are already familiar with an everyday part of the concept of a reference frame, which is that the coordinates of some event relative to you depend on where you are and which way you are facing, but it is easily understood that the events are the same even if the observer is in a different position or facing in a different direction. We know how to map between those frames using translations and rotations.

An extension of this concept also applies to the relationship between time and space in reference frames for observers traveling at different velocities. This mapping is called a "boost". It is mathematically closely related to rotations, but because of the difference between time and space the quantities involved are cosh and sinh rather than cos and sin. (Technically it is equivalent to a rotation through an imaginary angle).

 

[Prem1998]

The universe is the same regardless of reference frame, but the way of identifying events in space and time depends on the choice of reference frame.

It doesn't make any sense to say that the universe is same regardless of the reference frame. I mean, the universe is just the events which happen in it. And, if the events happening at any instant depend on the reference frame, then for different people, universe is different. Saying that the universe is same implies that all the relativistic effects are due to our incorrect observations of the universe due to our relative velocities but it's not the case.

 

[Jonathan Scott]

And, if the events happening at any instant depend on the reference frame, then for different people, universe is different.

The events which happen do NOT depend on the reference frame. The location and time used to identify an event depends on the reference frame, and aspects such as velocity, orientation and time dilation are relative to the reference frame, but the view from any reference frame can be systematically mapped to how it appears from any other reference frame by a suitable Lorentz transformation.

 

[Prem1998]

The events which happen do NOT depend on the reference frame. The location and time used to identify an event depends on the reference frame, and aspects such as velocity, orientation and time dilation are relative to the reference frame, but the view from any reference frame can be systematically mapped to how it appears from any other reference frame by a suitable Lorentz transformation.

I didn't say that for different observers different events happen. But, what one will observe in the universe when he pauses the time depends on which reference he was in when he paused the time. So, obviously different observers will see different universes, different lengths of the same thing, and different instants of the same happened event. So, the universe isn't the same and is different for different people.

 

[jbriggs444]

To most of us, the universe includes our past and our future [along with the "elsewhere" outside our light cone]. The fact that a snapshot of the universe taken at a particular instant does not include these pieces means that a snapshot of the universe not the same as the whole universe.

Yes, we all agree that the contents of two hyper-surfaces of simultaneity sliced out of the universe at different angles will not be identical.

 

[Jonathan Scott]

... So, the universe isn't the same and is different for different people.

By that logic, you'd have to claim that the universe is different if you're facing in a different direction, which isn't a useful definition of "different".

If I'm looking at something from one direction I might claim that its "width" from my point of view is different from what someone sees when looking at it from a different direction, but that doesn't mean we are observing different events.

Similarly, if we are looking at things from different velocities, we may disagree about things such as lengths and time intervals, but those depend on our point of view and there's a simple rule for mapping between those points of view. So the view of the universe is different, but we all observe the same events and the same physics, and can easily map any view to any other.

 

[pixel]

How can time dilation be the same for both observers?

Your confusion is arising due to an incorrect interpretation of time dilation. I only read through about half of the responses so sorry if I'm repeating here.

Following your scenario, when Dick reads his clock at 07:05 AM, 5 minutes have elapsed in his frame of reference. This is what’s known as a "proper time interval" because the clock is in the same location in his frame at the beginning and end of the interval. The time dilation formula tells us what time interval would be measured in another frame of reference moving with respect to Dick’s, in which the clock is not in the same location at the start and end of the interval. For v = 0.8c, it's 1.67 x 5 = 8.35 minutes in Jane’s frame (hence “moving clocks slow down.”).

Now the fallacy is to say, well if it 8.35 minutes in Jane’s frame, then it must be 1.67 x 8.35 minutes in Dick’s frame etc. etc. ad infinitum, but the 8.35 minutes in Jane’s frame is not a proper time interval so you can't apply the time dilation formula to it.

 

[PeroK]

It doesn't make any sense to say that the universe is same regardless of the reference frame. I mean, the universe is just the events which happen in it. And, if the events happening at any instant depend on the reference frame, then for different people, universe is different. Saying that the universe is same implies that all the relativistic effects are due to our incorrect observations of the universe due to our relative velocities but it's not the case.

This is where a knowledge of classical physics is an advantage rather than learning relativity first. A simple example is that the sun rises at different times around the world. If I'm in London and phone someone in New York, it may be dawn in London, but it won't be in New York. So, does that make the universe "different" for us?

The frequency of a siren on an ambulance depends on how fast you are moving with respect to the ambulance.

Or, I travel to New York on one flight and record the distance as 5,600km. Someone else travels on a different flight on a different path and measures the distance as 5,900km. So, that's two different distances from London to New York. You're familiar with the concept that we can both start from A and travel to B and measure different distances we have travelled. Once you understand relativity, you find we may also have measured different elapsed times since we last met. What seems obvious for distances may seem world-shattering for time. But, once you understand the structure of spacetime, it's not so world-shattering.

The point is that you really have to think through what you need to be "different" to get an inconsistency. None of these examples shows an inconsistent universe. Measurements are different, but not in an inconsistent way. Measurements, even in classical physics, depend on your reference frame.

 

[stevendaryl]

Okay, so when Jane sees 07:03 AM on her clock is not 'the same time' when Dick sees 07:03 AM on the same clock. Then, what is the time on Jane's clock according to Jane when there is 07:03 AM on her clock according to Dick?

Let me make this a little more concrete by imagining that Jane and Dick are each in a long spaceship, and the spaceships are passing each other. There is a clock in the front of each spaceship, and a clock in the rear of each spaceship. Let's suppose that the spaceships are so long that it takes 5 minutes for the front of Dick's spaceship to travel from the rear of Jane's spaceship to the front of Jane's spaceship, and let's suppose that Dick's spaceship is traveling at 0.8 c, relative to Jane, and that Dick's front clock and Jane's rear clock both say 7:00 when they pass each other. Then the time dilation rules tell us the following:

1. When Dick's front clock passes Jane's rear clock, they both say 7:00.
2. When Dick's rear clock passes Jane's rear clock, Dick's rear clock says 7:05, while Jane's rear clock says 7:03.
3. When Dick's front clock passes Jane's front clock, Dick's front clock says 7:03, while Jane's front clock says 7:05.

Jane and Dick have two different explanations for these phenomena. The following picture illustrates Jane's story. According to Jane, there are three odd facts about Dick's ship:

• Dick's clocks advance slower than Jane's--they only advance 3 minutes for each 5 minutes that Jane's clocks advance.
• Dick's clocks are not correctly synchronized: His rear clock is set 3 minutes and 12 seconds ahead of his front clock.
• Dick's ship is shorter than Jane's, by the factor 0.6.

She explains the three facts above in the following way:

1. Initially, both of Jane's clocks are set to 7:00. Dick's front clock is set to 7:00, but his rear clock is set to 7:03:12
2. By the time Dick's rear clock has reached Jane's rear clock, her clocks have advanced 3 minutes, while Dick's clocks have only advanced 1 minute and 48 seconds. So Dick's rear clock advanced from 7:03:12 to 7:05, while Jane's rear clock advanced from 7:00 to 7:03.
3. By the time Dick's front clock has reached Jane's front clock, her clocks have advanced 5 minutes, while Dick's clocks have only advanced 3 minutes. So Dick's front clock shows time 7:03, while Jane's front clock shows time 7:05.

Dick has a completely different explanation. From his point of view:

• Jane's clocks run slower, advancing only 3 minutes to his 5.
• Jane's clocks are not synchronized; her front clock is ahead of her rear clock by 3 minutes and 12 seconds.
• Jane's ship is shorter than Bob's by a factor of 0.6.

His story explaining the same three events is illustrated by the following picture:

His explanation of the three facts is:

1. Initially, Dick's two clocks are set to 7:00. Jane's rear clock is set to 7:00, but her front clock is set to 7:03:12
2. When Dick's front clock reaches Jane's front clock, his front advances 3 minutes, to 7:03, while Jane's front clock advances only 1 minute 48 seconds, to 7:05
3. When Dick's rear clock reaches Jane's rear clock, his rear clock will have advanced 5 minutes, from 7:00 to 7:05, while Jane's rear clock will have advanced 3 minutes, from 7:00 to 7:03.

So each can consistently believe that it is the other ship that has time-dilated, out-of-synch clocks.

 

[pervect]

For a good reference on the following exercise, try Bondi's old (but very good) book, "Relativity and Common Sense". The numbers may be slightly different than yours, but they're easy to work with.

Dick and Jane are together at 7:00. At 7:01, Dick sends out a bright red flash, which illuminates Jan'es clock. This signal returns to Dick at 7:04, at which point Dick "sees" Jane's clock, and jots down the reading.

Dick concludes that at 7:02.30, Jane was 90 light seconds away (1 minute and 30 seconds, one half of the round-trip time of 3 light minutes). This is because, in Dick's frame, light travels at a constant velocity, so the time it took the light to get to Jane must be the same time it took for it to return.

This is worth working out for yourself, it's one of the main points of this thought exercise. Using mathematical notation, If Dick sends a signal at time t1, and receives it at time t2, dick concludes that at time (t1+t2)/2, the signal was reflected off an object that was at a distance of c*(t2 - t1)/2 , c being the speed of light.

Going back to the example - we can say that Dick "sees" Janes clock illuminated at 7:04, but the time, according to Dick, at which it was illuminated was 7:02.30. The difference is due to the travel time of light.

GIven this information, you can work out Jane's speed relative to Dick - you should get 3/5 the speed of light.

Now, here's the kicker. According to relativity, when Jane's clock was illuminated, it reads 7:02, not 7:02:30.

The argument works just as well if Jane sends out a blue flash of light towards Dick at 7:01 according to Jane's clock.

 

[Prem1998]

Thanks a lot for all the answers. It was really helpful.