Take this scenario - An oscillator that goes back and forth at .99 C , where c = speed of light. A laser shines light onto this oscillator with a specific frequency. Due to the relativistic doppler shift the wavelength changes. A change in wavelength means photons are more energetic, right ? This would amount to something like 14 x the energy absorbed by the oscillator than an object at rest affected by the same laser. Even if the laser was 20% efficient you could power the laser with the extra energy form the oscillator. The most immediate argument against this is that the oscillator will be slowed down when it absorbs photons from the laser. This way all that extra energy has actually been there all along as kinetic energy. I do not think that this is the case since (photon momentum) = (photon energy) / c [courtesy of wikipedia] and so the oscillator would only be slowing down a negligible amount What am I missing ?? Is the equation on hyperphysics inaccurate ?
What do you mean by this? Don't just wave your hands. Give me the math. You are ignoring that the light falling on the oscillator will be red shifted half of the time. Hand-waving. Show the math. The most immediate argument against this is that you don't have an isolated system. Something is powering the laser, after all. Energy is not conserved in an open system, even in classical (non-relativistic) mechanics.
Hi DrZion, I think that before considering this problem, you would do well to consider a low speed case. Replace the high speed shuttle with a kid on a bike (total mass 50kg, riding at 5m/s). Replace the laser with a thrown ball (mass 0.1kg, thrown at 10m/s). If the kid is riding toward the ball, the ball has more kinetic energy if the bike is the rest reference than if the ground is the rest reference. If the kid is riding away from the ball, the ball has more kinetic energy if the ground is the rest reference than if the bike is the rest reference. What gives?
In engineering a perpetual machine I must rely at least to some degree on things other than mathematics - a diagram, an explanation, even some hand waving to get the point clearly across. Assume that the oscillator is connected to a stirling engine so that when it heats up it produces electricity for the laser to run on. I'll just draw a schematic for you. Hi Pete, I was looking at the relativistic doppler shift which we were discussing earlier and I realized that relativity opens new doors for perpetual motion machines. The one I described above is just one example. The first idea involved thermal radiation and time dilation - which would lead to two identical bodies radiating different amounts of heat as long as one was moving fast relative to the other. This would only violate the second law of thermo- it would decrease entropy, the basis of perpetual motion machines of the second variety. The one above, to me, seems like it is violating the first law in that it outright creates energy - something I don't really believe in.:bugeye: The difference here is that the ball is a massive object while a photon has only energy. If the kid kept catching the balls and putting them in a big basket on the back of the bicycle then after catching 500 balls his mass would be doubled. If he was riding on a frictionless bicycle his velocity would be reduced by half (ignoring the fact that the balls are also transferring kinetic energy). Photon momentum is negligible - its energy/299 792 458 . So the kid could fly into the sun without getting repelled by the photons it produces . If the sun radiated energy in the form of balls, the kid would very quickly get repelled because those balls carry a lot more (momentum) / (joule of energy) than photons.
He doesn't have to keep them. Let him drop them by the wayside. Very small isn't always negligible, as many perpetual motion aspirers have discovered. Pretend that the world is Newtonian (ie no upper speed limit, and Newton's laws rule). Let's give the balls the same kinetic energy, but make them very light and very fast - say 299 792 458m/s. Now, the momentum of each ball is apparently negligible - it's energy/299792458 (in SI units). What gives?
So only momentum is transferred , not mass? This doesn't work as an analogy to photons because photons can affect the thermal energy or energetic orbitals/vibrational levels while barely altering momentum/kinetic energy, something balls cannot. In other words the photons have a higher energy : momentum ratio . Yes, but perpetual motion experts agree that energy could not be created even if there were perfect materials. http://www.lhup.edu/~dsimanek/museum/people/people.htm But this way the relativistic doppler effect doesn't apply. The balls would still transfer momentum and contribute to acceleration of the object affected by the ball-radiation but they would not contribute to the thermal energy of the object - while photons do.
But they can. High speed, light weight. The balls (in the Newtonian world) would transfer momentum, contribute to acceleration, and contribute to thermal energy just like absorbed photons (in the real world). There would indeed be slight differences when transforming to different reference frames, but not enough to matter at the low speed of the bicycle. I just don't think you're ready to tackle relativistic kinematics before you have a proper grasp of Newtonian kinematics.
Simplest way to see if energy is conserved is to write down the Lagrangian for the system and if its explicitly independent of t, ie \(\frac{\partial \mathcal{L}}{\partial t} = 0\) then you have energy conservation via Noether's theorem. If you want Newtonian mechanics then you check its invariant under the Galilean group and if you want relativistic invariance the Poincare group.
Imagine that a laser pulse has been fired head-on at a absorbent boulder moving at .99 c through space. The pulse contains 100 joules of energy . However, the boulder will perceive this pulse to be greater than 100 joules due to the doppler shift. In order for conservation of energy to remain intact, the kinetic energy of the boulder has to be related to the energy of the pulse. Any excess energy over 100 joules has to come from the boulder.
DrZion, that situation is not significantly different to this: Imagine that a small marble has been fired head-on at 100,000km/s at an absorbent boulder moving at .99 c through space. The marble's kinetic energy is 100 joules. However, the boulder will perceive this energy to be greater than 100 joules due to it's frame of reference. In order for conservation of energy to remain intact, the kinetic energy of the boulder has to be related to the energy of the marble. Any excess energy over 100 joules has to come from the boulder. Really, you should try to tackle Newton first.
Seriously, DrZion, think about it. You launch a 0.02 nanogram marble (2e-14 kg)at 100,000km/s. The marble's kinetic energy is 100 joules. It's momentum is 2e-6 Nm (2 millionths of a newton metre - negligible, right?) But a 10 kilotonne boulder (1e7 kg) cruising at 100,000km/s in the opposite direction will perceive the marble's kinetic energy to be 400 joules. Where does the extra 300 joules come from? Do you think it could be used to power a perpetual motion machine?
Indeed not .. at least not a perpetual motion machine of the first kind. Momentum scales differently than kinetic energy and so a minor change in momentum can have large effects on kinetic energy. Something to put in my 'learned' folder ! Please Register or Log in to view the hidden image! If the marble underwent a perfectly inelastic collision with the boulder, most of that kinetic energy would be converted to heat , while momentum would barely change. momentum p = vm kinetic energy .5 m v^2 so it is not linear Relativistically, going close to c, pc = e for a photon p = e/c Relativistic momentum is similar to newtonian momentum, except that mass is changed due to relativistic effects. Going closer and closer to c, it is mass that starts to change much more rapidly than velocity, and so it is mass that gets expressed the most in momentum. The kinetic energy is basically the new mass M C^2 - Mo C^2 minus the rest mass, and going close to c it is also mass that becomes expressed the most. So, going close to the speed of light momentum scales almost linearly with kinetic energy and the difference in energy is entirely drawn from kinetic energy.
All right. This might not be easy, but it should be resolvable assuming perfect materials and perfect efficiency - it's an energy conservation question, not a thermodynamics question. Questions: How much momentum does a receiver on the rim lose in catching a given amount of laser energy? How much energy is required to accelerate that receiver back up to speed? I think it should be the same as the difference in light energy between emission and capture. Possible complication: Do we need to consider the mass added to the receiver when it absorbs the light? I don't think so, but I'm not sure. What about the recoil of the laser as it emits the light? I think that could be neglected by saying that the laser is rigidly connected to the wheel axis, and that everything is very massive compared to the receivers on the wheel. But, I could be wrong.
Has the answer something to do with the fact that DRZions thought experiment shows that energy is not invariant but it does not prove that conservation of energy does not stil hold in his example?
