Relativity Paradox

[impwarhamer]

This is a paradox to do with time dilation and relativity that I just can't get my head around.

Let's say a rocket takes off from earth and flies at some significant proportion of the speed of light, then returns to earth.

This is a classic style question, the normal answer would be that because of time dilation, less time would pass for the rocket than for the people on earth.

To put some numbers on it, lets just say the rocket has experienced 1 year, and the earth 100 years.

However, the problem is, this assumes the earth is the observer.

If we take the rocket as the observer, the earth has flown away at near-light speed and returned, and this should give that the earth is now 1 year older while the rocket is 100.

Of cause, they can't be both 1 and 100 years old at the same time, it doesn't work, so what happens?

[Chakat Firepaw]

The "can't" is Newtonian thinking. One of the important results of Special Relativity is that "the same time" is something that is different for different observers.

For the Earth, after 1 year has passed the rocket has experienced 100 years when viewed from the Earth's frame of reference.

For the rocket, after 1 year has passed the Earth has experienced 100 years when viewed from the Rocket's frame of reference.

This is why faster than light travel inherently means that you have time travel, you can bounce between two moving objects in a way that you benefit from the time dilation both ways and arrive before you left.

If you go to the classic rail car example, it would be quite possible to have a set of three photographs where one shows the front door open and the back door closed, one showing the back door open and the front door closed and one showing them both opening.

Now, the next step in your thinking would be to go to the twin paradox. This is where the rocket goes out and comes back with the question becoming "who's older, the twin who stayed on Earth or the one on the ship?" This is a far more complicated issue because it is actually a general relativity problem, (while turning around, the rocket's frame of reference is non-inertial).

[GeneralVeers]

One frame of reference (Earth) is static.

The other (the rocket) is not. The rocket applies a significant force (its engine) that propels it away from Earth. Then it turns around and applies that force again to return. Meanwhile the only significant force the Earth experiences is that of the Sun, which doesn't change (much--cue various complications such as the planet's linear momentum changing since its orbit is an ellipse, and other such obfuscations).

[Gaarst]

The famous "Twin Paradox" is actually very straighforward to solve in special relatvity: the twin on the rocket ages slower.

This is because usual postulates of special relativity are made under inertial frames. Even if you skip acceleration of the rocket, the one on the rocket is in two different inertial reference frames: the rocket going away from Earth, and the rocket going towards Earth.

When this change of frame occur, the notion of simultaneity as it was in the first frame (here the twins at the start of the experiment) is lost and redefined according to the second frame. The best way to visualise this is using a Minkowski diagram

Doing a little maths then concludes the resolution of the problem.

You could also end up with the same result calculating proper time for both twins, and seeing that the stationary twin's proper time is greater.

If you want to to consider continuous acceleration, then you'll have to use General relativity and time dilation under accelerations; and again the problem gets solved the same way.

Edited November 4, 2015 by Gaarst

[WestAir]

Impwarhamer,

The way I understand it is that each observer experiences time differently, but never less than 0. If you were the only bit if information in the Universe I assume it would be impossible to know how fast and in what direction you were traveling (and therefore how much time dilation you were experiencing), but I'm assuming physics still has a way of knowing because you will still have properties that describe your velocity and direction through spacetime. I also assume that means there is a method physics uses to determine which observer - Earth or the Rocket - will experience time dilation. I want to say it's the acceleration and high velocity through spacetime that is the key factor here, but I can't honestly admit to knowing enough to make that claim.

Also, to Wedge, is there even such a thing as a static reference frame? I was under the impression that the Universe has no base reference frame, only reference frames relative from one to the next.

[Gaarst]

One frame of reference (Earth) is static.

Also, to Wedge, is there even such a thing as a static reference frame? I was under the impression that there is no static or base frame of reference in the Universe and it's all just relative from one to the next.

The Earth is static in its own reference frame, that's it.

A reference frame can't be absolutely static, because that would imply they are not all equivalent and this violates Einstein's first postulate of special relativity.

Here, inertial is what is important.

- - - Updated - - -

If you were the only bit if information in the Universe I assume it would be impossible to know how fast and in what direction you were traveling (and therefore how much time dilation you were experiencing), but I'm assuming physics still has a way of knowing because you will still have properties that describe your velocity and direction through spacetime. I also assume that means there is a method physics uses to determine which observer - Earth or the Rocket - will experience time dilation. I want to say it's the acceleration and high velocity through spacetime that is the key factor here, but I can't honestly admit to knowing enough to make that claim.

A direct consequence of Einstein's first postulate is that, in an inertial frame, there is no way to say if you're moving or not without any interaction with another reference frame. And even then, the inertial frames' displacement is relative to each other.

Speed and time dialtion don't matter either because as long as no external force is applied (ie: velocity changing), your reference frame is inertial.

[Nikolai]

as long as no external force is applied (ie: velocity changing), your reference frame is inertial.

Right. And that's the critical difference between the Earth and the rocket in this example. Even though all inertial reference frames are relative -- that is, it's impossible to say who's moving when all reference frames are moving at constant velocity with respect to one another -- accelerating reference frames are not relative. If the rocket accelerates to a decent fraction of light speed, then turns around, then comes back, then matches speed with Earth, it will experience a series of accelerations that Earth does not.

[GeneralVeers]

Also, to Wedge, is there even such a thing as a static reference frame? I was under the impression that the Universe has no base reference frame, only reference frames relative from one to the next.

When you're sitting in a car, or a bus or a jet plane or a rocket or any other vehicle, and the driver steps on the accelerator, you feel it. You know your reference frame is changing relative to what it was a moment ago, even if you're in a closed cabin and can't hear the engine.

[problemecium]

Yep. It's all about the fact that the rocket turned around.

Time dilation can occur in two ways in Special Relativity: You go really fast, or you're in a gravitational field.

General Relativity points out that being in a gravitational field and accelerating are fundamentally equivalent. So here's the real sequence of events:

- Rocket launches from Earth and acclerates for a bit. By the time it reaches top speed, it has aged maybe five minutes while on Earth an hour has gone by. Even aboard the rocket, people will agree that time is going faster on Earth than in the rocket, which is "experiencing a strong gravitational field" as it accelerates.

- Rocket flies off at near light speed. As it does so, looking back, Earth appears to age in slow motion to people aboard the rocket, while the rocket appears to age in slow motion to people on Earth. Naturally the two will start to disagree on what time, day, and eventually year it is.

- Rocket accelerates again so as to turn around, placing itself in a "strong gravitational field" for twice as long as before. Everyone agrees time slows down for the rocket - a LOT.

- Rocket approaches Earth. As it does so, Earth once again seems to be in slo-mo from the rocket and vice versa. As time goes by, of course, this will make the people aboard the rocket perceive Earth as slipping ahead of the rocket in time, helping to make up for their period of slo-mo while accelerating. However, it never makes up for it all the way before they arrive at Earth again.

- Rocket hits the brakes upon return, once more putting itself in slow-motion in time.

If you carefully add up all the time differentials from the periods of acceleration and inertial motion, you'll find that the time differential caused by the acceleration is always more significant than the time differential built up or taken away during inertial motion. Traveling at a greater speed to increase the dilation in inertial motion (in an attempt to restore the paradox) simply means you have to do more accelerating. Traveling a greater distance will only make the time differentials bigger in all phases of the experiment.

At best, you could get the rocket to age only a tiny bit less than Earth by running the experiment very quickly or moving very slowly, but of course this just brings us back toward everyday situations of short distances and slow movement.

P.S.:

- Earth is of course also experiencing a gravitational field, but this is very small (we're not orbiting anywhere near lightspeed and neither Earth nor the Sun is massive enough to cause a large time dilation).

- If instead of turning around, you stopped the rocket and then took Earth and pushed it toward the rocket, Earth would end up experiencing the twins paradox itself and both would arrive together at the same age.

Edited November 4, 2015 by parameciumkid

[PB666]

One frame of reference (Earth) is static.

The other (the rocket) is not. The rocket applies a significant force (its engine) that propels it away from Earth. Then it turns around and applies that force again to return. Meanwhile the only significant force the Earth experiences is that of the Sun, which doesn't change (much--cue various complications such as the planet's linear momentum changing since its orbit is an ellipse, and other such obfuscations).

The more obvious confuscation is that the earth is, in its orbit within its inertial reference frame, and so the sun, other than hv, which is either indirect or direct depending whether you believe light is a particle or wave, is applying a net force on the earth. The solar wind is acting on earths magnetic field the net affect is largely negligable.

The answer to the OP is here:

http://www.sciencealert.com/all-of-richard-feynmans-physics-lectures-are-now-available-free-online

[Alias72]

This gave me an interesting thought...

Since we know that the earth is moving at 1,000s of mph at all times, shouldn't relativity apply more in our rocket if it launches prograde vs retrograde?

harder to answer. All measurements are made from a reference frame.

If we accelerate from earth away from the reference frame then relativistic effects will increase.

If we accelerate from earth towards the reference frame then relativistic effects decrease.

If our frame of reference is the earth then it does not matter. Both rockets accelerate away from the earth at the same rate and experience the same amount of relativistic effects.

Another way of putting it is that all objects experience relativistic effects relative to all other objects in existence. If the objects are moving in the same direction with the same magnitude then the relativistic effects are zero.

This is the equation of time dilation.

dt is the time measured by the moving object.

dt' is the time the observer measures

v is the velocity of the moving object compared to the observer.

because v is the difference in velocity between the observer and the moving object it is different from every conceivable point at which you could measure it.

[impwarhamer]

Thanks for all your comments guys!

So what I'm getting here is that its the acceleration of the rocket that cancels out the paradox?

Gonna have to think about that for a bit, but its a start.

[K^2]

So what I'm getting here is that its the acceleration of the rocket that cancels out the paradox?

In a nut shell. I could write out formulas for what happens to time flow from perspective of accelerated observer on the rocket, but math gets hairy fast. If you haven't done any tensor calculus, it will be Greek to you.

The simplest thing you can do is keep in mind that Special Relativity always works for any inertial observer. So any predictions you make from inertial frame will be valid without having to do complex math.

[GeneralVeers]

This gave me an interesting thought...

Since we know that the earth is moving at 1,000s of mph at all times, shouldn't relativity apply more in our rocket if it launches prograde vs retrograde?

Very slightly. But the difference in relativistic distortion is insignificant.

Earth is orbiting at 30km/sec. Light travels ten thousand times faster than that. When you plug that into the formula Alias72 provided, you're taking that fraction, one ten thousandth, and squaring it. When you square "very small" the calculator spits out "extremely small". That's the amount by which the temporal distortion changes if you flip the rocket around and launch it in the other direction.

[K^2]

Depends on what you need it for. GPS satellites take into account time dilation due to their orbital velocity and due to Earth's gravitational field. Did you know that there exists an orbit where time advances at exactly the same rate as on Earth's surface, because orbital time dilation equals the difference in gravitational time dilation due to different elevations? IIRC, GPS satellites are above that altitude, so their clocks run faster. In contrast, ISS is well bellow that altitude, and their clocks run slower. And when you need to be able to position your satellites just right, it's the sort of thing you have to keep in mind.

[YNM]

Thanks for all your comments guys!

So what I'm getting here is that its the acceleration of the rocket that cancels out the paradox?

Gonna have to think about that for a bit, but its a start.

Well, in any case, the fact that you have to change between two (or more) inertial reference frame destroys all kind of continuity given by usual special relativity. Even if we ignore general relativity, distance plays in the reality (like, how when you're 100 lightyears away, you look at your home 100 years ago, wrt your "current" time). But a proper explanation does need general relativity, the key idea is that accelerating makes your time slower.

This gave me an interesting thought...

Since we know that the earth is moving at 1,000s of mph at all times, shouldn't relativity apply more in our rocket if it launches prograde vs retrograde?

We tried special (maybe with some influence of general) relativity in planes that goes east and west, so on satelites it should be more pronounced.