# Speed and Time

I am a high-school student, and while I was eating at MCD I suddenly realised that one of two things I had accepted as general concepts had to be at least partially incorrect. Here are the two concepts:

1. Speed is relative - Ex: In a universe empty of all but two objects, if they begin to move apart it is impossibe to tell which is moving. They could both be moving, depending on your point of view.

2. Time slows down the faster you travel - Ex: If something moved at the speed of light, time would stop for it. Lesser speeds yield a slowing of time. (I think this was actually proved, and was a logistical hurdle for the GPS system…)

So… where have I gone wrong? I keep thinking of a universe empty of all but two clocks that move apart fast enough to yield a reduction of one second on one clock. Which clock shows the reduction? Arrg!

You’re pretty much on the money on the relative motion bit but not so clear on time dilation.

When you move faster your clock slows as seen by outside observer. Say you travel around the world a few times at 0.9c. You will see no change in your clock but a ground observer will see your clock slow down. This increases the closer you get to the speed of light. Going as fast or faster than light has no meaning.

This has been experimentally observed by taking an atomic clock on a 747 and flying around the world then comparing it to another clock on the ground that was in sync before the trip. Since GPS satellites are going much faster and precise time it critical to their function they take relatavistic time dilation into account in their programming.

Thanks for the quick response Padeye, but the end result would still be the same: a difference in the two times. It makes sense that the person looking at a slowed clock would notice no change; they would be slowed too, right?

The other clock shows the reduction. You are correct that velocity is relative. What you are not considering is that time is also relative. If you are moving relative to me, your clock appears to run slow from my point of view. But from your point of view, you are stationary and I am the one moving. My clock appears to run slowly from your point of view.

I’ve shot myself in the foot by responding to relativity questions before, but it’s an interesting topic, and I love mind puzzles like this one introduces.

There’s no absolute frame of reference for time any more than there is for space. Any measurements you take have to be relative to something. If you’re in a lab on one spaceship viewing a lab on another spaceship, your frame of reference for direct measurements is where, and when, you are.

Its perhaps counterintuitive that you, in your lab, see your brother Bob’s clock running slowly, and Bob, viewing you, sees your clock running slowly, even though as far as you can measure locally your clock is running just fine. But it’s true.

It should at least be reasonably obvious that if your question #1 is true, that all motion is relative, that the situations should be symmetric. If you are moving away from Bob and see his clock moving more slowly, then Bob is moving away from you, and he should see your clock moving more slowly. This defies common sense in that it means each person is running at ‘normal speed’ while the other clock is slowed down.

Relativity is chock full of surprises like this. What it shows is that time, like space and motion, are local concepts. It also shows that there’s no sense of a universal clock. That means there’s no tick-tick-tick out there by which you can universally measure when something happens.

The problem with assuming that you don’t see Bob’s clock slow down because you’re also slowed down is that it assumes there’s a universal clock by which you can measure that both you, and Bob, are slowed down. It assumes that you are slowed down by exactly the same amount as Bob no matter which frame of reference you’re in. But the moment you start making assumptions about absolute frames of reference, you start running into trouble with the speed of light being constant, which we’ve demonstrated in many experiments to rather stupifying amounts of accuracy.

Actually, time dilation itself has been directly measured, so the question has to become: Well, this suggests I can detect Bob’s clock slowdown, so what’s wrong with my assumptions?

Here’s a thought experiment which partially shows that time can seem to slow down for someone other than yourself. For those relativity/math gurus, note that I’m ignoring Lorentz contraction in the direction of motion for the moment.

Assume that the speed of light is constant.

Imagine that you’re in a spaceship and you pass by Bob, in another identical spaceship at a point p0. You’ve arranged to pass by him at 50% of the speed of light. You’ll pass by him at a time we’ll call t0.

Let’s assume you both have a clock on board which, from the point of view of the person on that spaceship ticks away every second. At every tick of Bob’s clock, Bob sends out a light pulse to you (and you to him each one of your pulses, but ignore that for the moment).

The clock indicates a passage of time. In the normal course of operations, you can measure everything Bob is doing according to the passage of time of his clock, as much as you can measure everything you’re doing in terms of the passage of time of your clock. If it takes you ten minutes to play a game of solitaire by your clock, and Bob’s your (twin, identical) brother, he can play the same game of solitaire in the same time. That is, your local clock can be considered a measurement of the continuous passage of time even though the ticks are discretely 1 second apart.

So, you’re watching light pulses from Bob as you pass by him. Those light pulses correspond to ticks of Bob’s clock. That means the light pulses are indicative of the continuous passage of time on his ship. That game of solitaire you play that takes 600 ticks of your clock (10 minutes) should take Bob 600 ticks of his clock, or 600 pulses of light from his ship.

But you’re rocketing away from him. Even if your time frame is personally slowing down locally on your spaceship you’d be unaware of it because you’re slowing down too, so your game of solitaire is still 600 ticks of your clock long. So local time on your spaceship is still ticking away merrily at the normal rate.

So the only thing that matters is relative time – that is, how are you perceiving Bob’s passage of time? Your own time must by definition seem normal, so the only question of interest is what’s happening aboard Bob’s ship as viewed from yours.

You know your ship and Bob’s are identical, as are the clocks. So you know his clock should be ticking away at what Bob considers to be the normal rate.

So let’s assume for the moment that Bob’s rate of time passage is the same as yours.

This means that he’s sending out pulses every second. At your time t1, one second after passing him, he sends out a pulse. This pulse races away from him at a fixed speed, the speed of light. The pulse is heading towards you. You’re at P1 at the time this happens.

At your time t1, though, you’ve moved away from him by a distance of 0.5C1 (pick your units… 0.5C meters per second, times 1 second, equals 0.5C1 meters). So the light’s going to have to travel this distance to get to you, and since the speed of light is constant, this will take a known amount of your time. In fact, it will take 0.5C*1/C, or 0.5 seconds.

But in 0.5 seconds, you’ll have moved away from P1. You’ll be slightly further, so the light will have to travel even further. This additional distance will take 0.25 seconds. But in that time you’ll have moved even further… and so on. The gap is getting smaller, of course, and the light pulse from Bob will eventually reach you, but it won’t reach you at a distance which is 1 of your seconds of travel at 0.5C. Wherever this point is (you could work it out, but it doesn’t really matter where it is), it is after P1. Call this point P1’.

P1’ is strictly further along in space than P1. Since P1 is the distance covered by your spaceship at half the speed of light, P1+dP represents P1’. dP represents this extra distance. But the speed of light is constant, and your velocity is well known relative to Bob. (P1+dP)/C gives the time it takes, therefore, for the pulse to reach you.

But C is constant, and the distance you must have covered before the pulse got to you is greater than the distance you should have covered in 1 of Bob’s pulses if your clocks are passing time at the same rate. Bob’s pulse is covering longer distance because you’re moving away from Bob. But because the speed of light is constant, this implies it takes a longer time for the pulse to reach you.

This implies either that your velocity isn’t what you thought it was, or that Bob’s pulses aren’t coming out every second.

However, your velocity was established as part of the thought experiment, and you never underwent acceleration.

Therefore, Bob’s clock isn’t ticking every second, from what you can tell. It’s ticking slower because the light is taking extra time to reach you, so the events of the pulse reaching you are separated by more than a second.

You can argue the same thing about the arrival of the second, and all subsequent pulses.

Bob’s pulses, however, are indicative of Bob’s clock, and the ticks of Bob’s clock describe his continuous passage of time. It’s not just that the pulses seem to arrive slower… this implies that the clock itself is ticking slower, and so everything that Bob does that you could possibly witness from your vantage point must be moving slower.

You can also argue about all of this about Bob’s interpretation of your time relative to his (its symmetric). This means that Bob must perceive your clock as running slowly.

This just goes to show that the apparent passage of time is a local phenomenon, and that there is no universal clock. Each of you is no more right than the other. If you’re both passing time normally (to yourselves), but the other appears to be passing time slowly, then there can be no universal measure by which to analyse events anywhere in the universe.

Sadly, this isn’t quite the way the universe works. The math above is slightly bogus. If you analyse the light pulses in this way, you’ll come to the incorrect conclusion that if you’re both heading towards each other that you’ll see each other running faster. This isn’t the case. You’ll still see each other running slower.

The problem is that in the direction of motion, as you approach the speed of light, there’s an effect called Lorentz contraction that monkeys with relative distances – but only in the direction of motion. This means that some of the time dilation effect described above is accounted for by Lorentz contraction, and it is this effect that changes the nature of the approaching-your-brother case.

So all I meant to describe above is to show that you can have a situation that describes the fact that time dilation can exist. The math above is not meant to show how much dilation occurs.

You can, however, set up experiments (like the light-clock-on-a-train thought experiment) that shows the true time dilation effect by constructing light-motion measurement that is perpendicular to the direction of travel and therefore is not subject to Lorentz contraction effects.
There, I’ve put my understanding under the microscope of the dopers, who will most undoubtedly shoot many holes in this.

Ok, so both the clocks would show a slowdown. Presumably this would be of the same amount considering that you both observe your relative speed difference as the same, right? Supposing after the initial burst of speed to reduce one or both of the clocks 1 second, they are slowed to a relative stop and their times compared. Will they match their readings or not? If they are the same, then there would have to be some point where an observer from one of the clocks would see the other accelerate to match. If they are different, then on what basis are they changing?

Going back to the 747 post… If the clock on the 747 (or the GPS) slowed down after traveling quickly and then being brought to rest. The change in time is obviously permanent, right? If both observers see the other clock slowing, then when they stop and compair their readings they would have to agree at some point. They can’t BOTH be slower than the other!!

Takje a look at Mach’s principle. This might suggest that there is such a thing as an inertial frame of reference, due to the gravitation of all the other objects in the universe, and so speed, or velocity, is not in fact relative, as it would be in the OP thought experiment.

Both… (From the other’s POV)

No. Speed is not relative in the real universe because of inertia and Mach’s principle.
We do not live in a universe with just two objects in it- the inertia of the objects would be defined by relation to each other if we did, and very different rules would apply.
In our universe only the moving object undergoes time dilation.

Well that’s different, because the 747 has accelerated.

If two people on spaceships are travelling apart from each other, they will see the other’s time as being slower because it takes longer and longer for the signal from the other ship to reach them.

The 747 case is like if you had two spaceships travelling apart from each other, and one turned around and caught up to the other. When they meet, the clock of the spaceship that turned around would be behind that of the one that didn’t.

Speed is relative but acceleration is not (or not as it relates to this explanation). So you can tell which is moving from the intitial frame.

When the two objects move away from each other, at least one of them have to accelerate (read, “Twins paradox”). The change in intertial frames when accelerating will mean that when the two clocks are brought back to relative stop to check the clocks, the object that endured the larger acceleration will have had a shorter experience.

If they both accelerate the same amount, I assume the clocks would match if they both go through the same acceleration profiles and end up where they start.

That’s right, sorry, it is acceleration that is the key…
that and the inertial frame defined by the gravity of distant galaxies.

You must must must remember that the clock you’re carrying on your spaceship always appears normal, so you can’t ‘slow it down’ to a stop. You could slow the other one down to a stop by travelling away (or towards) at the speed of light, but you can’t do that, so that too is moot.

Nothing you can do can change the rate your local clock appears to be running.

As has already been pointed out, acceleration bolluxes up the whole discussion. Acceleration is indistinguishable from a gravitational field (the equivalence principle), and this has an effect on time dilation too. If you’re in a heavy gravitational field, or under acceleration, the situation is no longer symmetric.

If you’re at a great distance from a black hole, and Bob’s near the event horizon and hovering there (by applying a goofy amount of thrust), then he and you can be stationary relative to one another. Yet, you perceive him as having a slow clock, and he perceives you as having a fast clock. Suddenly, the situations aren’t symmetric.

The difference is that in the velocity-only case described before, neither you nor Bob was accelerating, so the situations were entirely symmetric.

But this introduces an interesting problem. You can end up with different ‘permanent’ passages of time by using acceleration.

Let’s say you and Bob perform another experiment, and when you start, you’re exactly the same age.

Let’s assume you’re in deep space so the effects of gravitational fields are negligible. You apply your super thrusters and head away from Bob, and you achieve 50% of the speed of light in a few seconds (you’ve got technology to prevent you from becoming chunky salsa on the aft end of your spacecraft). Then, you stay at 50% of the speed of light for 10 years, and you think Bob’s clock is slow, and he thinks your clock is slow, for the entire 10 years, except for the few seconds you were accelerating, which we’ll get back to in a moment.

At the 10 year point, you turn around, applying thrust, and come back to Bob in another 10 years (plus the handful of seconds to apply thrust).

When you get back to Bob, and stop (taking another few seconds), you find that Bob has aged more than you.

What gives?

The situations should be totally symmetric. If you were heading away from Bob, Bob was heading away from you, right? And when you were heading towards Bob, Bob was heading towards you, right?

The thing is, though, while Bob didn’t change his original position, didn’t apply any acceleration, you did. You applied thrust, gave yourself additional velocity, and even did it twice more before you finished.

While you weren’t thrusting, it’s true that Bob and you saw each others’ clocks as running slow, even though you each saw your own clocks running perfectly normally.

However, acceleration means that you changed from one ‘non-inertial’ reference frame (Bob’s, stationary relative to him) to another ‘non-inertial’ frame (heading away from Bob at 50% of the speed of light).

Bob did no such thing. He stayed in the original reference frame.

This is often called the Twin Paradox, but it’s not really a paradox. Ultimately, applying acceleration can produce an asymmetry in the passage of time.

What does this mean for clocks?

It means Bob’s clock will actually read a different time than yours, when you return.

Your clocks do not necessarily agree.

This has been experimentally proven with real clocks using gravity instead of acceleration. It has been shown, using super accurate atomic clocks, that time way way up in a high-altitude aircraft does indeed pass time more quickly than time down here in the gravity well.

So whenever you have one clock or the other undergoing acceleration, you can introduce asymmetries in clock behaviour.

Your post was well put but I think you meant “inertial”, not, “non-inertial.”

If it helps you to visualize things, consider velocity as a rotation (barring a few mathematical nitpicks, this is actually a pretty good analogy). The example I like to use has two trains, stopped at an intersection.

There’s an intersection of two train tracks, at a 60[sup]o[/sup] angle. Two identical trains (each 100 meters long) are stopped at the intersection, while the engineers get out and argue over who has the right-of-way. So the fronts of the trains are at the same point (the intersection), and the angle between the trains is 60 degrees. You’re sitting right in the middle of one of the trains, and looking straight out your window, you see the back end of the other train. Since you’re 50 meters from the front of your train, and the back of the other train is directly to the side of you, you say that the other train is only 50 meters long. It’s awfully wide, as well, but that’s another matter. In your reference frame, the length is only 50 meters.

Meanwhile, another fellow in the other train measures his own train to be 100 meters long and essentially 0 width, but he measures yours to be 50 meters long and 86.6 meters wide. In his frame of reference, he’s right, too. But, if you take the width squared plus the length squared, you’ll get the same value no matter what reference frame you’re in (10000 m[sup]2[/sup]).

The situation in relativity is similar. If you look at two points (actually, events, since time matters, too), then the spatial distance between them will depend on your reference frame, and the time difference will also vary. But, if you take the right combination of the spatial and time difference, you’ll get something (called the “proper interval”) which does not depend on reference frame. The only weird part is that, instead of it being [symbol]D[/symbol]x[sup]2[/sup] + [symbol]D[/symbol]t[sup]2[/sup], it’s [symbol]D[/symbol]x[sup]2[/sup] - [symbol]D[/symbol]t[sup]2[/sup].

Whoops, sorry, not enough coffee prior to posting. You’re right. I meant that you were moving from one inertial frame to another. Otherwise, read my post as stated.

Sorry, very nervous. Trying to get relativity right. Already got beaten up in another thread on the topic.

Ok, so the change in the clock would be based on which accelerated, which can be determined by inertia. This is mostly making sense now… but just one more question.

There are three clocks, named C1, C2, and C3. C3 is considered “stationary”. C1 and C2 start out moving at 75% the speed of light away from C3, side by side showing no difference in relative position between them. Halfway through our observation C2 slows to only 25% the speed of light on an otherwise unchanged course. The observation continues to an end, and each clock comes to a halt in relation to each other (either “instant brakes” or just a snapshot of the moment is up to your preference). To C3, both C1 and C2 started out running slower than C3, but halfway through the period C2 slowed in speed and “sped up” in relation to C1’s time (but was still slower than C3). From C3’s point of view, from fastest clock to slowest the order is C3, C2, C1. However, from C1’s point of view C3 began moving at 75% the speed of light away from the stationary C1 and C2. Halfway through the period C2 accelerates along C3’s course at 50% the speed of light. From C1’s point of view from fastest clock to slowest is C1, C2, C3.

Now, assuming that either C3 or C1 can be proved correct, would one of the observers at the clocks be able to determine past acceleration by differences in the expected results of time dilation? Or is it just a mistake to try to perceive both of their points of view at the same time?

Wow William, I was thinking about time dilation along exactly those lines recently and was planning to post basically the same way you did.

At least if one of us is wrong, then both of us are. Unless “wrong” is also relative, of course.

This is true for the symmetric case you probably mean. It’s not quite that simple in general. For example, suppose you have two clocks, the first on a rotating space station (which is orbiting Earth) experiencing 1g of acceleration. The other is always accelerating at 1g in a straight line until it gets to, say 0.99 C, then decelerating at 1g to a stop, then doing the same thing on the way back. The second clock will be running slow when it gets back, but the first won’t be (It will only be fast or slow by around the amount satellites experience, a small effect, but without looking at it more carefully, I couldn’t even say whether it’d be slow or fast.)

This is true for the symmetric case you probably mean. It’s not quite that simple in general. For example, suppose you have two clocks, the first on a rotating space station (which is orbiting Earth) experiencing 1g of acceleration. The other is always accelerating at 1g in a straight line until it gets to, say 0.99 C, then decelerating at 1g to a stop, then doing the same thing on the way back. The second clock will be running slow when it gets back, but the first won’t be (It will only be fast or slow by around the amount satellites experience, a small effect, but without looking at it more carefully, I couldn’t even say whether it’d be slow or fast.)