Exploring the Wonders of Special Relativity

[Saw]

When you hear for the first time the idea that the speed of light is the same in any direction for any observer, regardless his state of motion wrt the source, it sounds as if you were being told that light magically adjusts itself to tease you. Today I pass by a guy who emits a beam of light. I am riding on a monkey: I measure c. Tomorrow I pass by riding on the fastest vehicle in the universe: I still measure exactly c. You think: obviously, the measurement conditions are different, why should I obtain the same result? If I did change the measurement frame, does it mean that light also reacted to change its behaviour so as to fool me?

As obviously that cannot be, I’ve given myself, based on my current understanding of SR, this explanation: what I measure is not necessarily what light “does” or what “happens”. What I measure is what the instruments of my frame (held by myself and my lattice of clocks and rulers at rest with me) do, while light passes by. Due to a complex compensation of effects, I always measure c for light speed, but that does not mean that the same thing has “happened” on each occasion: the proof is that each day the frequency of the light beam is different, so my efforts have not been totally in vain.

Actually, the reasoning is similar to the classical one, it’s just that the actors have changed:

- In classical physics it was thought that a clock would give off the same reading in any frame; certainly, the “ticker” of my clock was not doing the same thing as the ticker of your clock (there is relative motion between the two of us), but since the “counter” of each clock did a different thing as well in the same proportion, a fortunate compensation of effects arose, we all got the same readings and this enabled different frames to transform their measurements leaning on such common element. (The same could be said of rulers.) However, the fact that our respective tickers do different things would still be apparent: if your ticker escapes your clock and hits my nose or your nose, we will feel different impacts (energy, isn’t it?).

- SR stems from the observation that nature works differently. Clocks do not work the same way in any frame (just as rulers). But a similar compensation of effects still takes place in favour of light speed, which thus takes the role formerly held by homogeneous time (and length). However, the fact is that light is not really behaving wrt me as wrt you and that is why, if light escapes out of your clock, I will measure the same speed but different frequency (energy, isn’t it?).

Well, is this more or less on track? Many mistakes…?

 

[Integral]

... SR stems from the observation that nature works differently. Clocks do not work the same way in any frame (just as rulers). But a similar compensation of effects still takes place in favour of light speed, which thus takes the role formerly held by homogeneous time (and length). However, the fact is that light is not really behaving wrt me as wrt you and that is why, if light escapes out of your clock, I will measure the same speed but different frequency (energy, isn’t it?).

Well, is this more or less on track? Many mistakes…?

Time dilation and length contraction are not based on observation, they are predictions made by SR.

You cannot learn anything new about the behavior of light from SR. Einstein postulated the constancy of c. So the Behavior of light is assumed in SR.

To learn about the behavior of light you must come to grips with Maxwell's equations (where the constancy of c originates) and Quantum Mechanics.

 

[Dale]

Time dilation and length contraction are not based on observation, they are predictions made by SR.

Yes, but they are predictions that have been confirmed by observation.

 

[Integral]

Yes, but they are predictions that have been confirmed by observation.

Absolutely!

 

[atyy]

what I measure is not necessarily what light “does” or what “happens”. What I measure is what the instruments of my frame (held by myself and my lattice of clocks and rulers at rest with me) do, while light passes by. Due to a complex compensation of effects, I always measure c for light speed, but that does not mean that the same thing has “happened” on each occasion: the proof is that each day the frequency of the light beam is different, so my efforts have not been totally in vain.

Yes, something like that. The laws of physics, such as Maxwell's equations which govern both light and the measuring instruments, have the special relativistic symmetry. http://books.google.com/books?id=FGnnHxh2YtQC&printsec=frontcover#PPA67,M1

 

[belliott4488]

Saw - I don't know if this will help, but here's another thought: It's not really light that is adjusting itself so that you will always measure its speed as c - it does nothing to "compensate". Rather it simply moves from point A to point B (in any reference system) in an amount of time t, where (B-A)/t = c.

Where the compensating adjustments enter in order for this to be true is in the nature of space and time, i.e the distance B-A and the time interval t adjust themselves (in whatever frame they are measured) in exactly the right way to maintain (B-A)/t = c.

In short, it's space and time that do the compensating (if you want to think of it that way), not light.

 

[Saw]

Thanks to all. Hmm, there are several things I could say in answer to your comments but in my post there is a point that nobody addressed in particular, which especially worries me and about which I’d like to hear your opinion.

Any transformation, I understand, requires some common element that the two frames rely on for mutual conversion. I said that:

- In classical relativity, that was the role of time and length.
- In SR, it is… the speed of light = c? Or could we say that it is c and v, since the latter is also a common element? Or perhaps c/v? Or proper time, which is an invariant?

I suppose that has a clear mathematical answer.

The same applies to geometry. In a classical ST diagram, you flip the drawing of one frame over the time axis and you get it superimposed over the drawing of the other frame. On what do you lean to do the rotation over spacetime in SR?

 

[belliott4488]

Any transformation, I understand, requires some common element that the two frames rely on for mutual conversion. I said that:

- In classical relativity, that was the role of time and length.
- In SR, it is… the speed of light = c? Or could we say that it is c and v, since the latter is also a common element? Or perhaps c/v? Or proper time, which is an invariant?

I'm not sure what you mean by that. The transformation between two coordinate systems requires that you are able to express the coordinates of a point in one system as functions of the coordinates in the other system (and vice versa). There are parameters that will enter into these functions, which can include all those you mention. For example, in SR you can express the transformation between two frames using only v and c as parameters, but only if the origins of the two frames coincide at some point. If they don't, then there will be some distance, time, or both that enter into the transformations, i.e. they will include both translation and rotation (the latter being how a basic Lorentz tranformation affects the coordinates). Is that all consistent with what you were describing?

The same applies to geometry. In a classical ST diagram, you flip the drawing of one frame over the time axis and you get it superimposed over the drawing of the other frame. On what do you lean to do the rotation over spacetime in SR?

Now I'm really confused. In classical Gallilean relativity, the time axes are all parallel (since everyone measure time intervals equally), but different frames can be related by translations in the time direction, a space direction, or both, depending on how their origins are related; also one frame can be moving at some velocity v relative to another (time-dependent translation).

In SR, as I alluded to above, Lorentz translations correspond to rotations of the axes, but these rotations are not the same as in real space - rather, they correspond to rotations by imaginary angles (or equivalently, by hyperbolic trig functions of real angles), which has the effect that if you rotate the x-axis clockwise, for example, then the t-axis rotates counterclockwise.

 

[Phrak]

SR stems from the observation that nature works differently. Clocks do not work the same way in any frame (just as rulers)...

Space and time did not work as expected.

 

[MeJennifer]

In SR, as I alluded to above, Lorentz translations correspond to rotations of the axes, but these rotations are not the same as in real space - rather, they correspond to rotations by imaginary angles (or equivalently, by hyperbolic trig functions of real angles), which has the effect that if you rotate the x-axis clockwise, for example, then the t-axis rotates counterclockwise.

True. Note that interestingly this model breaks down in curved spacetimes, because, in trying to stick to this description of rotation by imaginary angles, the imaginary axis splits into an infinite number of different axes. This in effect prevents quantum interactions to be described analytically in curved spacetimes, at least not with the currently available mathematical tools.

 

[Saw]

Thanks. Before going into more complicated subjects, I would like to clarify the basic point:

I'm not sure what you mean by that. The transformation between two coordinate systems requires that you are able to express the coordinates of a point in one system as functions of the coordinates in the other system (and vice versa). There are parameters that will enter into these functions

Yes, but I mean something more than that. The very fact that you can “express the coordinates of a point in one system as functions of the coordinates in the other system (and vice versa)” forcefully requires that the two systems make at lest “some” homogeneous or frame-invariant measurements. Otherwise the transformation would be impossible.

Take this simple example of systems that have a single X axis (positive and negative) and are at rest with each other:

I am the origin of system A. In my coordinate system, my friend A’ is located at x = 2 m.
Instead, for my friend, who is the origin of system A’, I am located at x’ = -2 m.

The transformation is (x for A’) = (-x’ for A) = -(-2) = +2

or vice versa (x’ for A) = (-x for A’) = = - 2

Really, the distance between A and A’ is only one reality. The two observers label it with different X values (x and x’) because they have different perspectives: for A, A’ is located to the right; for A’, A is located to the left. However, the two perspectives only differ in that respect: A and A’ measure from different positions. Hence their values differ in that point, the direction in which they look at each other, which is mathematically represented by a plus or minus sign. But their perspectives are not different as to the numeric value of the distance: actually, A and A’ measure with the same rod and so they obtain the same result in terms of that numeric distance.

That is why the transformation is a simple multiplication by (-1) = you change what is affected by perspective (direction, the sign) and you leave unaltered that other element (the numeric value of the distance), which is not affected by perspective.

If we introduce a third object A’’ and A’ claims that A’’ is located at x’ = 1 m, then the transformation for A would be:

(x for A’’) = -(x’ for A) + (x’ for A’’) = -(-2) + 1 = +3 m

In this new example, A has reasoned by:

a) changing what is affected by perspective: A’ measures from its position but A measures from its own; therefore in A frame we must add (x’ for A) and change its sign again…

b) leaving unaltered what is unaffected by perspective = the absolute value of (x’ for A) and the distance (x’ for A’’), including its sign.

So, yes, frame A has obtained the coordinate of A’’ as a function of the A’ values, but not all values are the same: there are values that A accepts as valid for its purposes (the b values) and others that have to be changed (the a values).

If you introduce motion, you get the Galilean transformation, where v plays the part of the mutual distance between A and A’.

But the Galilean transformation has the mathematical form it has because it assumes that the measurements of length and time of both frames are not affected by perspective. (Please note: how much a third object has displaced in each frame is frame-variant in classical relativity as well. That is why I only refer to length.)

Finally, SR postulates and experiment proves that also length and time intervals and even the “spacetime point” where the interval start are frame-variant. That means TD, LC and, intertwined with them, RS. That implies many more “a values” (affected by perspective), but still “b values” (unaffected by perspective) must exist; otherwise the transformation would be impossible.

So I was wondering: coming back to the simplest example, where the “b element” (frame-invariant) is the numeric value of the distance between A and A’…, what is, technically speaking, the equivalent of such element in SR?

I thought there would be a standard answer for that. Can it be simply c, the speed of light, for example?

 

[MeJennifer]

Finally, SR postulates and experiment proves that also length and time intervals and even the “spacetime point” where the interval start are frame-variant.

Distance and duration are not invariant under SR, what is invariant is the spacetime distance between two events.

 

[Saw]

Distance and duration are not invariant under SR

Did I say the contrary...?

what is invariant is the spacetime distance between two events.

For sure. My question is only: can we thus say that the spacetime interval plays in SR the role of the absolute value of the distance between A and A' in my above example?

 

[MeJennifer]

can we thus say that the spacetime interval plays in SR the role of the absolute value of the distance between A and A' in my above example?

The spacetime interval describes distances between events.

 

[Saw]

The spacetime interval describes distances between events.

For sure. The situations are different, but they are analogous. I am just wondering how to construct the analogy.

 

[Ich]

I am just wondering how to construct the analogy.

Let A and A' be situated at the same point, but looking into different directions, at an arbitrary angle. Let them describe the world in "forward" (x) and "to the right" (y) coordinates.
What is invariant? What not?
Let A and A' be situated at the same event to a certain time, but moving with different speed, at an arbitrary velocity. Let them describe the world in "timeward" (t) and "spaceward" (x) coordinates.
What is invariant? What not?