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This paper will focus on the relation between gauge-invariant energy density of gravitational field and electromagnetic field. After the brief review the history of finding the analogy between gravitation and electromagnetism, we recall the two different ways to remove the gauge degrees of freedom, Physical decomposition and gauge-fixing. Then, by using the gauge-fixing approach, we get the gauge-invariant energy density (Hamilton density) of gravitational field. By using the similar method into electrodynamics, we can also get the gauge-invariant energy density of electromagnetic field. Finally, while comparing with the two gauge-invariant energy density of gauge-invariant energy density of gravitational field and electromagnetic field, we find there is a similar electromagnetic energy density does appear in gravitational field. The amazing conclusion is that although the rotating electric charge with constant speed can produce static magnetic field energy, the rotating mass with constant speed cannot produce any similar "magnetic" energy in gravitational field. Only accelerated rotating mass can produce the similar "magnetic" energy, this is a peculiar property of gravitational field never reported before.

Research on the Gauge-Invariant Energy Density of Similar Electromagnetic Properties of Gravitational Field