Take two uncharged particles (e.g., neutrons) at rest with respect to one another in empty space. Now apply 1 Joule of energy to accelerating them apart. They will eventually come again to rest with 1 Joule of gravitational potential and accelerate back together due to gravity. Their kinetic energy when they return to their original position will be the same 1 Joule that we originally gave them.(adsbygoogle = window.adsbygoogle || []).push({});

According to Special Relativity, energy and mass are equivalent and transmutable, so now instead of accelerating them apart, convert them each to mc^{2}worth of energy (a hard gamma ray), and allow it to depart in opposite directions at the speed of light. Some arbitrary time later, convert each gamma ray back into a neutron. The two particles will then have gravitational potential that apparently came from nowhere, and will accelerate together yielding kinetic energy that they apparently should not have.

The potential solutions I've come up with so far are:

1. A limit on the conversion efficiency such that the "extra" energy must be put in as part of the mass/energy conversions, but that doesn't really make sense.

2. The energy of the photons somehow dissipates over distance at the same rate the gravitational potential increases. But this would make it impossible for us to see stars that are billions of light-years away.

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# Apparent paradox: e=mc^2 violates conservation of energy?

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