Meassuring time in two inertial frames of reference

Question

I've recently read this experiment:

Consider two observers, one standing still and the other one driving down the drag strip in a car at 120mph. According to special relativity the if standing observer meassures 30 seconds on his watch the one in the car will meassure 29.99999999999952 seconds. (Paraphrased Brian green - The Elegant Universe)

The author doesn't state if the car is standing still at time $t=0$ or if it's already going at 120mph.

Assuming the car undergoes no acceleration and thus both observers are in a inertial frames of reference ( assume flat spacetime). Why are we able to say, by comparing the times on the watches, which observer is going at a speed 120mph? I would have thought that one observer $A$ going in one direction at 120mph while observer $B$ is standing is the same as observer $B$ going 120mph while $A$ is standing.

The only explanation I have for this is that we can't really compare both times without unless one observers undergoes some sort of acceleration or a light signal is sent from one observer to another. This is just a hunch and I'm not sure how would I go around showing this.

Could anyone please provide some further explanation?

Edit: I've realized they both stop the time after crossing the finish line so my explanation certainly don't hold, the question still stands.

Edit #2: Thank you very much for all your answer, sadly I can't mark multiple answer as right even though you all provided valuable insight. I've done the some calculation and It appears that is that to a "standing obsever" (relative to earth) who meassures 30 seconds on his watch, in flat spacetime that the moving observer meassures 29.99999999999952 seconds by the time dialation formula $$\Delta t' = \frac{\Delta t}{\sqrt{1-\frac{v^2}{c^2}}}$$ this roughly agrees with the numbers given. From this we can assert that the observer was really already going at 120mph at $t = 0$, another thing that this is that this setup doesn't consider the time it will take light to travel from the finish line to the standing observer.

Answer 1

It means that the standing observer clocks the car at 120mph, ie the car would travel 120 miles on the ground in one hour according to the clocks on the ground. In the frame of the car, the observer on the ground is also moving at 120mph, ie the observer would move 120 miles in one hour according the the clocks in the frame of the car.

The time dilation effect in SR arises because of the relativity of simultaneity- you should read up on that topic, because it is key to understanding most of the other effects in SR.

In the example you quote the drag-racer arrives at the end of the strip in slightly less time (in the racer's frame) than it does in the ground frame. That's because to the racer the strip is length contracted. If the racer sees 30 seconds displayed on a clock at the finishing line, the racer will think that clock is out of synch with the clock at the start of the strip.

You don't really need clocks at all- they are just devices for showing the time and making the explanation easier to understand. The underlying cause is the geometry of spacetime. There are two significant events in your example, namely the car crossing the start line and the car crossing the finishing line. The time between those two events is 30s in the ground frame and ever so slightly under 30s in the frame of the car. That's a property of the geometry of spacetime, and is true whether or not you have clocks to measure what's happening.

The effect is symmetrical in the following sense. Let's suppose the drag racer has another car following behind at the same speed, and in the drag racer's frame the distance between the two cars is the same as the length of the strip in the ground frame. In that case it will take 30s, in the racer's frame, for the ground observer to pass from the first car to the second, while in the ground observer's frame it will take 29.999999999999952 seconds (check the number of nines I typed!). So the drag racer seems time dilated in the frame of the ground, and the ground observer seems time dilated by exactly the same amount in the frame of the racer.

Answer 2

"Why are we able to say, by comparing the times on the watches, which observer is going at a speed 120mph? I would have thought that one observer A going in one direction at 120mph while observer B is standing is the same as observer B going 120mph while A is standing."

Quite right. There's no difference. Both observers see the other clock running slightly slower than their own.

You can understand this intuitively by thinking about two people walking across a field at the same speed in slightly different directions. Each of them measures time (called 'proper time') in the 'forwards' direction, and space in the 'sideways' direction. Because they're each facing in a slightly different direction, they each measure time slightly differently. Both observers see the other observer falling behind themselves. Both observers see the other's lengths shrink.

The geometry of spacetime is different to Euclidean space, so this analogy is not quite accurate. But it is correct in the sense that time dilation and length contraction are purely geometrical effects from rotating the time/space coordinate system, like in this example. When you walk in different directions (move at different velocities), your measurements of distances forwards and sideways (in time and space) are different, but symmetric. Both of you see the other one moving at 120 mph. Both of you see the other falling behind. Both of you see the other clock running slow. If one of you turns and walks back to meet your friend, you will have travelled different distances when you meet up again, but this is not because of 'acceleration', but simply because one path through spacetime is longer than the other.

Answer 3

As you say, we shouldn't be able to tell which observer is the one moving at 120mph if they were always moving at that speed. Or even further, you could add another observer C that is moving at 250mph, which would make it even more complicated!

However, we know that they haven't been moving at that speed always. The universe started as an homogeneous fluid of fields (we believe) and it was from more complicated interactions that different objects formed and acquired their own velocities. Therefore, as we need to accelerate in order to move at 120 mph, we break that symmetry between observers and so it is clear that it is who accelerated the one that will experience the time dilation.

Extra: Note that given that the expansion of the universe is homogeneous and isotropic, the relative motion given by this expansion doesn't lead to any time dilation. That is why two galaxies can move apart faster than the speed of light without violating special relativity.