Originally posted by Peter_C
One thing I may be forgetting in the above post is that the molecular energy is stored similar to kinetic energy... it is potential energy waiting to be released... For example a compressed spring holds no more mass than the same spring uncompressed.
Not correct, as shown by the following example:
Assume that you have two springs: A and B. The springs are compressed, so that when triggered, the two springs are pushed apart. As a result, spring B is pushed "backward" as spring A is pushed "forward" by the release of compression.
Basic Parameters:
Springs: 1kg each.
Velocity each attains from the energy release: .01c
We will also assume that there is a third reference C moving in the opposite direction of A at .99c relative to the two springs before were released (the original rest frame).
First we analyse it using classical mechanics:
For either mass to attain .01 c, it must have an KE of mv²/2 or:
1(300000)²/2 = 4.5e12 joules
Therefore, the total energy of the system as measured from the rest frame is 9e12 joules, either as stored energy before being triggered, or as the combined KE of both masses after being triggered.
Now, let's look at things from the frame of C.
At the start, both A and B are moving at .99c relative to C, so the total KE of the system at this point is:
2(297000000)²/2 = 8.8209e16 joules
plus the energy store of 9e12 joules equals
8.8218e16 joules total energy.
After the springs are triggered:
A will have a velocity of .01+.99c =1c relative to C
This gives it a KE relative to C of 4.5e16 joules.
B will have a velocity of .99c-.01c = .98c relative to C
This gives B a Relative KE to C of 4.3218e16 joules
Adding the KE's of A and B together gives us a total energy of 8.8218e16 joules, the same value we got for total energy before the springs were triggered. Thus energy is conserved at all times from all frames, as it should be.
Now by relativity:
The Relativistic KE formula is KE = mc²(1/sqrt(1-v²/c²)-1)
Therefore, in order for the masses to attain .01 c wrt to the original rest frame the total KE is
(2)c²(1/sqrt(1-(.01c)²/c²)-1) = 9.0006750563e12 joules wrt to the rest frame.
This is also the amount of stored energy needed for the compressed springs. Thus it is also the total energy of the system at any point during the test, relative to the rest frame.
Again, at the start, both A and B are moving at .99c relative to C.
So we just use the relativistic KE formula to find the KE relative to C.
Thus
KE = 2c²(1/sqrt(1-(.99c)²)-1) = 1.095986169e18 joules
Plus the 9.0006750563e12 joules of the stored energy equals
1.09599517e18joules
In order to determine A and B's relative velocity wrt C after the springs are triggered, we must use the addition of velocities theorem W = (u+v)/(1+uv/c²)
This gives us a relative velocity of A wrt C of
(.99c+.01c)/(1+.99c(.01c)/c²) = .99019705c
The KE of A relative to C using the Relativistic KE formula is:
5.5434143392e17 joules
B's relative velocity is found by
(.99c-.01c)/(1-.99c(.01c)/c²) = .98979901
B's relative KE wrt C is
5.4170853919e17 joules
Adding these two KE's together gives us
1.0960499731e18 joules,
Which is 5.48e13 joules more than what we got for the total energy of the system before the springs were released. We have more energy after than before, and energy is not conserved.
The problem comes from not taking into account the mass of the stored energy of the springs. This works out to:
9.0006750563e12/c² =.00010001 kg, which has to be added to the 2kg of our springs before we can calculate the KE of the system wrt C
Thus:
KE = 2.00010001c²(1/sqrt(1-(.99c)²)-1) = 1.096040974e18 joules
Plus the 9.0006750563e12 joules of the stored energy equals
1.0960499731e18 joules.
Which equals the total energy of the system after the springs are released, and energy is conserved.
Therefore, one can see that "stored" or potential energy
does add to the mass of an object. If if didn't, energy wouldn't be conserved in all frames.