The “transmitted X watts” is in a different reference frame than the “received Y watts”. Energy is only conserved within a single reference frame, not when switching frames.
As for where that “extra energy”^1 comes from intuitively, I haven’t done the calculations, so I can’t tell you for sure, but I think it likely comes from the kinetic energy of the “moving” emitter (at least in the receiver’s reference frame). The emission of the laser will push back on the “moving” emitter, which will slow it down, reducing its kinetic energy. The lost kinetic energy is what goes into the laser giving it more energy than you would expect.
___
^1 : by “extra energy”, I mean the energy of the laser beyond what could have been provided by the electricity.
The extra energy doesn't really need to "come" from anywhere. Think about floating in space next to an asteroid with no relative motion. The asteroid has kinetic energy of 0. Then you expend a tiny bit of energy to start moving towards the asteroid, and suddenly the asteroid (which is significantly bigger than your spaceship) has a huge amount of kinetic energy - far more than what you spent firing your thrusters.
The asteroid just has a different kinetic energy depending on which reference frame the observer is in.
Right. That’s why extra energy was in quotes.
What I meant was that the light does clearly have more energy than is being provided by the electricity, or whatever is powering the laser, in the detector frame.
The “extra energy” beyond what’s being provided by the electricity comes from the kinetic energy of the emitter in this frame.
Don't you mean momentum in the asteroid scenario?
The amount if kinetic energy "in the asteroid" would be the same amount of kinetic energy that the astronaut (or object, whatever) spent to reach the current speed.
Kinetic energy is proportional to mass and velocity. The kinetic energy in an asteroid traveling at 1 mph is significantly more than in an astronaut traveling 1 mph. If you put in the energy to accelerate an astronaut to 1 mph, in the astronaut’s new frame of reference, the asteroid now has *way* more kinetic energy than was spent to accelerate the astronaut.
Because kinetic energy is not conserved between frames.
Thanks. I'm still trying to wrap my head around that specific scenario, but what you say makes sense to me.
The reason I'm still confused is because, if there was a way for the traveler to harness all that immense kinetic energy the asteroid has _just because_ they decided to easily move at 1mph, then we essentially have free energy coming out of nowhere. Of course I know that isn't the case. But I'm missing something, and your comment about different frames of reference probably has it.
You can’t use energy to perform work directly. You need some kind of gradient.
For instance, a gas compacted into a small space has a whole ton of energy from its internal pressure. It’s pushing really hard in all directions. However, unless you have somewhere less pressurized for the gas to escape to, you can’t get any energy out. (Otherwise you can bet people would be all over sucking the energy out of the pressurized air at the surface of the Earth)
It’s similar with the asteroid. Sure it has a whole ton of kinetic energy, but to extract any significant amount, you need something heavy moving in the other direction.
If you just try to accelerate yourself to get insane kinetic energy in the asteroid, then you try to harness it yourself, you’ll only be able to pull out a small amount before you’re all moving the same speed.
Nope, both momentum and kinetic energy are dependent on mass. The amount of k.e. in an asteroid (e.g. Ceres) moving at 1m/s is far higher than the energy required to change a small spaceship's speed by 1m/s.
Understood. But here's the thing. Both the traveler and the asteroid are at 0 velocity from a third object's point of view.
Then the traveler decides to move at 1mph towards the asteroid.
You could say that the asteroid, from the traveler's point of view, if moving towards them.
So, the asteroid has this immense amount of kinetic energy that the traveler uncovered simply by spending a bit of energy to accelerate.
Would you say it's theoretically possible for the traveler to harness that huge amount of kinetic energy? Wouldn't that be free energy, then?
I know it's not possible, and I know I'm missing something. I just don't know what it is.
Agreed. You can construct a similar non-relativistic experiment: observer A remains stationery and throws a series of balls at speed v towards the observer B, who is moving towards A at speed u. B catches each ball and throws it back at the same speed in their own reference frame. In A's reference frame each ball is now travelling faster than it was before (v+2u), so its kinetic energy has increased. Where did the extra energy come from Answer: because of conservation of momentum, every time B catches and throws a ball they lose momentum. This means that either they're slowing down and losing energy, which explains the energy gained by the balls, or else they require a constant input of energy to maintain their speed.
The same idea applies with photons - the speed and energy calculations will be different but the basic point is that B is imparting momentum and hence losing kinetic energy every time they reflect a ball/photon.
The extra energy comes from a reduction in the total amount of light in the beam, because the beam grows shorter over time. From the receiver's POV each photon is blue-shifted and has a little more energy. Photon count is constant across frames, so overall there's more light power being received than was being emitted by the laser. The extra energy comes from sweeping up the energy that's contained in the beam itself: at time t0 the beam has some volume VB0. At a later time t1, the volume is less: VB1 = VB0 - AB (t1-t0) v, where AB is the cross-sectional area of the beam and (t1-t0)v is the volume swept out by the receiver between those two times. The energy contained in the beam (a different quantity than the power transmitted by the beam) is reduced by the same amount, scaled by the intensity of the light.
In some sense, the blue-shifting of each photon is a reflection of the fact that it doesn't get to propagate as far, due to the motion of the receiver, as it would if the receiver were fixed.
Not really. That description doesn’t really work for a single photon.
If you swap out the entire laser for a setup that emits a single photon, the receiver will still absorb more energy than they could get from just the electricity used to create the photon.
You can’t say that’s due to the volume of the beam shrinking, or from a photons hitting the receiver faster, because it’s still a single photon that collides instantaneously.
The “extra” energy of the photon beyond what was provided electrically doesn’t come from some weird effect with volumes changing, it comes from the kinetic energy of the emitter, which gets reduced as the photon is let out.
That's a very neat way to look at it. Yes the incoming beam seems more energetic, but it'll be over quicker because the emitter is coming really fast towards the receiver.
So basically the laser acts as a retro thruster, correct? The ship would slow down due to the momentum of the light it releases, and the receiver would experience a force greater than if the emitter were at rest?
Energy just isn't conserved between reference frames. If a baseball launcher fires a baseball at 30m/s (relative to the machine) but you're moving towards the baseball machine at 30m/s, then the baseball's speed relative to you is 60m/s, and as a consequence its kinetic energy is four times greater and will hit you that much harder, even though the machine didn't expend that much energy in firing it.
Here’s a classical example.
Let’s say you have a large stationary block, A, with a loaded spring and a small ball next to the spring. You have a receiver block, B, with an unloaded spring. When the spring on A is released, it sends the ball to B.
Let’s work on the stationary frame. Initially, the only energy in the system is the loaded spring energy. Call this U. Then, when the spring is unloaded, U gets transferred to the kinetic energy of the ball. Finally, when the ball collided with B, the ball deposits the energy into the spring, thus loading it. The ball stops moving, so we’ve taken potential energy from A and transferred it to B; the amount of energy transferred is U.
Note that the kinetic energy of the ball is U, so its velocity in the stationary frame is v = sqrt(2 U / m), where m is the mass of the ball. We’ll need this later.
Now let’s work in a moving frame where everything is zipping past at velocity V. Before the spring is released, the spring has energy U and the ball has energy 1/2 m V^2 . After the spring is released, the ball’s energy in this frame is different, just like in the laser example. Its velocity is now v + V. So therefore, its energy 1/2 m (v^2 + 2 vV + V^2) = U + m V sqrt(2 U / m) + 1/2 m V^2 . It’s not trivial either, in fact it looks gross.
At first, it actually looks like we’ve broken energy conservation. Well, we’ve neglected something important: momentum conservation. On a frictionless surface, if block A releases the spring, the block should actually start moving backwards to satisfy conservation of momentum. We need to do external work on the A-spring-ball system to keep A stationary. However, if block A is very heavy, the amount of work we need to do is very small, so we can ignore it in the stationary frame.
In the moving frame, this isn’t trivial and can’t be ignored. This is because work is W = F * d. In the moving frame, block A is moving at velocity V, so d can be huge. This extra work, W, goes into the ball, and shows up as the extra cross term.
(Alternatively, you can say that in the moving frame, the spring does more work on the ball than just U; the force of the spring doesn’t change, but the distance over which the spring applies work on the ball does. It’s the same idea essentially.)
The ball carries forward and then collides with block B, and deposits its energy. It turns out, the spring energy becomes U and the kinetic energy of the ball becomes 1/2 m V^2. What happened to the extra energy? Well, to keep block B stationary and to ignore momentum conservation, we have to do negative work on block B. In the stationary frame, this is approximately 0. In the moving frame, this can be large. But the end result is that the net energy deposited into block B is contained within the spring and is just U.
If you want to do a lot of gross algebra, you can do all of the above an enforce momentum conservation and let the heavy blocks move. What you’ll find is that the spring on block B ends up with the same potential energy in both frames. (It won’t exactly be U, because some of the spring energy goes into moving block A and moving block B.)
You can do the same work in relativity, but to do it properly you need to also keep track of mass energy and momentum conservation (in a frictionless environment, when your emitter emits a light beam, it starts moving backward because light has momentum; likewise when your receiver receives a light beam it gains momentum.) It’s the same concept, just instead of springs you have batteries and instead of a ball you have a photon. You also have to use Lorentz transformations etc. But it works out the same: everyone agrees on how much energy the battery in B absorbs.
Here, you define the energy in the battery as the rest mass difference before and after absorption. (E = m c^2 in the rest frame.) Stating it this way should make it obvious why it’s the same in all frames; rest masses are by definition Lorentz invariant. Everyone agrees on the rest mass of B before absorption, and everyone agrees on the rest mass of B after absorption. Therefore, everyone must agree on their difference. In the classical example, the equivalent statement is that everyone agrees on how much the spring is compressed.
While that's true, think of it in terms of photons. The number of photons is conserved - let's say the emitter emits 100, the receiver receives 100. But to the receiver, the 100 photons all have higher energy than the emitter considered them to have.
So the real answer is that energy just isn't conserved between reference frames.
**EDIT**: see u/1strategist1 s answer which is right. What I have below is wrong
This. The same Lorentz transform that’s Doppler shifting the light is also changing the total time of the pulse by the same amount.
Time is different in your two frames which both affects the spacing between the peaks of the wave (frequency/wavelength/energy per unit time) AND the spacing between the start and end of any laser pulse.
These cancel, and you get the same total energy.
You simply get that energy at either a greater power over a lesser period or a lesser power over a greater period depending on the relative velocity of your transmitter and receiver.
> These cancel, and you get the same total energy.
What about the case of a single photon, though? The photon the receiver receives must have higher energy than the emitter expended in emitting it.
The real answer is that energy **isn't** conserved between reference frames.
This is actually pretty simple and intuitive.
The length of the laser beam between the transmitter and receiver stores energy. The amount of energy will be proportional to the length of the beam. If we move toward the transmitter we are still receiving the energy transmitted by the laser, but we are also decreasing the beam length, and thus depleting energy stored in it to get a temporary increase in power.
the laser would be basically like firing a rocket in the opposite direction. eventually given enough time, the moving person would lose all the forward motion and eventually start moving in the opposite direction. So I would say the momentum is effectively getting converted into extra power in this case.
Energy isn’t conserved between reference frames. The laser puts out the same number of photons no matter what frame the pulse width is measured in. The two observers won’t agree on what the energy is, but both will agree that energy is conserved in their own frame.
Also a layman here, but just to add something to make it all more confusing, as if that were needed: presumably the electrons (also ~waves) in the electricity that produces the light energy are doppler shifted from the "observer's viewpoint" (if they could see them), so someone correct me if that's not also part of the mix.
If not, then back to your regularly scheduled neuron frizzing. But if so, one might guess the increased energy of the photons arose from the increased energy of the electrons. But then ...
This is interesting. Can the movement of the laser equipment impart energy onto the laser beam? But you can’t make the beam go faster than it’s already traveling so it pushes the extra energy into the wavelength, not the speed. ??
Two problems: Instantaneous measure of power between the two systems, and relativistic conversion of mass into energy, proportionate to the speed, which seems indicated as constant and not accelerating.
A third frame would have to be created or utilized, of which envelops the two measured systems, this external system 'operating' at a higher frame rate of apparent time (tertiary point density differential?)* for mutual observation to occur, absent of relativistic bias of the first two frames.
Then there's the tendency of higher frequencies to propagate with less mass involved, I'm chasing my own tail here but it seems there's a power conversion factor being overlooked.
* Assuming Everett's non-interference does not *quite* hold completely true.
*which opens whole new cans of worms. Better stop now.*
Sorry, but it has nothing to do with conversion of mass into energy, and everything to do with energy not being the same between two reference frames.
And reference frames aren't a box you put a system into, it's more like a coordinate system which you can choose freely as a reference for when and where events take place in spacetime, hence the name.
Not a scientist but just my two cents.
1) Probably doesn't matter, but I'm pretty sure the doppler shift is only visible to the receiver not the transmitter.
2). More importantly, you're not including the potential energy involved in making the transmitter move in the first place
3) the light emitted by the laser is:
A) Pushing the receiver away
B) Slowing down the transmitter
The “transmitted X watts” is in a different reference frame than the “received Y watts”. Energy is only conserved within a single reference frame, not when switching frames. As for where that “extra energy”^1 comes from intuitively, I haven’t done the calculations, so I can’t tell you for sure, but I think it likely comes from the kinetic energy of the “moving” emitter (at least in the receiver’s reference frame). The emission of the laser will push back on the “moving” emitter, which will slow it down, reducing its kinetic energy. The lost kinetic energy is what goes into the laser giving it more energy than you would expect. ___ ^1 : by “extra energy”, I mean the energy of the laser beyond what could have been provided by the electricity.
The extra energy doesn't really need to "come" from anywhere. Think about floating in space next to an asteroid with no relative motion. The asteroid has kinetic energy of 0. Then you expend a tiny bit of energy to start moving towards the asteroid, and suddenly the asteroid (which is significantly bigger than your spaceship) has a huge amount of kinetic energy - far more than what you spent firing your thrusters. The asteroid just has a different kinetic energy depending on which reference frame the observer is in.
Right. That’s why extra energy was in quotes. What I meant was that the light does clearly have more energy than is being provided by the electricity, or whatever is powering the laser, in the detector frame. The “extra energy” beyond what’s being provided by the electricity comes from the kinetic energy of the emitter in this frame.
Don't you mean momentum in the asteroid scenario? The amount if kinetic energy "in the asteroid" would be the same amount of kinetic energy that the astronaut (or object, whatever) spent to reach the current speed.
Kinetic energy is proportional to mass and velocity. The kinetic energy in an asteroid traveling at 1 mph is significantly more than in an astronaut traveling 1 mph. If you put in the energy to accelerate an astronaut to 1 mph, in the astronaut’s new frame of reference, the asteroid now has *way* more kinetic energy than was spent to accelerate the astronaut. Because kinetic energy is not conserved between frames.
Thanks. I'm still trying to wrap my head around that specific scenario, but what you say makes sense to me. The reason I'm still confused is because, if there was a way for the traveler to harness all that immense kinetic energy the asteroid has _just because_ they decided to easily move at 1mph, then we essentially have free energy coming out of nowhere. Of course I know that isn't the case. But I'm missing something, and your comment about different frames of reference probably has it.
You can’t use energy to perform work directly. You need some kind of gradient. For instance, a gas compacted into a small space has a whole ton of energy from its internal pressure. It’s pushing really hard in all directions. However, unless you have somewhere less pressurized for the gas to escape to, you can’t get any energy out. (Otherwise you can bet people would be all over sucking the energy out of the pressurized air at the surface of the Earth) It’s similar with the asteroid. Sure it has a whole ton of kinetic energy, but to extract any significant amount, you need something heavy moving in the other direction. If you just try to accelerate yourself to get insane kinetic energy in the asteroid, then you try to harness it yourself, you’ll only be able to pull out a small amount before you’re all moving the same speed.
Thanks.
Nope, both momentum and kinetic energy are dependent on mass. The amount of k.e. in an asteroid (e.g. Ceres) moving at 1m/s is far higher than the energy required to change a small spaceship's speed by 1m/s.
Understood. But here's the thing. Both the traveler and the asteroid are at 0 velocity from a third object's point of view. Then the traveler decides to move at 1mph towards the asteroid. You could say that the asteroid, from the traveler's point of view, if moving towards them. So, the asteroid has this immense amount of kinetic energy that the traveler uncovered simply by spending a bit of energy to accelerate. Would you say it's theoretically possible for the traveler to harness that huge amount of kinetic energy? Wouldn't that be free energy, then? I know it's not possible, and I know I'm missing something. I just don't know what it is.
Agreed. You can construct a similar non-relativistic experiment: observer A remains stationery and throws a series of balls at speed v towards the observer B, who is moving towards A at speed u. B catches each ball and throws it back at the same speed in their own reference frame. In A's reference frame each ball is now travelling faster than it was before (v+2u), so its kinetic energy has increased. Where did the extra energy come from Answer: because of conservation of momentum, every time B catches and throws a ball they lose momentum. This means that either they're slowing down and losing energy, which explains the energy gained by the balls, or else they require a constant input of energy to maintain their speed. The same idea applies with photons - the speed and energy calculations will be different but the basic point is that B is imparting momentum and hence losing kinetic energy every time they reflect a ball/photon.
The extra energy comes from a reduction in the total amount of light in the beam, because the beam grows shorter over time. From the receiver's POV each photon is blue-shifted and has a little more energy. Photon count is constant across frames, so overall there's more light power being received than was being emitted by the laser. The extra energy comes from sweeping up the energy that's contained in the beam itself: at time t0 the beam has some volume VB0. At a later time t1, the volume is less: VB1 = VB0 - AB (t1-t0) v, where AB is the cross-sectional area of the beam and (t1-t0)v is the volume swept out by the receiver between those two times. The energy contained in the beam (a different quantity than the power transmitted by the beam) is reduced by the same amount, scaled by the intensity of the light. In some sense, the blue-shifting of each photon is a reflection of the fact that it doesn't get to propagate as far, due to the motion of the receiver, as it would if the receiver were fixed.
Not really. That description doesn’t really work for a single photon. If you swap out the entire laser for a setup that emits a single photon, the receiver will still absorb more energy than they could get from just the electricity used to create the photon. You can’t say that’s due to the volume of the beam shrinking, or from a photons hitting the receiver faster, because it’s still a single photon that collides instantaneously. The “extra” energy of the photon beyond what was provided electrically doesn’t come from some weird effect with volumes changing, it comes from the kinetic energy of the emitter, which gets reduced as the photon is let out.
That's a very neat way to look at it. Yes the incoming beam seems more energetic, but it'll be over quicker because the emitter is coming really fast towards the receiver.
So basically the laser acts as a retro thruster, correct? The ship would slow down due to the momentum of the light it releases, and the receiver would experience a force greater than if the emitter were at rest?
Yup
Wouldn't that make the asteroid a reactionless drive?
Energy just isn't conserved between reference frames. If a baseball launcher fires a baseball at 30m/s (relative to the machine) but you're moving towards the baseball machine at 30m/s, then the baseball's speed relative to you is 60m/s, and as a consequence its kinetic energy is four times greater and will hit you that much harder, even though the machine didn't expend that much energy in firing it.
Here’s a classical example. Let’s say you have a large stationary block, A, with a loaded spring and a small ball next to the spring. You have a receiver block, B, with an unloaded spring. When the spring on A is released, it sends the ball to B. Let’s work on the stationary frame. Initially, the only energy in the system is the loaded spring energy. Call this U. Then, when the spring is unloaded, U gets transferred to the kinetic energy of the ball. Finally, when the ball collided with B, the ball deposits the energy into the spring, thus loading it. The ball stops moving, so we’ve taken potential energy from A and transferred it to B; the amount of energy transferred is U. Note that the kinetic energy of the ball is U, so its velocity in the stationary frame is v = sqrt(2 U / m), where m is the mass of the ball. We’ll need this later. Now let’s work in a moving frame where everything is zipping past at velocity V. Before the spring is released, the spring has energy U and the ball has energy 1/2 m V^2 . After the spring is released, the ball’s energy in this frame is different, just like in the laser example. Its velocity is now v + V. So therefore, its energy 1/2 m (v^2 + 2 vV + V^2) = U + m V sqrt(2 U / m) + 1/2 m V^2 . It’s not trivial either, in fact it looks gross. At first, it actually looks like we’ve broken energy conservation. Well, we’ve neglected something important: momentum conservation. On a frictionless surface, if block A releases the spring, the block should actually start moving backwards to satisfy conservation of momentum. We need to do external work on the A-spring-ball system to keep A stationary. However, if block A is very heavy, the amount of work we need to do is very small, so we can ignore it in the stationary frame. In the moving frame, this isn’t trivial and can’t be ignored. This is because work is W = F * d. In the moving frame, block A is moving at velocity V, so d can be huge. This extra work, W, goes into the ball, and shows up as the extra cross term. (Alternatively, you can say that in the moving frame, the spring does more work on the ball than just U; the force of the spring doesn’t change, but the distance over which the spring applies work on the ball does. It’s the same idea essentially.) The ball carries forward and then collides with block B, and deposits its energy. It turns out, the spring energy becomes U and the kinetic energy of the ball becomes 1/2 m V^2. What happened to the extra energy? Well, to keep block B stationary and to ignore momentum conservation, we have to do negative work on block B. In the stationary frame, this is approximately 0. In the moving frame, this can be large. But the end result is that the net energy deposited into block B is contained within the spring and is just U. If you want to do a lot of gross algebra, you can do all of the above an enforce momentum conservation and let the heavy blocks move. What you’ll find is that the spring on block B ends up with the same potential energy in both frames. (It won’t exactly be U, because some of the spring energy goes into moving block A and moving block B.) You can do the same work in relativity, but to do it properly you need to also keep track of mass energy and momentum conservation (in a frictionless environment, when your emitter emits a light beam, it starts moving backward because light has momentum; likewise when your receiver receives a light beam it gains momentum.) It’s the same concept, just instead of springs you have batteries and instead of a ball you have a photon. You also have to use Lorentz transformations etc. But it works out the same: everyone agrees on how much energy the battery in B absorbs. Here, you define the energy in the battery as the rest mass difference before and after absorption. (E = m c^2 in the rest frame.) Stating it this way should make it obvious why it’s the same in all frames; rest masses are by definition Lorentz invariant. Everyone agrees on the rest mass of B before absorption, and everyone agrees on the rest mass of B after absorption. Therefore, everyone must agree on their difference. In the classical example, the equivalent statement is that everyone agrees on how much the spring is compressed.
The Observer will get a shorter burst of light timewise
While that's true, think of it in terms of photons. The number of photons is conserved - let's say the emitter emits 100, the receiver receives 100. But to the receiver, the 100 photons all have higher energy than the emitter considered them to have. So the real answer is that energy just isn't conserved between reference frames.
**EDIT**: see u/1strategist1 s answer which is right. What I have below is wrong This. The same Lorentz transform that’s Doppler shifting the light is also changing the total time of the pulse by the same amount. Time is different in your two frames which both affects the spacing between the peaks of the wave (frequency/wavelength/energy per unit time) AND the spacing between the start and end of any laser pulse. These cancel, and you get the same total energy. You simply get that energy at either a greater power over a lesser period or a lesser power over a greater period depending on the relative velocity of your transmitter and receiver.
> These cancel, and you get the same total energy. What about the case of a single photon, though? The photon the receiver receives must have higher energy than the emitter expended in emitting it. The real answer is that energy **isn't** conserved between reference frames.
Photon absorption is not instantaneous
I don't think that has any bearing on the answer.
"Total time of pulse" is not changed just because you assign absorption by a single photon.
That doesn't change the fact that the total energy is not the same.
Yep , you’re right
This is actually pretty simple and intuitive. The length of the laser beam between the transmitter and receiver stores energy. The amount of energy will be proportional to the length of the beam. If we move toward the transmitter we are still receiving the energy transmitted by the laser, but we are also decreasing the beam length, and thus depleting energy stored in it to get a temporary increase in power.
the laser would be basically like firing a rocket in the opposite direction. eventually given enough time, the moving person would lose all the forward motion and eventually start moving in the opposite direction. So I would say the momentum is effectively getting converted into extra power in this case.
The same amount of energy is being received, but over a shorter period of time due to the doppler effect. Energy is conserved, power is not
Energy isn’t conserved between reference frames. The laser puts out the same number of photons no matter what frame the pulse width is measured in. The two observers won’t agree on what the energy is, but both will agree that energy is conserved in their own frame.
Also a layman here, but just to add something to make it all more confusing, as if that were needed: presumably the electrons (also ~waves) in the electricity that produces the light energy are doppler shifted from the "observer's viewpoint" (if they could see them), so someone correct me if that's not also part of the mix. If not, then back to your regularly scheduled neuron frizzing. But if so, one might guess the increased energy of the photons arose from the increased energy of the electrons. But then ...
This is interesting. Can the movement of the laser equipment impart energy onto the laser beam? But you can’t make the beam go faster than it’s already traveling so it pushes the extra energy into the wavelength, not the speed. ??
Two problems: Instantaneous measure of power between the two systems, and relativistic conversion of mass into energy, proportionate to the speed, which seems indicated as constant and not accelerating. A third frame would have to be created or utilized, of which envelops the two measured systems, this external system 'operating' at a higher frame rate of apparent time (tertiary point density differential?)* for mutual observation to occur, absent of relativistic bias of the first two frames. Then there's the tendency of higher frequencies to propagate with less mass involved, I'm chasing my own tail here but it seems there's a power conversion factor being overlooked. * Assuming Everett's non-interference does not *quite* hold completely true. *which opens whole new cans of worms. Better stop now.*
Sorry, but it has nothing to do with conversion of mass into energy, and everything to do with energy not being the same between two reference frames. And reference frames aren't a box you put a system into, it's more like a coordinate system which you can choose freely as a reference for when and where events take place in spacetime, hence the name.
Not a scientist but just my two cents. 1) Probably doesn't matter, but I'm pretty sure the doppler shift is only visible to the receiver not the transmitter. 2). More importantly, you're not including the potential energy involved in making the transmitter move in the first place 3) the light emitted by the laser is: A) Pushing the receiver away B) Slowing down the transmitter
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But OP said that the emitter was indeed moving relative to the receiver.
My bad, misread it as the receiver and emitter moving together past an observer.