Length and time comparisons in SR

[teodorakis]

Hi, when we say the distance in the direction of motion shrunks and time slows(sorry for my grammar), do we say that if we were in the moving reference frame(relative to us) the length we measure from our stationary frame(relative to the other one) gets bigger and same logic for the time slowing process?
Other way, let's say we see a fish moving in other frame we say it shrunks because in our stationary frame we know the "normal" size of the fish and when we observe the other frame the"normal" size shrunks. and by same logic the swimming process looks slowed because we know the "normal" process in our stationary reference frame.
Lastly to prove the vertical lengths to the direction of motion does not change we say that two observers agreed to make ticks each others vertical rulers' and when they compare the results they must agree the ticks must be same height by symmetry but i stucked at this comparing thing. Does one of them stops(she/he must be deaccelarate in this case) and compare the results which destroys the symetry, or do they compare when they passing each other?

 

[bcrowell]

Lastly to prove the vertical lengths to the direction of motion does not change we say that two observers agreed to make ticks each others vertical rulers' and when they compare the results they must agree the ticks must be same height by symmetry but i stucked at this comparing thing. Does one of them stops(she/he must be deaccelarate in this case) and compare the results which destroys the symetry, or do they compare when they passing each other?

I think there's definitely something missing logically in the argument given in the first sentence, because it doesn't make use of the assumption that the rulers are oriented perpendicular to the direction of motion, and that assumption is necessary. Stopping doesn't work, because if you stopped in the case where the rulers were oriented parallel to the direction of motion, you would also get equal lengths. I think what's missing from your argument is a discussion of the issues relating to time and simultaneity, as in the standard resolution of the garage paradox: http://en.wikipedia.org/wiki/Ladder_paradox . Alternatively, you can take a "ruler" to be shorthand for radar distance and work everything out in terms of the exchange of light pulses.

There are a lot of different ways of proving that there is no length contraction in the transverse direction. For instance, it's possible to prove geometrically that area in the (x,t) plane remains invariant under an x boost. (This follows from boosting a square shape, dissecting the resulting parallelogram into little squares, and then boosting the little squares back into the original frame.) A similar proof holds for volume in (x,y,t) space. Since both x-t area and x-y-t volume are preserved, it follows that lengths in the y direction are unchanged.

 

[teodorakis]

let me ask another way, when we proving there's no length contraction in the transverse direction, we can use thought experiment such as nails on a meter stick, as you know in this experiment each observer has vertical rulers with nails on them as they pass each other they should agree on the length of the vertical rulers one of them can not claim that my 1 meter mark coincides with others 0.8 meter mark, but i say that their nail system should be same to make such conclusin, for example one of them can use larger scale and can say my 0.8 meter coincides with other ones 1m, how do we accept or do we accept that each others' scaling system is same? Do we say that because in their own inertial reference frames,they form the famous clocks and rods lattice by using light beam? And the speed of light is constant for both of them so 1m is in same scale for both of them?

 

[ghwellsjr]

Yes, the meter sticks are identical and the clocks are identical for both observers. If they came to rest with respect to each other and compared them, they would be identical. That is, the meter sticks would all be the same length and the clocks would tick at the same rate.

When one or both of them accelerate so that there is a relative speed between them, they each can measure the relative speed of the other one and it will be identical (except the directions will of course be opposite) and if each one measures the other one's clock rate, they will each measure the other clock's tick rate as going slower than their own. If each one measures the other one's meter stick that is aligned along the direction of relative motion, they will each measure the other one's meter stick as shorter than their own. If each one measures the other one's meter sticks that are aligned perpendicular to the direction of relative motion, they will each measure the other one's similarly aligned meter sticks as identical in length to their own.

And the faster their relative speed, the greater will be the difference between the clock tick rates and the greater will be the difference between the lengths of the meter sticks aligned along the direction of relative speed, but the meter sticks perpendicular to the direction of their relative speed will always be identical.

And if they were to come back together again and compare their clock's tick rates and their meter stick's lengths, they would again all be identical.

And the same thing applies to all other time rates and lengths, the fish would age at corresponding rates as the clocks and swim at corresponding rates and their lengths would be affected in the same way as the meter sticks.

However, each observer cannot tell that their own clocks and meter sticks are in any way changed, it is only the other one's clocks and meter sticks that are changed with respect to their own.

Is all this very clear? Do you have any more questions?

 

[teodorakis]

waow, that was a hell of an answer :). Thank you for your interest.

 

[teodorakis]

When one or both of them accelerate so that there is a relative speed between them, they each can measure the relative speed of the other one and it will be identical (except the directions will of course be opposite) and if each one measures the other one's clock rate, they will each measure the other clock's tick rate as going slower than their own. If each one measures the other one's meter stick that is aligned along the direction of relative motion, they will each measure the other one's meter stick as shorter than their own. If each one measures the other one's meter sticks that are aligned perpendicular to the direction of relative motion, they will each measure the other one's similarly aligned meter sticks as identical in length to their own.

But when one of them accelerate the situation isn't symmetric anymore, can we still use special relativity?

 

[teodorakis]

Sorry to repeat this but i have to fully understand it, i think I'm talking about the proper length or proper time, which is the observers measure in their own resting frame of reference, so these lengths and times are identical because they are both measured using constant speed of light, right? I'm asking this because when we compare the lengths when two frames have a velocity relative to each other, we compare the length marks and this is only meaningfull when the scales are identical at rest.

 

[ghwellsjr]

It may not be symmetric while one of them is accelerating, but after the effect of the acceleration is done, all that is left is a relative speed between the two observers and it doesn't matter which one or both accelerate, it only matters the final speed difference.

We can also use special relativity while acceleration is going on but it's much more complicated to analyze. Most of the time, people ignore the immediate effect of acceleration and just analyze the final outcome because as long as the accelerations are quick, they don't significantly alter the final outcome and they have no effect on the final clock rate differences or meter length differences, except, of course, the accelerations cause the speed difference and it's the magnitude of the speed difference that matters, not how it got there.

Keep in mind, we're only talking about clock rate differences, not the actual time readings on the clocks--that's an entirely different matter.

 

[ghwellsjr]

Sorry to repeat this but i have to fully understand it, i think I'm talking about the proper length or proper time, which is the observers measure in their own resting frame of reference, so these lengths and times are identical because they are both measured using constant speed of light, right? I'm asking this because when we compare the lengths when two frames have a velocity relative to each other, we compare the length marks and this is only meaningfull when the scales are identical at rest.

I have not talked about frames of reference which is a different subject having to do with Special Relativity. I have discussed what each observer measures which is independent of the frame of reference that SR uses to analyze scenarios, so can you see that it really doesn't make sense to talk about what "observers measure in their own resting frame of reference"?

The reason why lengths contract and time dilates is because no matter what speed we have accelerated to, whenever we measure the round-trip speed of light (the only way we can measure it) we get the same constant value. This also has nothing to do with Special Relativity. I am merely describing for you what actually is observed and measured by observers at rest and in motion with respect to each other.

Special Relativity is a way of assigning co-ordinates to a scenario so that we can meaningfully analyze and describe what is happening in situations that are more than trivial, and it does this by arbitrarily assigning the same time interval to both halves of the round-trip measurement of the speed of light. In Special Relativity, measured values are attached the term "proper" to them because they are frame independent as opposed to purely calculated values based on the co-ordinate system in use which are attached the term "co-ordinate" to them and they are generally different when you look at the situation from a different co-ordinate system (also called a frame of reference).

 

[yuiop]

Lastly to prove the vertical lengths to the direction of motion does not change we say that two observers agreed to make ticks each others vertical rulers' and when they compare the results they must agree the ticks must be same height by symmetry but i stucked at this comparing thing. Does one of them stops(she/he must be deaccelarate in this case) and compare the results which destroys the symetry, or do they compare when they passing each other?

There is no need to stop as that complicates things. The important thing is to have a notion of "at the same time" and make make measurements "at the same time". Let us say we attach a vertical ruler to the fish. There are clocks at the top and bottom of the ruler and marker pens also at the top and bottom. These clocks are at rest with respect to each other and therefore tick at the same rate. It is an easy matter to synchronise these clocks. Let us also say we have another vertical ruler that is at rest in the ocean with similar synchronised clocks attached to this second ruler. Now as the fish passes the ruler in the ocean the marker pens leave marks on the ocean ruler. Some (intelligent :tongue2: ) parasites riding on the fish ruler note the time on their clocks as the marks are made. They check their notes later and agree that the marks were made at the same time. Some observers on the ocean ruler do the same and agree that the marks were made at the same time according to the clocks attached to the ocean ruler. Now a measurement of the distance between the marks on the ocean ruler should agree with the length of the vertical ruler attached to the fish.

A similar process is used for measuring horizontal lengths without stopping, but a method called "Einstein synchronisation" is used to define a notion of "at the same time". This notion of simultaneous is not universal for spatially separated clocks and observers with different velocities will disagree on what is simultaneous. This is known as the "relativity of simultaneity" and is an important concept to understand in a study of Special Relativity.

 

[jambaugh]

Let me give you an analogue which helped me better understand this.

First remember that in relativity space and time are unified into a single space-time. So picture a measuring rod existing over time as a ribbon through time. If you want to measure the spatial length of the rod you are speaking of the width of the ribbon. There is a proper width to the ribbon measured orthogonal to its edge but that is a specific frame of reference for the ribbon. Picture the ribbon at an angle and you are seeing it as an observer moving relative to the ribbon. Their "time direction" is at an angle relative to the measuring rod's. When you measure the width in your frame you see a different width /-----/ than someone moving with the ribbon. |---|. (Note this is not identical to relativity it is just an analogue to see why it changes not quantify how it changes.)

Similarly picture the ticking of clocks at various points on the measuring rod as sequences of dots drawn along the ribbon. Specifically imagine dots spaced 1 light-second apart along each edge corresponding to synchronized clocks ticking off seconds at each end of a measuring rod. Now to the oblique observer the vertical spacing is not 1 light-second apart but some other value because their "vertical" is at an angle to the ribbon. Likewise the oblique observer has a different notion of things "happening at the same time".

Understand these changes of perspective and then note that while ribbons in space obey Euclidean geometry, space-time has a hyperbolic (pseudo-euclidean) geometry so changes of perspective do not involve rotation x' = x cos(theta) + y sin(theta) but rather a weird hyperbolic pseudo-rotation, x' = x cosh(b) + t sinh(b), which reverses the contraction and dilation of measurements you would see with a real ribbon.

Where "length"=width expands under euclidean rotation of the ribbon, lengths contract for boosted rods. Where "periods between dots" contract for the rotated ribbon, periods between ticks contract for boosted clocks.

 

[teodorakis]

Let me give you an analogue which helped me better understand this.

First remember that in relativity space and time are unified into a single space-time. So picture a measuring rod existing over time as a ribbon through time. If you want to measure the spatial length of the rod you are speaking of the width of the ribbon. There is a proper width to the ribbon measured orthogonal to its edge but that is a specific frame of reference for the ribbon. Picture the ribbon at an angle and you are seeing it as an observer moving relative to the ribbon. Their "time direction" is at an angle relative to the measuring rod's. When you measure the width in your frame you see a different width /-----/ than someone moving with the ribbon. |---|. (Note this is not identical to relativity it is just an analogue to see why it changes not quantify how it changes.)

Similarly picture the ticking of clocks at various points on the measuring rod as sequences of dots drawn along the ribbon. Specifically imagine dots spaced 1 light-second apart along each edge corresponding to synchronized clocks ticking off seconds at each end of a measuring rod. Now to the oblique observer the vertical spacing is not 1 light-second apart but some other value because their "vertical" is at an angle to the ribbon. Likewise the oblique observer has a different notion of things "happening at the same time".

Understand these changes of perspective and then note that while ribbons in space obey Euclidean geometry, space-time has a hyperbolic (pseudo-euclidean) geometry so changes of perspective do not involve rotation x' = x cos(theta) + y sin(theta) but rather a weird hyperbolic pseudo-rotation, x' = x cosh(b) + t sinh(b), which reverses the contraction and dilation of measurements you would see with a real ribbon.

Where "length"=width expands under euclidean rotation of the ribbon, lengths contract for boosted rods. Where "periods between dots" contract for the rotated ribbon, periods between ticks contract for boosted clocks.

ok as i understand you say that things(events maybe) form some blob in spacetime and various observers see the different parts of this blob and note as their measured lengths and time, like in Newtonian universe(where time is constant) different observers see different aspects of objects in their own perspective and for one of them width of something is some value and for another it means another value.