High School Time dilation problem question

[Bartolomeo]

So it's not a state at all. It's a decision about what reference frame to use.

Well, this decision looks like that. If you choose a frame, in which you move with equal velocities, the source and you have to turn tubes at the same angle. If you choose a frame, in which you at rest, you look at right angle to direction of motion of the source. Angle depends on chosen reference frame.

 

[Ibix]

Proper rest is a state, when you consider yourself as being at rest and interpret observations from the point of view observer at rest. Imagine that you stay at certain point at axis y, at point Y. You posses a long thin tube. You ascribe yourself a state of proper rest. You want to measure frequency of moving source - green monochromatic lamp. You turn the tube into the origin. You stay and wait. The lamp moves along x in negative direction. At certain moment you see a short flash, when the lamp passes the origin and a photon goes through you tube. The flash will be red in color due to Transverse Doppler Effect.
Then you ascribe yourself small velocity. You repeat the experiment but do nothing special in this case - just turn the tube a bit into front. Why? Because if photon flies along y-axis up, you approach the point of "intersection from the left" and you have to take aberration into account. You tilt tube so as the photon would be able to pass through it. You see a flash again. The photon will be a bit less red, than it was in the first experiment.
Then you repeat the experiment again, but ascribe yourself even larger proper velocity. You think that since your velocity increased, you tilt you tube even further an see that color of photon moved into blue range of spectrum.
In fact you do nothing, just turn you tube further an further into front and interpret every observation as transverse doppler effect, since photon according to you comes to you along y-axis and due aberration becomes more and more blue with each experiment.

So you are just saying that if you do different experiments you get different results. This is true (and frame invariant) but trivial.

 

[Bartolomeo]

So you are just saying that if you do different experiments you get different results. This is true (and frame invariant) but trivial.

Yes, outcome of experiment depends on how I make it. and what I think about my own state of motion. If I am at rest, I do not consider aberration. If I am in motion, I consder aberration. You told me in your very first post, that outcome of experiment does not depend on how it is done, don't you? And what I think about my motion. No, it does. The faster I move, the blueish source will be.

 

[jbriggs444]

Well, this decision looks like that. If you choose a frame, in which you move with equal velocities, the source and you have to turn tubes at the same angle. If you choose a frame, in which you at rest, you look at right angle to direction of motion of the source. Angle depends on chosen reference frame.

When asked what "proper rest" means, you give a longwinded and incorrect response. When this is objected to, you try to clarify by invoking aliens and laser pointers.

I conclude that "proper rest' is not a meaningful term.

 

[Bartolomeo]

When asked what "proper rest" means, you give a longwinded and incorrect response. When this is objected to, you try to clarify by invoking aliens and laser pointers.

I conclude that "proper rest' is not a meaningful term.

Especially for you I even posted a video.

 

[Ibix]

Yes, outcome of experiment depends on how I make it. You told me in your very first post, that outcome of experiment does not depend on how it is done, don't you?

No. I told you that the outcome of one experiment did not depend on which frame you use to describe it. See the second quote and my response in this post.

 

[Bartolomeo]

No. I told you that the outcome of one experiment did not depend on which frame you use to describe it. See the second quote and my response in this post.

I have nothing to say.

 

[Ibix]

I have nothing to say.

To expand a little - in the post I linked in my last, you wrote "And what will a co-moving inertial observer (an observer of a co-moving inertial frame of reference) see, who at some point in time ends up near one of the observers hurtling around the circle? This will depend upon what velocity it ascribes to itself." There is only one point in space where it makes any sense to compare the viewpoint of an instantaneously co-moving inertial observer and a circular orbiting observer: where the two are co-located and co-moving. At that point there is only one setting of the tube angle that will receive light, and the frequency that you measure is obviously frame invariant. It does not matter if the co-moving observer decides to use some other frame - they will conclude that the tube is set in the correct position to reveive light at the moment they co-move with the orbiter. If they move the tube they will be doing a different experiment (which may obviously have different results), an experiment which is irrelevant to the question at hand because it won't work when they are co-moving with the orbiter.

I now think you were referring to running multiple experiments which can, of course, have different results. But they are irrelevant to the circular motion thread as I just noted, and you are misapplying them in this one. When the two parties do the same experiment (i.e., set their receiving tube to the same angle to the other's direction of travel by their own local measurements) they receive the same frequency that the other received. This is the reciprocity that is expected, and is consistent with symmetric time dilation. The fact that, in any given realisation of the experiment, the two parties do not see the same frequency is not a counter-argument because there is no symmetry in that realisation. One of the parties is emitting a light pulse at a time of their choosing and the other is passively receiving. The symmetry is broken and no reciprocity is expected.

 

[Bartolomeo]

To expand a little - in the post I linked in my last, you wrote "And what will a co-moving inertial observer (an observer of a co-moving inertial frame of reference) see, who at some point in time ends up near one of the observers hurtling around the circle? This will depend upon what velocity it ascribes to itself." There is only one point in space where it makes any sense to compare the viewpoint of an instantaneously co-moving inertial observer and a circular orbiting observer: where the two are co-located and co-moving. At that point there is only one setting of the tube angle that will receive light, and the frequency that you measure is obviously frame invariant. It does not matter if the co-moving observer decides to use some other frame - they will conclude that the tube is set in the correct position to reveive light at the moment they co-move with the orbiter. If they move the tube they will be doing a different experiment (which may obviously have different results), an experiment which is irrelevant to the question at hand because it won't work when they are co-moving with the orbiter.

It is very difficult graphically demonstrate coincidence of commoving and rotating observer, we need spatial imagination. But I think you can understand that. Just try to move either observer relatively to rotating circle and central lamp or the whole rotating establishment relatively to resting observer.
Fig. 1 – Commoving observer moves in the reference frame of the lamp (blue shift)
Fig. 2 – Commoving observer and the lamp are at equal velocities to each other (no shift)
Fig. 3 – Lamp is in motion in the reference frame of commoving observer (reds shift)
Fig. 4 – Rotating observers in the same spatial positions as commoving ones.

Length of arrow V reflect "proper velocities"
http://[url=https://postimg.org/image/8mvqlxk0z/][PLAIN]https://s29.postimg.org/8mvqlxk0z/Fig_1.jpg

 

[Bartolomeo]

I now think you were referring to running multiple experiments which can, of course, have different results. But they are irrelevant to the circular motion thread as I just noted, and you are misapplying them in this one. When the two parties do the same experiment (i.e., set their receiving tube to the same angle to the other's direction of travel by their own local measurements) they receive the same frequency that the other received. This is the reciprocity that is expected, and is consistent with symmetric time dilation. The fact that, in any given realisation of the experiment, the two parties do not see the same frequency is not a counter-argument because there is no symmetry in that realisation. One of the parties is emitting a light pulse at a time of their choosing and the other is passively receiving. The symmetry is broken and no reciprocity is expected.

Relative velocity is the same in every give experiment. It doesn’t change.

Experiment is to show the independence of observations from the arbitrary choice of the reference system. Of course, if you do not change anything, nothing changes.

The purpose of any given experiment is to confirm or deny certain suggestions. Suggestion is the following: observation of clock rate of relatively moving clock does not depend of state of rest or motion of any relativistic observer. If Ibix and Bartolomeo conduct certain experiment, Ibix must measure dilation of Bartolomeo’s clock (to register redshift) and Bartolomeo has to measure dilation of Ibix’s clock (to register redshift) in any given experiment despite of what they think about their state of motion or rest.

Amount of redshift must depend on their relative velocity solely. Let’s say 0.9 c.

They conduct experiment at 10 AM and write a report. Ibix releases a photon along y axis, Bartolomeo reflects it and Ibix gets it back. Report claims, that Ibix measured redshift and Bartolomeo blueshift. Relative velocity was 0.9 c.

They conduct another experiment after lunch at 2 PM. Relative velocity was 0.9 c again. velocity solely. Let’s say 0.9 c. Bartolomeo releases a photon along y axis, Ibix reflects it and Bartolomeo gets it back. Report claims, that Bartolomeo measured redshift and Ibix blueshift.

They can conduct 1000 experiments or even 1000000 experiments. Relative velocity is still 0.9 c. They change angle of rotation of their tubes in every experiment at 0.01 degree. In any experiment, they cannot find, that Bartolomeo fixes redshift and Ibix redshift. It is ABSOLUTELY IMPOSSIBLE.

Amount of time dilation, which corresponds to relative velocity 0.9 c is the same in every experiment. It is mutual property. The more Ibix takes for himself, the less remains to Bartolomeo.

Time dilation and length contraction is like a short blanket. You cover your head but it is too short to cover legs and vice versa.

Well, Ibix had finally found THE brilliant solution! Ibix saw time dilation at 10 AM, Bartolomeo at 2 PM. They publish a report, that they both saw time dilation fully in accordance with SR.

 

[Battlemage!]

Amount of redshift must depend on their relative velocity solely. Let’s say 0.9 c.

The Doppler shift formula recognizes the notion of the source of the emitted frequency, so how can the source-emitted frequency be arbitrarily interpreted as the received frequency simply by changing reference frame?

 

[Battlemage!]

To expand a little - in the post I linked in my last, you wrote "And what will a co-moving inertial observer (an observer of a co-moving inertial frame of reference) see, who at some point in time ends up near one of the observers hurtling around the circle? This will depend upon what velocity it ascribes to itself." There is only one point in space where it makes any sense to compare the viewpoint of an instantaneously co-moving inertial observer and a circular orbiting observer: where the two are co-located and co-moving. At that point there is only one setting of the tube angle that will receive light, and the frequency that you measure is obviously frame invariant. It does not matter if the co-moving observer decides to use some other frame - they will conclude that the tube is set in the correct position to reveive light at the moment they co-move with the orbiter. If they move the tube they will be doing a different experiment (which may obviously have different results), an experiment which is irrelevant to the question at hand because it won't work when they are co-moving with the orbiter.

I now think you were referring to running multiple experiments which can, of course, have different results. But they are irrelevant to the circular motion thread as I just noted, and you are misapplying them in this one. When the two parties do the same experiment (i.e., set their receiving tube to the same angle to the other's direction of travel by their own local measurements) they receive the same frequency that the other received. This is the reciprocity that is expected, and is consistent with symmetric time dilation. The fact that, in any given realisation of the experiment, the two parties do not see the same frequency is not a counter-argument because there is no symmetry in that realisation. One of the parties is emitting a light pulse at a time of their choosing and the other is passively receiving. The symmetry is broken and no reciprocity is expected.

They can conduct 1000 experiments or even 1000000 experiments. Relative velocity is still 0.9 c. They change angle of rotation of their tubes in every experiment at 0.01 degree. In any experiment, they cannot find, that Bartolomeo fixes redshift and Ibix redshift. It is ABSOLUTELY IMPOSSIBLE.

Hmmm... Let me take a gander at making them see the same Doppler effect, in my obviously unqualified noobish ways... What about this scenario?Ibix shoots the light wave. Bartolomeo sees Ibix moving away at 0.9c. Since Ibix is moving away according to Bartolomeo's frame, Bartolomeo sees the following redshift:

$$f_O = f_E \sqrt{\frac{1-0.9}{1+0.9}}$$

Now, according to Ibix, when the light wave comes back, to him it is emitted from the mirror. As the mirror is moving away from him, he'll see the following Doppler shift:

$$f_O = f_E \sqrt{\frac{1-0.9}{1+0.9}}$$

Looks like it's exactly the same for Ibix if the reflected light can be treated as if it emitted from the mirror.

Unless... that assumption is not valid?

 

[Bartolomeo]

The Doppler shift formula recognizes the notion of the source of the emitted frequency, so how can the source-emitted frequency be arbitrarily interpreted as the received frequency simply by changing reference frame?

Very good question! I have exact answer in my files, just copy - paste:

In this case you can say that source is in motion and dilates, or observer is in motion and dilates. Doppler Shift will be the same as measured by observer. Graphically:

If an observer and a source approach each other head – on, the observer will see Doppler blueshift. In this case the observer may explain his observations in different ways depending on chosen reference frame. He may consider himself at rest, and wave fronts gather „on the nose“ of the source then, but because of dilation of the source‘s clock it will be „less blue“ than it were in classical case. In classical case wave fronts would hit him almost instantly. Either he may ascribe to himself a state of proper motion and source is at rest then. In this case maximum frequency he may register (in classical case) has to be 2f. But, since his own clock dilates, as he approaches the speed of light the frequency he measures will tend to an infinitely large value.

Sure, he can use a frame in which they have equal proper speeds or any other. Measured frequency will be the same, but relative contributions of time dilation will be different and frame dependent.

Amount of time dilation every observer possesses is in accordance with his "proper velocity". Total amount of time dilation is in accordance with relative velocity 0.9 c

This is purely longitudinal Doppler Shift, not Transverse.

All the ideas prior to these considered Transverse effect, when beam of light travels up and down at normal to direction of travel of observers.

 

[stevendaryl]

It is the core of special relativity. Einstein's work of 1905. Observer introduces a reference frame and allocates AT LEAST two clocks in different spatial positions. AT LEAST two clocks.
Observer in SR is not a "physical person". It is the whole reference frame, which is filled with many people and each of them has a clock. They synchronize clocks by light signals and make judgement then. John is certain point of reference frame K. John has a clock. Bill is far away. Bill has a clock too. John and Bill are the relativistic observer together. They synchronize clocks by light. Jim passes by John first and compares his clock with Johns at this moment in immediate vicinity. Then Jim passes by Bill. They again compare clocks in immediate vicinity. John and Bill together make conclusion, that Jim's clock runs slower.

Right, but that conclusion requires synchronization of different clocks. For clocks at rest relative to Jim and synchronized according to Jim, the opposite conclusion follows: that John's and Bill's clocks are running slow. There is no situation in which Jim "sees" the other clocks to be running faster than his. What he sees is a sequence of different clocks, not a single clock running fast or slow.

 

[Ibix]

Fig. 1 – Commoving observer moves in the reference frame of the lamp (blue shift)

Blue shift due to time dilation. No shift due to relative motion of source and receiver. Net result is that the receiver measures a blue shift. I assume here that the co-moving observer is co-located and co-moving with the orbiting observer.

Fig. 2 – Commoving observer and the lamp are at equal velocities to each other (no shift)

No shift due to time dilation. Blue shift due to relative motion of source and receiver. Net result is that the receiver measures a blue shift. I assume here that the co-moving observer is co-located and co-moving with the orbiting observer, the same as in figure 1.

Fig. 3 – Lamp is in motion in the reference frame of commoving observer (reds shift)

Red shift due to time dilation. Blue shift due to relative motion of source and receiver. Net result is that the receiver measures a blue shift. I assume here that the co-moving observer is co-located and co-moving with the orbiting observer, the same as in figure 1 and 2.

Fig. 4 – Rotating observers in the same spatial positions as commoving ones.

...but not actually co-moving here. The rotating observer's velocity is tangent to the circle and a single inertial frame cannot have a velocity that is tangent to the circle in more than one place. So at least two of these measurements are irrelevant - they are comparing different experiments, as I have been saying for some time now.

Length of arrow V reflect "proper velocities"

This is a bizarre terminology. My proper time is the time measured with a clock attached to me. My proper length is length measured with a ruler attached to me. Proper velocity, by extension, is the distance I travel measured with a ruler attached to me (i.e. zero) divided by the proper time. So proper velocity is always zero, which is why you never see the term. All of these are invariants since I've specified what is being used to make the measurements.

You seem to arbitrarily choose a set of rulers and clocks that aren't attached to me, change sets when you feel like it and call whatever result you get my proper velocity. It's not a proper velocity. It's just my velocity in some frame you've picked. Please stop using the term, or provide a valid reference for its use.

Amount of redshift must depend on their relative velocity solely.

This is incorrect in general. The redshift measured in a given frame also depends on the angle between the 3-velocity vector of the emitter and the spatial component of the line joining the emission and reception events. Hence the necessity for radial and transverse expressions for red shift.

They conduct experiment at 10 AM and write a report. Ibix releases a photon along y axis, Bartolomeo reflects it and Ibix gets it back.

"Along the y-axis" according to who? Bartolomeo and Ibix do not, in general, agree that the photon was emitted along the y-axis and hence are unsurprised by differing readings.

They conduct another experiment after lunch at 2 PM. Relative velocity was 0.9 c again. velocity solely. Let’s say 0.9 c. Bartolomeo releases a photon along y-axis

"Along the y-axis" according to who? Bartolomeo and Ibix do not, in general, agree that the photon was emitted along the y-axis and hence are unsurprised by differing readings.

In any experiment, they cannot find, that Bartolomeo fixes redshift and Ibix redshift. It is ABSOLUTELY IMPOSSIBLE.

Indeed. Because there is no symmetry in a given experiment - as I said in my last post.

Well, Ibix had finally found THE brilliant solution! Ibix saw time dilation at 10 AM, Bartolomeo at 2 PM.

I wouldn't put it like that, but you can if you like. Your experimental descriptions are incomplete, but if I infer correctly what you mean, in the 10am experiment Ibix saw pure transverse motion but Bartolomeo did not and vice versa in the 2pm experiment. The experiments are anti-symmetric setups, so anti-symmetric results are to be expected.

 

[Ibix]

Hmmm... Let me take a gander at making them see the same Doppler effect, in my obviously unqualified noobish ways... What about this scenario?Ibix shoots the light wave. Bartolomeo sees Ibix moving away at 0.9c. Since Ibix is moving away according to Bartolomeo's frame, Bartolomeo sees the following redshift:

$$f_O = f_E \sqrt{\frac{1-0.9}{1+0.9}}$$

Now, according to Ibix, when the light wave comes back, to him it is emitted from the mirror. As the mirror is moving away from him, he'll see the following Doppler shift:

$$f_O = f_E \sqrt{\frac{1-0.9}{1+0.9}}$$

Looks like it's exactly the same for Ibix if the reflected light can be treated as if it emitted from the mirror.

Unless... that assumption is not valid?

Not quite. Look at it from Bartolomeo’s frame. He receives a frequency $f_O$ and reflects back the same frequency. So what Ibix must see is the same as if Bartolomeo were emitting at $f_O$ - which means that the final frequency must be $$f_R=\sqrt {\frac {c-v}{c+v}}f_O=\frac {c-v}{c+v}f_E $$You can check this by letting Ibix be at rest at the origin and having Bartolomeo be at $x=vt$. At time $T$ and $T+1/f_E$ Ibix emits light pulses. Work out when they reflect and return back to the origin. The time difference between their return times ought to be $1/f_R$.

 

[Battlemage!]

Not quite. Look at it from Bartolomeo’s frame. He receives a frequency $f_O$ and reflects back the same frequency. So what Ibix must see is the same as if Bartolomeo were emitting at $f_O$ - which means that the final frequency must be $$f_R=\sqrt {\frac {c-v}{c+v}}f_O=\frac {c-v}{c+v}f_E $$You can check this by letting Ibix be at rest at the origin and having Bartolomeo be at $x=vt$. At time $T$ and $T+1/f_E$ Ibix emits light pulses. Work out when they reflect and return back to the origin. The time difference between their return times ought to be $1/f_R$.

So Ibix will see the light pulse even more strongly redshifted than Bartolomeo when it reflects back, correct?

Also, this should mean that each time the pulse reflects back it should continue to become even more red shifted until it's no longer visible, right?

 

[Bartolomeo]

@Ibix . You were right. Only blueshift when rotating and commoving coincide in one spatial position. A bit later regarding the test.

 

[Ibix]

So Ibix will see the light pulse even more strongly redshifted than Bartolomeo when it reflects back, correct?

Also, this should mean that each time the pulse reflects back it should continue to become even more red shifted until it's no longer visible, right?

Yes.

 

[Bartolomeo]

Yes.

And what had happened to the energy?
It seems that if the mirrors "run away" from each other, i.e photon travels longer and longer patch between mirrors with every oscillation, the photon makes more reddish at every turn. In case if distance between mirrors changes with time. It is like photon gas in a closed container. If the container expands, photons turns reddish.
But if distance between mirrors doesn't change with time, each reflection doesn't lead to change of color. It means if it was released green to the mirror, it will come back green.
Distance which photon travels between every reflection, to be exact.
In case if photon travels between "two parallel lines" distance he travels at every oscillation is the same, despite of how we see it - either up and down or zigzag.

 

[Battlemage!]

And what had happened to the energy?
It seems that if the mirrors "run away" from each other, i.e photon travels longer and longer patch between mirrors, the photon makes more reddish at every turn. In case if distance between mirrors changes with time. It is like photon gas in a closed container. If the container expands, photons turns reddish.
But if distance between mirrors doesn't change with time, each reflection doesn't lead to color. It means if it was released grin to a mirror, it will come back green.
Distance which photon travels between every reflection, to be exact.

Isn't there some energy change due to the interaction of the light and the mirrors?

I found this interesting article from the American Journal of Physics: On energy transfers in reflection of light by a moving mirror.Unfortunately I'm not at school right now so I don't have access to the full article, but the abstract is interesting, and it'd be interesting to see all the factors tied together in one treatment (but that's probably way above my pay grade).

 

[stevendaryl]

And what had happened to the energy?

When a pulse of light bounces off a mirror, it imparts energy and momentum to the mirror.

 

[Bartolomeo]

Isn't there some energy change due t

o the interaction of the light and the mirrors?

I found this interesting article from the American Journal of Physics: On energy transfers in reflection of light by a moving mirror.

Unfortunately I'm not at school right now so I don't have access to the full article, but the abstract is interesting, and it'd be interesting to see all the factors tied together in one treatment (but that's probably way above my pay grade).

When a pulse of light bounces off a mirror, it imparts energy and momentum to the mirror.

There are two mirrors and photon gas between them. Someone wants to bring the mirrors closer to each other. He has to make some work and transfer some energy into system, because photons tend to push the mirrors away from each other. That is, if the mirrors are not rigidly fixed photons push them apart and transmit some energy to mirrors. Thus, if mirrors approach each other, photons will gain energy and will be more blueish. If mirrors recede, even by inertia, photons will transfer some energy to them.
But if mirrors are rigidly fixed, color doesn't change after reflection.
Of course, we assume that the mirrors are "ideal".

 

[stevendaryl]

But if mirrors are rigidly fixed, color doesn't change after reflection.
Of course, we assume that the mirrors are "ideal".

That's not 100% true. Every time a light pulse bounces off a mirror, then it imparts momentum and energy to the mirror. Of course, the energy loss is negligible in the limit of a very massive mirror or very low-energy light pulse.

 

[Bartolomeo]

That's not 100% true. Every time a light pulse bounces off a mirror, then it imparts momentum and energy to the mirror. Of course, the energy loss is negligible in the limit of a very massive mirror or very low-energy light pulse.

Obviously in real life. It "heats up" mirror. I mentioned that mirror is ideal, so us to omit "heating up"
If "ideal" mirrors move parallel to each other. One mirror "on the floor" is in motion and sends green photon. Photon travels together with this mirror "up", while mirror moves to the "right" in positive direction. When mirrors at the points of closest approach, the photon hits the mirror "on the top". Mirror "on the top" sees blue photon, because source approaches and immediately reflects it. Photon approached this mirror at oblique angle and leaves at oblique angle again. Photon comes back to "floor mirror". Since this mirror recedes from the mirror "on the top" photon redshifts, i.e. blue photon turns green again. Photon left the "floor mirror" at right angle and came back at right angle again.
This is consideration from longitudinal perspective. We can describe the same situation as the Transverse effect. In this case mirror "in the bottom" will be at rest, but mirror "on the top" in motion. Mirror in the bottom (at origin) sends GREEN photon straight up. Mirror "on the top" at that moment was at point (-X, Y) . When "top" mirror comes into point (Y,0) the mirrors at the closest approach. Due to ABERRATION photon blueshifts, i..e. turns BLUE. It means. that "top" mirror sees acceleration of clock, which is at rest. "Top" mirror reflects blue photon. Photon comes back to bottom mirror "straight from the top", from the point (Y,0)
Photon redshifts, since was released by moving source. "Bottom" mirror "sees" green photon which was BLUE at the source. It is in accordance with time dilation of a moving clock.
By the way, do you see reciprocity of observations in both cases. I don't.

 

[stevendaryl]

That's not 100% true. Every time a light pulse bounces off a mirror, then it imparts momentum and energy to the mirror. Of course, the energy loss is negligible in the limit of a very massive mirror or very low-energy light pulse.

If you work out the energy/momentum for an elastic collision between a photon and a perfect mirror, what you find is that if the momentum of the photon is p before the collision, then after the collision, it will be:

p' = \frac{-p (E-Pc)}{E+Pc}

where E is the total energy (mirror plus photon) and P is the total momentum. In the case where the mirror is very massive, so that the energy due to the photon is negligible, P \approx \gamma m v where m is the mass of the mirror, and v is its speed, and E \approx \gamma mc^2. So you get:

p' = -p \frac{c-v}{c+v}

The energy of the photon changes from pc to |p'|c = pc \frac{c-v}{c+v}

 

[stevendaryl]

Obviously in real life. It "heats up" mirror.

No, I'm not talking about that. I'm talking about the case of an ideal mirror. When a light pulse bounces off an ideal mirror, it imparts energy to the mirror.

 

[Bartolomeo]

No, I'm not talking about that. I'm talking about the case of an ideal mirror. When a light pulse bounces off an ideal mirror, it imparts energy to the mirror.

What that energy does with that "perfect mirror"? What that energy turns into?
Which amount of energy the "perfect" mirror has to consume? What that amount depends on?
Is it perfect, if it consumes energy?

 

[stevendaryl]

What that energy does with that "perfect mirror"? What that energy turns into?
Which amount of energy the "perfect" mirror has to consume? What that amount depends on?
Is it perfect, if it consumes energy?

Saying it's a perfect mirror says that all the energy of the mirror is in the form of kinetic energy (no energy due to heating up the mirror).

Let E be the total energy of mirror plus photon. Let P be the total momentum of mirror plus photon. Both of these quantities are conserved in a collision. Let M be the mass of the mirror. It is unchanged by a collision in the case of a perfect mirror. (If the mirror heats up, then M increases.) Let p be the momentum of the photon, and let p_m be the momentum of the mirror. Then we have:

Then what we have is:
p + p_m = P
|p|c + \sqrt{p_m^2 c^2 + M^2 c^4} = E

Rearrange and square the second equation, to get:
p_m^2 c^2 + M^2 c^4 = E^2 - 2 E |p| c + p^2 c^2

Now, we use the equation for momentum to rewrite p_m:
p^2 c^2 - 2 P p c^2 + P^2 c^2 + M^2 c^4 = E^2 - 2E|p|c + p^2 c^2

So: E^2 - P^2 c^2 - M^2 c^4 = 2(-Pc^2 \pm Ec) p
(where \pm is the sign of p)

So: p = \pm \frac{E^2 - P^2 c^2 - M^2 c^4}{2c (E \mp Pc)}

So for a fixed P and E, there are two solutions for the momentum of the photon:

p_{+} = \frac{E^2 - P^2 c^2 - M^2 c^4}{2c (E - Pc)}

p_{-} = -\frac{E^2 - P^2 c^2 - M^2 c^4}{2c (E + Pc)}

\frac{p_{-}}{p_{+}} = -\frac{E-Pc}{E+Pc}

If the energy of the photon is negligible compared to the rest energy of the mirror, then \frac{E-Pc}{E+Pc} \approx \frac{c-v}{c+v} where v is the original velocity of the mirror.

p_{+} is the momentum before the collision, and p_{-} is the momentum after the collision. The change in momentum of the photon goes into making the mirror go a tiny bit faster. It's a negligible change to the speed of the mirror, but just enough to change the total energy of the mirror by exactly the amount that is lost by the photon.

 

[Bartolomeo]

Saying it's a perfect mirror says that all the energy of the mirror is in the form of kinetic energy (no energy due to heating up the mirror).

Let E be the total energy of mirror plus photon. Let P be the total momentum of mirror plus photon. Both of these quantities are conserved in a collision. Let M be the mass of the mirror. It is unchanged by a collision in the case of a perfect mirror. (If the mirror heats up, then M increases.) Let p be the momentum of the photon, and let p_m be the momentum of the mirror. Then we have:

Then what we have is:
p + p_m = P
|p|c + \sqrt{p_m^2 c^2 + M^2 c^4} = E

Rearrange and square the second equation, to get:
p_m^2 c^2 + M^2 c^4 = E^2 - 2 E |p| c + p^2 c^2

Now, we use the equation for momentum to rewrite p_m:
p^2 c^2 - 2 P p c^2 + P^2 c^2 + M^2 c^4 = E^2 - 2E|p|c + p^2 c^2

So: E^2 - P^2 c^2 - M^2 c^4 = 2(-Pc^2 \pm Ec) p
(where \pm is the sign of p)

So: p = \pm \frac{E^2 - P^2 c^2 - M^2 c^4}{2c (E \mp Pc)}

So for a fixed P and E, there are two solutions for the momentum of the photon:

p_{+} = \frac{E^2 - P^2 c^2 - M^2 c^4}{2c (E - Pc)}

p_{-} = -\frac{E^2 - P^2 c^2 - M^2 c^4}{2c (E + Pc)}

\frac{p_{-}}{p_{+}} = -\frac{E-Pc}{E+Pc}

If the energy of the photon is negligible compared to the rest energy of the mirror, then \frac{E-Pc}{E+Pc} \approx \frac{c-v}{c+v} where v is the original velocity of the mirror.

p_{+} is the momentum before the collision, and p_{-} is the momentum after the collision. The change in momentum of the photon goes into making the mirror go a tiny bit faster. It's a negligible change to the speed of the mirror, but just enough to change the total energy of the mirror by exactly the amount that is lost by the photon.

If mirror gains kinetic energy, it will start moving and recede from another mirror. But, if the mirrors are rigidly fixed (on the opposite sides of a room) the same amount on energy it had swallowed on reception of a photon it will transfer to the photon when spitting it out.
Otherwise it is not a mirror, but a well-polished boots, for example.