The Paradoxical Nature of Frame-Dependent Energy

[Punkanzee]

Rocket A and Rocket B are traveling next to each other in a straight path away from the earth. Both rockets are so far from earth that the affects of gravity are negligible, we only need an arbitrary reference point in space.

 

Both rockets are initially moving at 10 m/s away from the earth and the mass of each rocket is 20 kg.

 

The equation for kinetic energy is given by 1/2 x mass x velocity^2. Thus an observer on earth, the kinetic energy of each rocket is 1/2 x 20 x 10^2 = 1000 joules. From Rocket A's perspective, Rocket B has 0 kinetic energy because the two are moving side by side at the same rate. The reverse is true of Rocket A from Rocket B's perspective. As you can see kinetic energy for is completely frame-dependent.

 

We need Rocket A to accelerate to 20 m/s from earth's perspective. Rocket A going 20 m/s would have 1/2 x 20 x 20^2 = 4000 joules and since it has 1000 joules already it is assumed that 3000 more joules is all that is required in order to accelerate Rocket A from 10 m/s to 20 m/s.

 

However, 20 m/s on earth translates to 10 m/s from Rocket B's perspective. We have already calculated that 1000 joules is the kinetic energy of Rocket A going 10 m/s and thus it is assumed that 1000 joules is all that is required to accelerate Rocket A from 0 m/s to 10 m/s.

 

Imagine that Rocket A can absorb microwave energy and convert it into electricity and the electricity is used to power an ion propulsion system to accelerate Rocket A. Assume all transfers of energy are 100% efficient. Earth and Rocket B are capable of emitting microwave energy to Rocket A in any amount necessary.

 

The earth fires a 3000 watt microwave laser to Rocket A for one second in the expectation that Rocket A should be able to accelerate to 20 m/s from it.

 

Or

 

Rocket B fires a 1000 watt microwave laser to Rocket A for one second in the expectation that Rocket A should be able to accelerate to 10 m/s from it.

 

Rocket B can also absorb microwave energy and store the energy but it doesn't have an ion propulsion system. Now imagine that earth fires a 3000 watt microwave laser for one second to Rocket B instead of Rocket A. Rocket B now holds 3000 joules. Rocket B still only has to give Rocket A 1000 joules from it's perspective. So Rocket B fires a 1000 watt microwave laser to Rocket A for one second as stated earlier. Rocket A accelerates to 10 m/s. Rocket B has 2000 joules left over and sends the 2000 joules back to earth in the form of a 1000 watt microwave laser fired for 2 seconds. Using Rocket B as an intermediary, the earth accelerated Rocket A to 20 m/s for 1000 joules, less energy than was thought to be necessary.

[swansont]

Energy isn't an invariant; there is no paradox. Within a given frame, however, it is conserved, which makes it a very useful quantity.

[kleinwolf]

Is there an invariant in classical mechanics made out of energy ?

 

As about relativity, I think the quantity E^2/c^2-p^2 remains invariant (towards this time a Lorentz transformation), is it true ?

[ajb]

Is there an invariant in classical mechanics made out of energy ?

 

As about relativity, I think the quantity E^2/c^2-p^2 remains invariant (towards this time a Lorentz transformation), is it true ?

 

The magnitude of the 4-momentum is a Lorentz scalar.

[swansont]

Is there an invariant in classical mechanics made out of energy ?

 

As about relativity, I think the quantity E^2/c^2-p^2 remains invariant (towards this time a Lorentz transformation), is it true ?

 

That's because it's the rest mass (multiplied by constant). I should have said kinetic energy isn't an invariant.

 

I may have omitted that because I ran into this just recently on a blog ( http://scienceblogs.com/builtonfacts/2009/03/a_connundrum.php ). "Invariant" and "conserved" aren't the same thing, and that can be confusing. KE is not, in general, conserved (i.e. it can be conserved, but only under certain circumstances), and it is not invariant. Total energy is conserved, but is not invariant. Rest mass is not conserved, but is invariant.

[Punkanzee]

Regardless, a beam of light that contains 1000 joules of energy will remain 1000 joules of energy. The Doppler effect will not change the overall energy. A 1000 watt laser may become a 990 watt laser but instead of lasting for 1 second it will last for 1.0101... seconds and thus energy is conserved. Power in this case is frame-dependent but energy is conserved. Transferring universally non frame-dependent energy into a system inside the universe where the energy is frame-dependent is a paradox. Furthermore the Doppler effect is so small in this example that it is practically non-existent.

 

Also, how can energy not be invariant and remain conserved? If the overall energy is changing then it is not conserved.

Edited March 23, 2009 by Punkanzee

[ajb]

 

Also, how can energy not be invariant and remain conserved? If the overall energy is changing then it is not conserved.

 

 

In a given frame the (total) energy will remain constant over the course of some process. However, the actual value of the energy will change if you move to another frame.

[swansont]

Regardless, a beam of light that contains 1000 joules of energy will remain 1000 joules of energy. The Doppler effect will not change the overall energy. A 1000 watt laser may become a 990 watt laser but instead of lasting for 1 second it will last for 1.0101... seconds and thus energy is conserved. Power in this case is frame-dependent but energy is conserved. Transferring universally non frame-dependent energy into a system inside the universe where the energy is frame-dependent is a paradox. Furthermore the Doppler effect is so small in this example that it is practically non-existent.

 

Also, how can energy not be invariant and remain conserved? If the overall energy is changing then it is not conserved.

 

 

No. Count photons instead. The number of photons will be the same, but their individual (and total) energy will be different.

[Punkanzee]

The energy of a photon of light = Planck's constant x the frequency of light.

So the only way the energy of the photon can be different in the vacuum of space is because Doppler shift . However the frequency of light changing due to Doppler shift is so low when only going 10m/s faster that it is practically non-existent and furthermore there is no exponential factor in the equation for light energy.

So essentially the light energy will remain constant regardless of reference frame when not going incredibly fast.

 

The kinetic energy however will not remain constant regardless of reference frame.

 

So explain to my why it would not be possible using all of the 3000 joules sent to Rocket B, Rocket B couldn't send all of it to Rocket A and thus cause Rocket A to have a velocity of 17 m/s(Rocket B) / 27 m/s(earth). Rocket A going 27 m/s from 3000 extra joules is completely ridiculous from earth's pov but makes perfect sense from Rocket B's pov because it is only 17 m/s.

[Neil9327]

You say that rocket B only has to give rocket A 1000 joules to accelerate B from 10 m/s to 20 m/s relative to earth (0 to 10 relative to it). But this isn't true, due to the conservation of momentum: What would happen is that rocket B would be slowed to 5 m/s and rocket B to 15 m/s because the effort of pushing forward A would equally push back B.

So the combined kinetic energy of the two rockets relative to earth would be:

 

0.5 * 20 * (0.5)^2 + 0.5 * 20 * (1.5)^2

= 0.5 * 20 * 0.25 + 0.5 * 20 * 2.25

= 2500 Joules.

 

Hmmm seem to have got something wrong here. But there is more to it than you have said. Got to go - friend here.