Question

If gravitational field has negative energy density, how does gravitational radiation carry positive energy?

Answer 1

In order to force two like electric charges nearby, some work has to be done. This implies that the following integral for the total energy increases:

∫ϵ0|E|22dV∝∫|E|2dV∫ϵ0|E|22dV∝∫|E|2dV

Now, as EM's and GM's laws are analogous, the following similar integral must also increase in GM (E⃗ E→ is just multiplied by some constant to obtain g⃗ g→):

∫|g|2dV∫|g|2dV

The problem is, however, that two like mass-charges attract. Thus total energy must decrease:

∫K|g|2dV∝∫|g|2dV∫K|g|2dV∝∫|g|2dV

implying K<0K<0 - all gravitational energy is negative. I strongly suspect that the constant for gravitomagnetic field will also be negative.

Answer 2

The observed fact that the orbits of binary pulsars decay, and the assumption that the decay is due to emission of gravitational waves, is interpreted by some to imply that the energy density of the gravitational field is positive.

HOPE IT HELPS YOU.
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