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It depends on which theory of relativity you mean. The special theory of relativity has only one assumption, namely that the speed of light is constant (in vacuum!) and has the same value for each non-accelerating observer. Thus you can say the speed of light must always be conserved.

In the general theory of relativity (the one with the curved spacetime and the geometric picture of gravity), one further assumption is made; that the inertial mass (resistance to acceleration) of every object is equal to its gravitational mass (source of gravity). This is called the Equivalence Principle and is sometimes explained as the requirement that all things fall at the same rate (if there is no air resistance) regardlass of mass. In this theory the equivalance of the two types of mass is always to be conserved.

Also note that although these are the main assumptions it is usually also assumed that energy and momentum are conserved and that there is no violation of causality.

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