Mass to energy conversion at fraction of speed of light

[San K]

Lets say we have a frame of reference moving at, say, 10% speed of light.

In it is a nuclear reactor (fusion or fission etc) that can be used to convert tiny amounts of mass into energy.

Since the reactor is moving at, say, 10% the speed of light, the mass of the nuclear material would increase by lorentz transformation formula.

The nucleus of the atom, being split, would have more mass (relative to... if it was at rest).

Now would more energy be released (per E = mc2)? relative to if the nuclear reaction was performed at "rest"...

 

[Bill_K]

Since the reactor is moving at, say, 10% the speed of light, the mass of the nuclear material would increase by lorentz transformation formula.

Sorry San, you have been seriously misled. "Relativistic mass" is a mistaken concept and should never be used, because it leads to confusion like this. A fissioning uranium atom releases just as much energy on a spaceship as it does on Earth.

 

[jtbell]

The energy "released" is the same in any inertial reference frame. Here's a numerical example that I once worked out for a similar situation:

https://www.physicsforums.com/showthread.php?p=2585116#post2585116

[added] Note that the total "relativistic mass" stays constant in each reference frame: it's 1000 MeV/c^2 in frame S, and 1667 MeV/c^2 in frame S'. Likewise, the total energy stays constant in each reference frame: 1000 MeV in frame S and 1667 MeV in frame S'. There is no "conversion of relativistic mass to energy!"

Similarly, the rest-mass (more properly called "invariant mass") of the system stays constant in each reference frame, and is the same in both frames. Before the decay, the invariant mass of the parent is 1000 MeV/c^2. After the decay, the invariant mass of the system is calculated from the momentum and energy of the two daughters as:

$$m_0 c^2 = \sqrt{E_{total}^2 - (p_{total} c)^2}$$

In frame S this is $\sqrt{(500+500)^2 - (300-300)^2} = 1000$.

In frame S' this is $\sqrt{(433+1233)^2 - (-167-1167)^2} = 1000$.

So there is no conversion of rest-mass (of the system) to energy, either! ("Conversion" means some of the rest-mass has to disappear, right?)

What really happens is that some of the rest-energy $m_0 c^2$ of the parent particle gets converted into kinetic energy of the daughter particles. It's simply one of the many ways that energy can be converted from one form to another.

Note that in general, the invariant mass of a system of particles does not equal the sum of the invariant masses of the individual particles. In this example, the invariant mass of the system is 1000 MeV/c^2, both before and after the decay; but the sum of the invariant masses of the daughters after the decay is 400 + 400 = 800 MeV/c^2.

 

[San K]

thanks Bill & Bell

so the remaining/balance 200 MeV/c^2 gets converted into (mainly) kinetic energy?

 

[jtbell]

Yes, in my example, the total kinetic energy increases by 200 MeV in both frames.

In frame S, K_total is 0 before, and 100 + 100 after the decay, for an increase of 200.

In frame S', K_total is 667 MeV before, and 33 + 833 = 866 after, for an increase of 199. The apparent difference of 1 MeV versus frame S is purely due to roundoff error in the calculations. The total before should really be 666.66..., and the total after should be 33.33... + 833.33... = 866.66...