Lorentz Question II: Doppler Shifted Freq. & Time Dilation

[Chrisc]

In free space, two frames A and B are at rest with respect to each other.
They each flash light signals toward the other at an equal and constant rate.
(i.e. on for one second, off for one second)
Each then begins to move toward the other at a significant constant speed.

••••• Assumption 1•••••
They cannot determine the equality of their motion in the equality of the (doppler shifted) frequency of the light signals measured by both.
Although it is reasonable to assume their procedure to acquire equal motion is verified in their measurements,
they can only agree their tests measure "relative" motion between them.

••••• Assumption 2•••••
Each now measures the (doppler shifted) frequency of the others light signals increased proportional their relative speed.


••••• Question 1•••••
Does the time dilation of either frame affect the frequency of light signals measured by the other?

 

[Doc Al]

••••• Assumption 1•••••
They cannot determine the equality of their motion in the equality of the (doppler shifted) frequency of the light signals measured by both.
Although it is reasonable to assume their procedure to acquire equal motion is verified in their measurements,
they can only agree their tests measure "relative" motion between them.

OK. Once they are done accelerating, the only thing that matters (or that they can determine) is their relative speed.

••••• Assumption 2•••••
Each now measures the (doppler shifted) frequency of the others light signals increased proportional their relative speed.

OK. The observed frequency increases with their relative speed, but is not simply proportional to it.

••••• Question 1•••••
Does the time dilation of either frame affect the frequency of light signals measured by the other?

The observed frequency of light is definitely affected by time dilation. According to frame A, frame B's clocks and light sources--since they are moving with respect to A--are subject to the usual time dilation. The relativistic Doppler shift takes all that into account. See: http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/reldop2.html#c1"

 

[Chrisc]

OK, so far so good.
••••• Assumption 3 •••••
I will take the position of frame A, I measure the relative speed of the distant frame B is 0.75c.
Calculating the time dilation and length contraction of frame B, I find for every second I measure, they measure 1.511 and for every meter I measure, they measure 0.661 meters.
Had I taken the position of frame B, the above would be correct with respect to frame A.

••••• Question 2 •••••
With this knowledge stated above, will any measurement B makes of the motion of a third frame be with respect to my frame, different by +0.511 meters per meter and -0.339 seconds per second, if the motion of the third frame is in the same direction as B?

 

[Doc Al]

OK, so far so good.
••••• Assumption 3 •••••
I will take the position of frame A, I measure the relative speed of the distant frame B is 0.75c.
Calculating the time dilation and length contraction of frame B, I find for every second I measure, they measure 1.511 and for every meter I measure, they measure 0.661 meters.
Had I taken the position of frame B, the above would be correct with respect to frame A.

OK. Sounds good. The "Lorentz factor" (usually called gamma) is 1.51.

••••• Question 2 •••••
With this knowledge stated above, will any measurement B makes of the motion of a third frame be with respect to my frame, different by +0.511 meters per meter and -0.339 seconds per second, if the motion of the third frame is in the same direction as B?

I think what you're asking is this. If B measures a moving clock and meterstick (moving in that third frame) to be dilated and contracted by some factor X, will frame A say measure them to be dilated and contracted by a factor of 1.51*X? The answer is no; it's more complicated than that.

To take a specific example, let's say that third frame was moving at the same speed 0.75c with respect to frame B. To find the Lorentz factor as seen by frame A, we first need the relative speed of the third frame with respect to A. That would be:
$$V' = \frac{V_1 + V_2}{1 + V_1 V_2/c^2}$$

When you plug in the numbers, you get V' = 0.96c. Which gives you a Lorentz factor of 3.57.

 

[Chrisc]

Thank you Doc Al.
You've gone one step further than I am ready to.
I want to make sure I understand the physical ramifications of this first
before I go continue, as my question is specifically the physical ramifications of the next step.
I will review the formulas again before posting.
I do not use LaTex so it might get messy.
If need be I will post the formulas in an image format.
Thanks for your patients.

 

[Chrisc]

I've studied the equations again and I have no issue with what you've explained, and I can now see you have already answered my original question re: priming the original measures of A or B with gamma.
This leads to my question regarding the physical meaning of the question and your answers.

••••• Assumption 4•••••
The equations of mechanics express no quantitative value without a bench mark such as rest. Absolute rest was assumed in order that the equations held throughout the universe. With the abolishment of absolute rest from the laws one of two dynamics was needed to uphold the laws.
1) a Lorentz transformation
2) a new absolute

•••••• Assumption 5•••••
Every measurement in inertial frames including the speed of light holds to the equations of mechanics because the Lorentz factor divides all measurements of the dimensions length and time "into" the speed of light: 1/sqrt 1-v^2/c^2. (ignoring mass for now)

••••• Question 3 •••••
It seems to me this means, in a physical sense, the Lorentz transformation is the formula by which we discover (through physically real measurements) the speed of light is a physically real absolute by which the equations of mechanics are upheld in all frames?

 

[Doc Al]

••••• Assumption 4•••••
The equations of mechanics express no quantitative value without a bench mark such as rest. Absolute rest was assumed in order that the equations held throughout the universe. With the abolishment of absolute rest from the laws one of two dynamics was needed to uphold the laws.
1) a Lorentz transformation
2) a new absolute

The notion of "absolute rest" in mechanics was abandoned long before Einstein. Before the Lorentz transformations there were the (still extremely useful) Galilean transformations. (At low speeds, the Lorentz transformations are well approximated by the Galilean transformations.)

•••••• Assumption 5•••••
Every measurement in inertial frames including the speed of light holds to the equations of mechanics because the Lorentz factor divides all measurements of the dimensions length and time "into" the speed of light: 1/sqrt 1-v^2/c^2. (ignoring mass for now)

Not sure what you're saying here. Realize that in the standard treatment, the Lorentz transformations are derived based upon the assumption that the speed of light is invariant.

••••• Question 3 •••••
It seems to me this means, in a physical sense, the Lorentz transformation is the formula by which we discover (through physically real measurements) the speed of light is a physically real absolute by which the equations of mechanics are upheld in all frames?

I don't understand what you're asking here. Again, the Lorentz transformations have the invariant speed of light built into them. They answer the question: How must position and time measurements transform in order to assure that the speed of light is the same for all observers?

 

[Chrisc]

Again, the Lorentz transformations have the invariant speed of light built into them. They answer the question: How must position and time measurements transform in order to assure that the speed of light is the same for all observers?

That is the essence of my question. They state how, but in stating how do they not at least hint why?
In that position and time "do" change by the amounts defined by the Lorentz transformations, are the Lorentz transformations defining a physical property and/or physical constant of space and time?

 

[matheinste]

Hello Chrisc.

Doc Al is of course correct but perhaps it is better for you to realize that the Lorentz Transforms were "designed by us" to to agree with the physics of the situation, the postulate that c is the same for all inertial observers. It is not the transforms that make the physics, but they reflect the "reality" of the situation.

Matheinste.

 

[Chrisc]

...It is not the transforms that make the physics, but they reflect the "reality" of the situation.

Thanks matheinste, that is the way I understand it.
Are you aware of any reasons given for the "reality" of the situation?

 

[matheinste]

Hello Chrisc.

---Thanks matheinste, that is the way I understand it.
Are you aware of any reasons given for the "reality" of the situation?----

No.

Matheinste.

 

[phyti]

Thanks matheinste, that is the way I understand it.
Are you aware of any reasons given for the "reality" of the situation?

It's geometry.