A fast-moving object moving at near to the speed of light cannot be accelerated to, or faster than, the speed of light, regardless of how much energy is put into the system. As a constant force is applied on such an object, and hence work is done on the object, its speed does not appear to increase by the amount specified by the
kinetic energy formula Ekinetic = 1/2 mv2. Instead, the energy provided to it continues to appear as mass, even as the rate of velocity increase nearly stops. The object's relativistic mass increases, in what is known as
mass dilation. The relativistic mass of an object is expressed as a function of its speed relative to the speed of light.
As discussed more fully in
mass in special relativity, the
relativistic mass which appears associated with a single fast-moving object is an observer-dependent quantity, and the part of it which is associated with a single object's kinetic energy is just as observer-dependent as the kinetic energy itself. In this case, either one may be made to disappear entirely for single objects, by proper choice of the
inertial frame. This choice is the frame in which the object is not moving. For this reason,
mass in special relativity is usually chosen to be
rest mass or
invariant mass, which is a quantity which is not
frame-dependent. In other words, there is no part of invariant mass for single objects which depends on kinetic energy, since this quantity is defined as the mass in the inertial frame where the object is not moving, and its kinetic energy is zero. In other interial frames, the equations for invariant mass correct for object speed, and thus again kinetic energy does not contribute to invariant mass.
In systems of objects by contrast, although a part of the invariant mass for systems of objects may depend on the kinetic energy of some of the objects in the system, this part of the mass is also constant, and not observer-dependent. This kinetic energy, unlike the case of single objects, cannot always be made to disappear by choice of observers, since for many systems there is no inertial frame where all objects are at rest. Thus, the best that can be done to minimize a system's mass is to pick an inertial frame in which kinetic energy is minimized-- but in that case, any residual minimal kinetic energy must be counted as part of the system's
invariant mass. The invariant mass (or energy) of a system is defined as that total energy which is present in the particular inertial frame where the contribution of kinetic energy to the total energy of the system is minimized (the
COM frame). The COM frame is chosen so that the momenta of objects in a system cancel, and add to zero, and this also minimizes the system's total kinetic energy. In other inertial frames where the system's objects are moving (on average) faster, the equations which define invariant mass correct for the increasing (and non-zero) momenta of the objects, and ensure that this quantity of invariant mass remains constant. Thus, some part of the kinetic energy of systems may contine to contribute a constant amount to the system's invariant energy and mass. However, this amount does not change, even when viewed from other inertial frames in which the kinetic energies of the various objects in systems may be very different
