Meaning of "Speed of Light is Constant": Explained

[roineust]

Isn't the meaning of speed, a variable of distance divided by a variable of time? Therefor isn't the meaning of a constant speed of light, a constant distance divided by a constant time? If there is any truth in this saying and there probably isn't, then what is the meaning of light constant distance and what is the meaning of light constant time?

These are probably only elementary school mathematics questions, but i guess it's better to ask 'dumb' questions than not to ask at all.

 

[fresh_42]

If you drive in a car at 50 mph and throw a ball from the rear to the windshield at 10 mph, then the ball has a speed of 60 mph relative to ground.

If you turn on your flashlight, then it travels at the same speed relative to the car or relative to the ground.

 

[Ibix]

Constant speed means that the speed does not change with time. You can take any distance you like, measure the time light takes to travel that distance, divide the two, and you will get the speed of light.

The interesting point about light's speed is not its constancy but its invariance - it is the same measured in all inertial frames.

 

[RPinPA]

Isn't the meaning of speed, a variable of distance divided by a variable of time?

Yes.

Therefor isn't the meaning of a constant speed of light, a constant distance divided by a constant time?

No.

Suppose I say an object traveled 10 m in 1 second. I say the speed is 10 m/s.
You say I'm wrong, according to your measurement the object only traveled 9 m in my experiment. And it took only 0.9 seconds to do so. You say the speed is 9/0.9 = 10 m/s.

We disagree on the distance and time in the experiment. But we agree on their ratio. This doesn't really happen at 10 m/s, but it does happen when measuring things traveling at light speed.

 

[Ibix]

This doesn't really happen at 10 m/s, but it does happen when measuring things traveling at light speed.

It does happen - you just worked out such a case. What doesn't happen at 10 m/s is me saying that your 1 m ruler isn't 1 m long, your clocks are ticking slow and you haven't zeroed them right anyway.

 

[roineust]

Yes, but if i say that my meter is not like your meter and my second is not like your seconds and all because we are traveling at different speeds, is it necessary that the change in meter and change in second, keep a constant ratio? Can't they change their ratio, but still keep the speed of light the same? This probably makes no sense and only results from basic arithmetic misunderstanding on my side.

 

[PeroK]

Yes, but if i say that my meter is not like your meter and my second is not like your seconds and all because we are traveling at different speeds, is it necessary that the change in meter and change in second, keep a constant ratio? Can't they change their ratio, but still keep the speed of light the same? This probably makes no sense and only results from basic arithmetic misunderstanding on my side.

If you can cope with some mathematics, then it goes like this. You have a Lorentz Transformation between inertial frames:
$$x' = \gamma(x - vt), \ \ t' = \gamma(t- \frac{vx}{c^2}), \ \ \gamma = \frac 1 {\sqrt{1- v^2/c^2}}$$
Where ##v## is the relative velocity between the reference frames. Now, one thing you can do with this is derive the velocity transformation law:
$$u' = \frac{u - v}{1- uv/c^2}$$
Where ##u, u'## are the velocities of a particle in the two frames. And if we take ##u = c##, i.e. a massless particle traveling at velocity ##c## in one frame we get:
$$u' = \frac{c-v}{1-v/c} = c$$
That shows that if a particle has a speed of ##c## in one frame, it has a speed of ##c## in every other frame.

 

[roineust]

I am probably mathematically confused between the relation of a meter to a second as in constant speed and the relation of meter to a second as in change of a meter and change of a second, as result of speed change. Since i can't understand mathematics and it is not accurate enough to write in words, this is probably a lost case.

 

[PeroK]

I am probably mathematically confused between the relation of a meter to a second as in constant speed and the relation of meter to a second as in change of a meter and change of a second, as result of speed change. Since i can't understand mathematics and it is not accurate enough to write in words, this is probably a lost case.

Ultimately, the precise meaning of the "invariance of the speed of light (across all inertial reference frames)" must be expressed mathematically. You can to some extent replace some mathematics by words and descriptions. But, if any arguments ensue about what those words mean, then you should be able unambiguously to point to the mathematics and say "that's what it means". That's the power of mathematics and its relationship to physics.

 

[roineust]

Is the ratio that time dilation and length contraction change in relation to one another, as a result of speed change, a linear ratio, such as in X2:

1/2
2/4
3/6
4/8

Or is it non-linear, such as in X^2:

1/2
2/4
3/9
4/16

Does this question make any sense?

 

[russ_watters]

Yes, but if i say that my meter is not like your meter and my second is not like your seconds and all because we are traveling at different speeds, is it necessary that the change in meter and change in second, keep a constant ratio? Can't they change their ratio, but still keep the speed of light the same?

Is the ratio that time dilation and length contraction change in relation to one another, as a result of speed change, a linear ratio, such as in X2:

1/2
2/4
3/6
4/8

Or is it non-linear, such as in X^2:

1/2
2/4
3/9
4/16

Does this question make any sense?

Speed is the ratio of distance to time. Which one of those cases is a constant ratio?

 

[Ibix]

I will say that your meter rule is wrong, your clocks are wrong and out of sync - but I note that those things combine to mean that your measurement of ##c## is always the same.

Does this question make any sense?

No, because you are missing the relativity of simultaneity. If you time a light pulse along a straight track then you need synchronised clocks at the beginning and end of the track - and I say they are out of sync. However, the amount they are out of sync by, coupled with their slow ticking, leads to the time you measure to be directly proportional to the distance measured by your ruler (which is moving and length contracted).

 

[vanhees71]

Let me try to explain it differently.

First of all the "speed of light" refers to the phase of a electromagnetic wave. A plane wave propagating in ##z## direction is described by (in Heaviside-Lorentz units)
$$\vec{E}(t,\vec{x})=\vec{E}_0 \cos(\omega t-k z), \quad \vec{B}(t,\vec{x})=\vec{e}_3 \times \vec{E}_0 \cos(\omega t-k z).$$
Maxwell's equations tell you that
$$\omega = c k$$
and this implies that the planes of constant phase, defined by
$$\omega t-k z=\text{const} \; \Rightarrow \; v_{\text{phase}}=\mathrm{d} z/\mathrm{d} t= \frac{\omega}{k}=c.$$
Now this plane wave must come from somewhere. Indeed it's a solution of the Maxwell equations describing an electromagnetic wave emitted from some very far source.

Now the remarkable thing with the Maxwell equations is that they contain as a paramater this phase-speed of electromagnetic waves (in vacuum), ##c##, which is considered a fundamental constant of nature.

At the same time from classical mechanics we expect that all the physical laws must look the same in any inertial frame of reference. Now suppose you observe the far-distant light source when moving with constant speed ##v## along the ##z## axis towards this light source. From naive Newtonian reasoning you'd argue that now the phase velocity must be ##v_{\text{phase}}'=c+v##, but that contradicts the invariance of the Maxwell equations when transforming from the coordinates as measured in the old frame to the coordinates as measured in the frame, where you are moving.

This was the problem physicists where struggling with since Maxwell has written down his equations, and the final solution was Einstein's famous paper about special relativity: We have to change the description of space and time in such a way that the Maxwell equations are unchanged when transforming from one inertial frame to another one, and thus the phase speed ##v_{\text{phase}}'## must be the same as in the original frame ##v_{\text{phase}}'=v_{\text{phase}}=c##.

In other words the phase speed of electromagnetic waves as measured from any inertial observer is independent of the velocity of the source of this electromagnetic waves.

This is achieved by the Lorentz transformations for time and spatial coordinates described in #7. It turns out that also ##\omega/c## and ##k## must transform as time and space coordinates, i.e., in our setup where the observer in the new frame moves towards the source
$$\omega' = \gamma (\omega + \beta c k), \quad k'=\gamma (k+\beta \omega/c).$$
Since now ##\omega=c k##, ##\beta=v/c## and ##\gamma=1/\sqrt{1-\beta^2}## you get
$$\omega'=\omega \frac{1+\beta}{\sqrt{1-\beta^2}} = \omega \sqrt{\frac{1+\beta}{1-\beta}}, \quad k'=k \sqrt{\frac{1+\beta}{1-\beta}}.$$
You thus get also for the observer moving towards the source
$$\omega'=c k',$$
but ##\omega'>\omega##. This means tha phase speed in the new inertial frame is indeed the same as in the original frame, i.e., ##v_{\text{phase}}'=\omega'/k'=c## but the frequency is larger than in the old frame. This is the Doppler effect for light: Moving towards the source shifts the light somewhat towards higher frequencies (i.e., it's "blue-shifted", because light that occurs blue to us is an electromagnetic wave with higher frequencies than, e.g., red light). In the same way you can show that for an observer moving away from the source the light gets red-shifted (here simply ##v## is replaced by ##-v## in the Lorentz transformation).

So the phase velocity of em. waves stays the same when changing from one inertial frame to another one, i.e., it's the same for any observer no matter, which speed the light source has in his rest frame, but the frequency gets shifted according to the Doppler effect for em. waves. This is in accordance with Maxwell's equations, which should not change by changing from one inertial frame to another moving with constant speed against the first frame, and thus the phase speed of light, ##c##, must be the same in all frames. This is achieved by using the Lorentz transformations for changing the time and space coordinates when transforming from one inertial frame to the other rather than the Galilei transformation valid in Newtonian mechanics.