Some questions about relativistic effects.

[politas]

There are some people on these forums who have a much better understanding of relativistic effects than myself, so I was wondering if anyone can answer these questions, which might help me to understand it.

Imagine a spaceship 189,000 miles wide, with a light source on one side and a mirror on the other side, arranged so that light from the light source can be seen in the mirror.

If the spaceship is at rest relative to the observer, the light from the light source should appear one second later in the mirror, right?

So, if D is the difference in seconds between the light reaching an observer from the light source and the mirror, at rest, D=1.

What happens to the value of D under the following circumstances, where speeds and accelerations are sufficient for measurable relativistic effects? (I'm looking for answers like "D<1","D>1","D=1"):

1) Light source and mirror are in a line perpendicular to a line drawn from ship to observer, and

a) Spaceship is accelerating away from the observer, and travelling away from the observer.

b) Spaceship is accelerating away from the observer and travelling towards the observer.

c) Spaceship is accelerating towards the observer and travelling towards the observer.

d) Spaceship is accelerating towards the observer and travelling away from the observer.

e) Spaceship is travelling towards the observer at a fixed velocity relative to the observer.

f) Spaceship is travelling away from the observer at a fixed velocity relative to the observer.

2) Light source and mirror are in a line parallel to a line drawn from ship to observer, with the mirror only half the above stated distance further away from the observer than the light source (so that at rest, D=1), and

a) Spaceship is accelerating away from the observer, and travelling away from the observer.

b) Spaceship is accelerating away from the observer and travelling towards the observer.

c) Spaceship is accelerating towards the observer and travelling towards the observer.

d) Spaceship is accelerating towards the observer and travelling away from the observer.

e) Spaceship is travelling towards the observer at a fixed velocity relative to the observer.

f) Spaceship is travelling away from the observer at a fixed velocity relative to the observer.

 

[Ziggurat]

What happens to the value of D under the following circumstances, where speeds and accelerations are sufficient for measurable relativistic effects? (I'm looking for answers like "D<1","D>1","D=1"):

Working out the details for each scenario takes some time, and I'm not actually sure how general an answer you can really get (it's easier to work with specific numbers). But I will point out that solving case #2 with acceleration is actually highly non-trivial. The problem is that accelerating large objects at relativistically significant speeds will necessarily run into the problem of the speed of sound in the ship. For example, let's say you've got thrusters on the back side of this giant ship which is one light-second long. You turn the thrusters on, and the ship starts accelarating. OK, but which PART of the ship? Well, the speed of sound in the ship cannot exceed c (in real materials, of course, it's not even close), which means that the front end of the ship CANNOT start accelerating until at least one second AFTER the back end of the ship has started to accelerate. So even assuming a perfectly stiff ship (speed of sound = c), calculating the trajectories of the front and back of the ship becomes quite difficult.

Say we want to try to simplify this. We stick thrusters on the front and back of the ship, with sufficient power such that both ends of the ship start accelerating together (let the middle part of the ship be stretchy for this part). Suppose they each accelerate for 1 second, and obtain some significant fraction of c. In the reference frame they started in, the front and back ends of the ship are both moving at the same speed, and are still the same distance apart. But in the ship's reference frame, you'll find that the distance between the front and back end of the ship has gotten larger. We've stretched the ship by accelerating both ends at once. While their trajectories are now easy to calculate, we've fundamentally changed the problem you were asking.

In other words, once you throw in acceleration, particularly for such a large object, you're forced to address a whole host of complicated questions, and figuring out the answers isn't easy. It's not so bad for case 1, since you can accelerate both the light and mirror side without worrying about length contraction between them, but I'm still not sure how general an answer you can give without more details of the speeds and the position of the observer. The observer can be placed exactly between the light and the mirror when there's no relative motion and no acceleration, but there isn't such an obvious place to position the observer in the other cases, particularly once you add acceleration, and the position of the observer is going to make a big difference in your answer (even with the ship at rest).

 

[politas]

The size of the ship isn't really important here, at least, not for what I'm trying to find out. I'm wondering about how time dilation works. I don't understand when things appear to speed up and down. The idea was to have some time-based event occur that is measurable and visible to an observer. Treat the spaceship as a point. D is the duration of a measurable, timed event that has a value of 1 when the ship is at rest relative to the observer. The light/mirror thing is an artificial construct to create such a measurable event, since it is based on a handy constant that doesn't require any masses moving relative to each other. Shrinking the distance between light and mirror reduces the base value of D; as long as variations are still measurable, it can be far smaller, if that helps.

What I want to know is under what conditions does D reduce, and under what conditions does it increase? (to the observer, obviously; it is always 1 for the moving object)

Case 2 I expected to be more complex than case 1; I didn't realise just how complex. I'm primarily interested in case 1.

See, I was reading some of the discussions about relativity, and the "twin paradox", and wondering about the revolutions. How fast are they occurring on the trips out and back, both while accelerating and "decelerating? I wanted to reduce the question to a simpler case, as a thought experiment to try to get a better grasp (if not a complete understanding; the maths is beyond my grasp without serious study I just don't have the time or inclination for)

 

[Ziggurat]

What I want to know is under what conditions does D reduce, and under what conditions does it increase? (to the observer, obviously; it is always 1 for the moving object)

If I understand you correctly, then what you're asking is basically when does it look like a clock on the ship is slowing down (you can turn the bouncing light problem into a clock problem if you ask how long it takes the reflected light to get back to the source, and not to some external observer whose position is then relevant). Clocks will always be *observed* to slow down for any moving object, regardless of direction or acceleration. But they can *look* like they're moving faster if the object is moving towards you, and *look* like they are slowed down even more than they are if the object is moving away from you: Doppler shift effects are added on top of time dilation when you want to know what you actually see.

Note that there's a distinction in relativity between what you see and what you observe. An example from everyday life is airplanes flying overhead at a significant fraction of the speed of sound. You hear the noise of the plane coming from some position behind where it is currently located, but knowing the speed of sound, you know that you're hearing sound that was emitted some time ago, and if you've been watching and calculate things correctly, you *observe* that the sound came from where the plane was, not from some point behind the plane. Observations in special relativity are independent of such optical illusions, but what you actually see can have very strong optical illusion effects. A famous example from astronomy is objects close to the speed of light moving towards you at an angle can *look* like they are moving sideways faster than the speed of light. But correctly accounting for their motion towards you (which can be hard to measure at intergallactic distances) will correctly produce an *observation* of the object moving at less than c.

 

[rwguinn]

If I understand you correctly, then what you're asking is basically when does it look like a clock on the ship is slowing down (you can turn the bouncing light problem into a clock problem if you ask how long it takes the reflected light to get back to the source, and not to some external observer whose position is then relevant). Clocks will always be *observed* to slow down for any moving object, regardless of direction or acceleration. But they can *look* like they're moving faster if the object is moving towards you, and *look* like they are slowed down even more than they are if the object is moving away from you: Doppler shift effects are added on top of time dilation when you want to know what you actually see.

Note that there's a distinction in relativity between what you see and what you observe. An example from everyday life is airplanes flying overhead at a significant fraction of the speed of sound. You hear the noise of the plane coming from some position behind where it is currently located, but knowing the speed of sound, you know that you're hearing sound that was emitted some time ago, and if you've been watching and calculate things correctly, you *observe* that the sound came from where the plane was, not from some point behind the plane. Observations in special relativity are independent of such optical illusions, but what you actually see can have very strong optical illusion effects. A famous example from astronomy is objects close to the speed of light moving towards you at an angle can *look* like they are moving sideways faster than the speed of light. But correctly accounting for their motion towards you (which can be hard to measure at intergallactic distances) will correctly produce an *observation* of the object moving at less than c.

I believe (that word again) that your reference to Doppler effect on time dialation is incorrect. I was under the impression that the value is a scalar, not a vector quantity.

Roger

 

[Ziggurat]

I believe (that word again) that your reference to Doppler effect on time dialation is incorrect. I was under the impression that the value is a scalar, not a vector quantity.

Roger

I'm using the term "added" rather loosely, and should have said something more like "you calculate doppler shift in addition to time dilation" (although both the Doppler effect and your time dilation numbers will both be scalar quantities, and I think you'll actually need a multiplication). What I meant was that you first calculate the time dilation using relativity to figure out how much a moving clock is truly slowed down in reference frame. This calculation only depends on the magnitude of the relative velocity of the clock. After you have calculated the time dilation, you then calculate a Doppler shift based upon not only the magnitude of the velocity, but also relative direction. If the object is moving away from you, it is red-shifted, if it is moving towards you, it is blue-shifted, if it is moving perpendicular to you there is no Doppler shift. But in all cases there is still a time dilation effect. It's the fact that you have to do the Doppler shift calculation as well as ("in addition to") the time dilation if you want to figure out what you would see (and not just observe) that I was trying to get at. Hope that clarifies it.

 

[rppa]

The size of the ship isn't really important here, at least, not for what I'm trying to find out.

But such things as the fact that the front and back of the
ship will be moving at different velocities, does come into play in resolving any paradoxes. As a general rule, when people create relativity paradoxes, apparent contradictions, there's usually something hidden which violates causality in some way (for instance, infinitely rigid sticks for which the far end moves at the exact same instant that the near end is moved).

BTW, one small nitpick: the number is 186,000, not 189,000.

I'm wondering about how time dilation works. I don't understand when things appear to speed up and down. The idea was to have some time-based event occur that is measurable and visible to an observer. Treat the spaceship as a point.

A red flag is going up in my mind: It's not a point, and the physical length may make a difference in what you're about to say. But let's reserve judgement...

D is the duration of a measurable, timed event that has a value of 1 when the ship is at rest relative to the observer. The light/mirror thing is an artificial construct to create such a measurable event, since it is based on a handy constant that doesn't require any masses moving relative to each other.

I'm confused about the setup, and one of the things that makes a big difference is how you measure D.

Here's how things might be phrased when you're being careful about stating the conditions: "Two clocks are synchronized and then separated and moved to the positions of the source and the mirror. When the light leaves the source, a time signal is sent from the source. When the light arrives at the mirror, another signal is sent from the mirror. How far apart is the time of arrival of these two signals at the observer? What is the difference between the timestamps sent in the two signals?"

Here are at least two important undefined things in your setup:
(1) Which direction is the motion, relative to light-mirror axis and position of observer?

(2) How does the observer measure what time the light left the (distant) source and what time it arrives at the (distant) mirror?

You can't just talk about things happening at exactly the same time at physically separated places, because different observers won't agree on which events are simultaneous.

I can say some general things, though. People have already mentioned the relativistic Doppler effect and that will come into play here. There are two parts to relativistic doppler: One is the standard doppler, which causes the times between events to appear shorter when an object is getting closer (blue shift) and longer when the object is receding (red shift). The other part is a smaller time dilation effect, which causes times to appear shorter and is independent of direction of motion.

When an object is moving perpendicular to the line of sight, there is no classical doppler, just the time dilation effect. This is called "transverse doppler", and is a purely relativistic prediction.

 

[politas]

The size of the ship isn't really important here, at least, not for what I'm trying to find out.

But such things as the fact that the front and back of the
ship will be moving at different velocities, does come into play in resolving any paradoxes. As a general rule, when people create relativity paradoxes, apparent contradictions, there's usually something hidden which violates causality in some way (for instance, infinitely rigid sticks for which the far end moves at the exact same instant that the near end is moved).

BTW, one small nitpick: the number is 186,000, not 189,000.

Whoops. I really didn't want to get into the physics of the moving object for this. Fascinating as it is, what I'm trying to do is get a grasp on the individual effects, before trying to understand how they interact.

A red flag is going up in my mind: It's not a point, and the physical length may make a difference in what you're about to say. But let's reserve judgement...

D is the duration of a measurable, timed event that has a value of 1 when the ship is at rest relative to the observer. The light/mirror thing is an artificial construct to create such a measurable event, since it is based on a handy constant that doesn't require any masses moving relative to each other.

I'm confused about the setup, and one of the things that makes a big difference is how you measure D.

Here's how things might be phrased when you're being careful about stating the conditions: "Two clocks are synchronized and then separated and moved to the positions of the source and the mirror. When the light leaves the source, a time signal is sent from the source. When the light arrives at the mirror, another signal is sent from the mirror. How far apart is the time of arrival of these two signals at the observer? What is the difference between the timestamps sent in the two signals?"

Ok, maybe you could phrase it that way, but then I'm no longer sure what's being measured.

Here are at least two important undefined things in your setup:
(1) Which direction is the motion, relative to light-mirror axis and position of observer?

perpendicular

(2) How does the observer measure what time the light left the (distant) source and what time it arrives at the (distant) mirror?

By tracking it very carefully with a really big telescope?

You can't just talk about things happening at exactly the same time at physically separated places, because different observers won't agree on which events are simultaneous.

I can say some general things, though. People have already mentioned the relativistic Doppler effect and that will come into play here. There are two parts to relativistic doppler: One is the standard doppler, which causes the times between events to appear shorter when an object is getting closer (blue shift) and longer when the object is receding (red shift). The other part is a smaller time dilation effect, which causes times to appear shorter and is independent of direction of motion.

Ok, so does the doppler effect act in much the same way as it does for sound?

When an object is moving perpendicular to the line of sight, there is no classical doppler, just the time dilation effect. This is called "transverse doppler", and is a purely relativistic prediction.

I think my set up was flawed for what I'm trying to understand. How about we forget about the enormous spaceship and try strobe lights?

Take two strobe lights, A and B, each producing a flash once per second. Start at time T0. Accelerate A away from B until time T1, then towards B until they are at rest relative to each other, at time T2. Continue accelerating A towards B until time T3, then accelerate A away from B until A and B are back together at time T4. Acceleration has a constant scalar value throughout the trip, only changing direction.
Scalar value of acceleration is such that at time T4, A has flashed half as many times as B.

If D1 is the duration between flashes emitted by A, as viewed from B, and D2 is the duration between flashes emitted by B, as viewed from A, how do D1 and D2 change throughout the journey?

 

[RussDill]

Take two strobe lights, A and B, each producing a flash once per second. Start at time T0. Accelerate A away from B until time T1, then towards B until they are at rest relative to each other, at time T2. Continue accelerating A towards B until time T3, then accelerate A away from B until A and B are back together at time T4. Acceleration has a constant scalar value throughout the trip, only changing direction.
Scalar value of acceleration is such that at time T4, A has flashed half as many times as B.

If D1 is the duration between flashes emitted by A, as viewed from B, and D2 is the duration between flashes emitted by B, as viewed from A, how do D1 and D2 change throughout the journey?

Thats the twin paradox

http://en.wikipedia.org/wiki/Twin_paradox

However, in the twin paradox, general relativity makes a back seat, and i think part of your question relates directly to general relativity. I think general relativity would just slow time further for the accelerator in relation to the one that does not accelerate.

[edited to add: I think the answers depend on what the rate of acceleration is. If you are just traveling away from a strobe source, the beats will slow down. If you are accelerating away from something, the beats will speed up. So, which net effect they add up to depends.

 

[Ziggurat]

However, in the twin paradox, general relativity makes a back seat, and i think part of your question relates directly to general relativity. I think general relativity would just slow time further for the accelerator in relation to the one that does not accelerate.

General relativity has nothing to do with any of this. You don't need it unless you deal with gravity, and nobody brought up gravity. Special relativity can handle acceleration quite fine on its own.

 

[RussDill]

General relativity has nothing to do with any of this. You don't need it unless you deal with gravity, and nobody brought up gravity. Special relativity can handle acceleration quite fine on its own.

acceleration and gravity are equivelent.

 

[politas]

Thats the twin paradox

http://en.wikipedia.org/wiki/Twin_paradox

Ok, the diagram on that page shows this big gap between the simultaneity planes for the stationary twin. What happens in between those times? What is seen?

 

[Atlas]

I've got a question too for you smart guys.

If the ship is at rest I think a strobe goes straight and hits the mirror 1 light second away.

But if the ship is traveling at c would a laser strobe have to lead its target. That is, shoot along the hypotenuse of a right triangle to where the refector target will be in, what would it be, the square root of 2 light seconds?

Does that calculation need to be part of the math to compute the answer?

 

[RussDill]

Ok, the diagram on that page shows this big gap between the simultaneity planes for the stationary twin. What happens in between those times? What is seen?

Basically, if you had a side view from the stationary point (B), as it's headed out, you'd see strobe spaceing like this:

B | | | | | | A->

So both parties would be receiving strobes at intervals longer than once per second.

When A turns around, you have a situation like this (strobes from A, B's frame of reference)

B | | | |||||A<-

As you can see, they are stacking up. The diagram on the wikipedia page represents the three frames of reference. It is trying to show the two different paths a and b traveled through spacetime to arrive at their final destination.

 

[RussDill]

I've got a question too for you smart guys.

If the ship is at rest I think a strobe goes straight and hits the mirror 1 light second away.

But if the ship is traveling at c would a laser strobe have to lead its target. That is, shoot along the hypotenuse of a right triangle to where the refector target will be in, what would it be, the square root of 2 light seconds?

Does that calculation need to be part of the math to compute the answer?

first of all, if the ship is traveling at c, you are screwed anyway, because by the time you see it, it's too late to fire a laser in order to hit it.

Second, I assume in this example, the ship is traveling past you, ie:

ship->


you

You need to do some math to calculate its current position and trajectory based on two observations. Once you do that, you can calculate where it would really be, since light takes time to reach you. Then, fire a laser beam so it intersects with the path of the ship, taking into account, again, the speed of light.

In this case, you don't need any relativity, just the speed of light.

 

[Atlas]

                Blink  *
                      /|
                     / |
                    /  | 
<------------------<   |------------------
                    \  |      <=== direction        
                     \ |           speed = c 
                      \|          
                mirror>-  1 light second to near wing tip
                          2 light seconds to far wing tip





                       O       
                       me

I was thinking something like this. The original ship was 1 light second across. If the blink occurs where its right in my line of sight I'll see it 2 seconds after the flash but it will have missed the mirror which has moved forward by 1 light second by the time the light reaches it.

The flash would have to be pointed forward at 45 degrees for it to hit the mirror if it were traveling at c. Am I thinking about that right?

 

[69dodge]

http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_paradox.html

 

[Stimpson J. Cat]

I've got a question too for you smart guys.

If the ship is at rest I think a strobe goes straight and hits the mirror 1 light second away.

But if the ship is traveling at c would a laser strobe have to lead its target. That is, shoot along the hypotenuse of a right triangle to where the refector target will be in, what would it be, the square root of 2 light seconds?

Does that calculation need to be part of the math to compute the answer?

It's not just a part of the math to compute the answer, in effect, it is the math for computing the answer. That is, you can derive the formulas for time dilation and length contraction from simple geometry, using the assumption that the speed of light is the same for all observers.

Consider the following triangle:

                                  ***
                               ***  *
                      z     ***     *
                         ***        * y
                      ***           *
                   ***              *
                *********************
                           x

Here x is the distance traveled by the light from the POV of the person on the ship, y is the distance traveled by the ship, as measured by the stationary observer, and z is the distance traveled by the light, as measured by the stationary observer. This gives us

z^2 = x^2 + y^2, or equivalently x^2 = z^2 - y^2

If we let t be the time as measured by the stationary observer, and t' be the time as measured by the person on the ship, then this gives us

(ct')^2 = (ct)^2 - (vt)^2

where v is the velocity of the ship. Solving for t' we get

t' = t sqrt(1-(v/c)^2)

which is the equation for time dilation. Keeping in mind that both observers agree on the relative velocity v, this means that

y' = y sqrt(1-(v/c)^2)

where y' is the distance traveled by the ship from the POV of the person on the ship. Since less time passes for the person on the ship, the distance he has traveled must be shorter by the same factor.


Dr. Stupid

 

[Ziggurat]

acceleration and gravity are equivelent.

No. Acceleration is equivalent to a UNIFORM gravitational field. Special relativity can handle a uniform gravitational field exactly the same as it can handle acceleration. You're just constantly changing reference frames. However, since every gravitational field of any interest is NOT uniform, you cannot treat gravity globally as a simple acceleration, and special relativity falls apart. Two objects sitting on opposite sides of a planet cannot be treated as if they are accelerating away from each other when they clearly aren't. It is this nonuniformity of gravity which requires general relativity. Absent that, you can do everything you want with special relativity.

 

[rppa]

How does the observer measure what time the light left the (distant) source and what time it arrives at the (distant) mirror?

By tracking it very carefully with a really big telescope?

That's perfectly legitimate, but you have to realize that you don't see events when they happen. You see them only when some sort of signal reaches you. In this case it's the light from the event, which travels to you at speed c and arrives at your telescope d/c seconds after the event happened.

If you further model some way of knowing d, then you can backtime your observation to say "from my point of view, that event happened at distance d, at time d/c ago". Now you have a way of assigning a time in your frame, to something that happened far away.

The point is that in all relativistic discussions, paradoxes are created by being vague about the information being measured (or by postulating faster than light communication), and paradoxes are resolved by acknowledging that information can't get from A to B faster than c.

There are two parts to relativistic doppler: One is the standard doppler, which causes the times between events to appear shorter when an object is getting closer (blue shift) and longer when the object is receding (red shift).

Ok, so does the doppler effect act in much the same way as it does for sound?

What I called the "standard doppler" above is exactly the same as the doppler effect for sound, except that in sound you can have a situation where the observer is moving relative to the medium, changing the apparent sound speed. That can't happen to light.

But the analysis of what happens to signals leaving a moving source is exactly the same as for sound. For a source moving toward you, emitting signals once per second, later signals travel less distance and get to you less than one second after earlier signals.

But there's this time dilation effect also. It's quite small. The size of the sound doppler effect, and the first-order light doppler, is v/c. The size of the time-dilation effect is v^2/c^2, much smaller. You don't see it unless there is no first-order effect (because the motion is perpendicular to line of sight). In that case for sound, there's no effect at all. Zero. But for relativity, there is a transverse doppler.

Take two strobe lights, A and B, each producing a flash once per second. Start at time T0. Accelerate A away from B until time T1, then towards B until they are at rest relative to each other, at time T2. Continue accelerating A towards B until time T3, then accelerate A away from B until A and B are back together at time T4. Acceleration has a constant scalar value throughout the trip, only changing direction.
Scalar value of acceleration is such that at time T4, A has flashed half as many times as B.

If D1 is the duration between flashes emitted by A, as viewed from B, and D2 is the duration between flashes emitted by B, as viewed from A, how do D1 and D2 change throughout the journey?

As somebody has already said, this is the twin paradox. It's actually an easy way to show the twins don't have identical experiences.

A is the accelerating twin. What B sees is that the flashes get farther and farther apart as A recedes. As A starts to decelerate, the corresponding flashes get closer together, but they're still red-shifted: D1 > 1.

When A turns around, the corresponding pulses will be blue-shifted, D1<1. But A is very far away. Those pulses will take awhile to get back to B. By the time B sees the pulses change from D1 > 1 to D1 < 1, A is already enroute back.

Meanwhile, A also sees a red-shift D2>1 as he recedes, and a blue-shift D2<1 when he turns around. But he sees the change as soon as he turns around, because the pulses that are red-shifted/blue-shifted are already nearby, have already travelled out to where he is.