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In relativistic mechanics, in order to be conserved, momentum must be defined as:

is the invariant mass of the object moving,

is the relative velocity between an object and an observer

is the speed of light.

Relativistic momentum can also be written as invariant mass times the object's proper velocity, defined as the rate of change of object position in the observer frame with respect to time elapsed on object clocks (i.e. object proper time).  Relativistic momentum becomes Newtonian momentum:  at low speed .

Relativistic four-momentum as proposed by Albert Einstein arises from the invariance of four-vectors under Lorentzian translation.  The four-momentum is defined as: