Energy conservation in relativity

[espen180]

I did a thought experiment for myself today to test whether energy is conserved in Lorentz transformations.

I chose to consider three objects lying on a horizontal line. In object 2's rest frame, object 1 has a velocity of 0.600c to the left, and object 3 has a velocity of 0.600c to the right. I let each object have a mass of 1kg.

The total kinetic energy in this system will be $$\Sigma E_k=2\cdot (\gamma-1)mc^2=2\cdot 0.25\cdot1kg\cdot c^2=0.5c^2\;[J]$$

Now consider the rest frame of object 1. Object 2 now has a velocity of 0.600c to the left and object 3 has a velocity of 0.882c to the left, according to the Lorentz transformation.

The total kinetic energy in this system is now $$\Sigma E_k=(\gamma_2-1)mc^2+(\gamma_3-1)mc^2=0.25\cdot 1kg\cdot c^2 + 1.125\cdot 1kg\cdot c^2=1.375c^2\;[J]$$

Why does the system have a different energy depending on the reference frame? Am i overlooking something? If energy is not conserved in Lorentz transformations, doesn't that raise the question of which reference frame displays the correct energy, violation the first postulate?

Any help is appreciated.

 

[Dale]

I did a thought experiment for myself today to test whether energy is conserved in Lorentz transformations.

Actually, your thought experiment was about the frame-invariance of energy, not about its conservation.

I chose to consider three objects lying on a horizontal line. In object 2's rest frame, object 1 has a velocity of 0.600c to the left, and object 3 has a velocity of 0.600c to the right. I let each object have a mass of 1kg.

The total kinetic energy in this system will be $$\Sigma E_k=2\cdot (\gamma-1)mc^2=2\cdot 0.25\cdot1kg\cdot c^2=0.5c^2\;[J]$$

Now consider the rest frame of object 1. Object 2 now has a velocity of 0.600c to the left and object 3 has a velocity of 0.882c to the left, according to the Lorentz transformation.

The total kinetic energy in this system is now $$\Sigma E_k=(\gamma_2-1)mc^2+(\gamma_3-1)mc^2=0.25\cdot 1kg\cdot c^2 + 1.125\cdot 1kg\cdot c^2=1.375c^2\;[J]$$

Why does the system have a different energy depending on the reference frame? Am i overlooking something?

Your conclusion that energy is different in different reference frames is completely correct, although your terminology is wrong. When a quantity, like the spacetime interval, is the same in different reference frames it is called "frame invariant" (or sometimes "absolute"). When a quantity is different in different reference frames, like length, then it is called "frame variant" (or sometimes "relative"). When a quantity is constant through time (dX/dt = 0) in a single reference frame then it is called "conserved". So "conserved" and "frame invariant" are completely different concepts. You have correctly demonstrated that energy is frame variant, but this does not imply that it is not conserved, two reference frames may disagree on the value of a quantity even though they both agree that it is constant. Note that energy is frame variant even in Newtonian physics.

As I am sure that you already know, time and space are related to each other through the Lorentz transform. You may not have been introduced to the http://en.wikipedia.org/wiki/Four-vector" ) related to each other through the exact same Lorentz transform. Energy is relative in the same way that time is relative.

 

[espen180]

I understand. Thank you for clearing it up for me.

 

[jtbell]

Why does the system have a different energy depending on the reference frame?

For the same reason it has a different energy depending on the reference frame in classical (non-relativistic) mechanics.

 

[espen180]

I never thought about that before.