This article is from the Relativity and FTL Travel FAQ, by Jason W. Hinson jason@physicsguy.com with numerous contributions by others.
It is important to note that the mass, m, in the above equations has a
special definition which we will now discuss (by "mass", we generally mean
the property of an object that indicates (1) how much force is needed to
cause the object to have a certain acceleration and (2) how much
gravitational pull you will feel from that object in Newtonian gravitation).
First, note what happens to the relativistic momentum (Equation 1:6) of an
object as its speed approaches c with respect to some observer. In that
observer's frame of reference, its momentum becomes very large (because
gamma goes to infinity), especially compared to the old definition of
momentum, p = m*v. However, if we define a property called "observed mass"
as being gamma * m, then we see that the momentum can be written as
(Eq 1:10)
p = (observed mass) * v
We see that the momentum can be written exactly as it was in Newtonian
physics, except that it seems the mass of the object as seen by an outside
observer is larger than its "rest mass" (m). Further, if we take the
relativistic equation for the energy of an object, Equation 1:8, we see it
too can be written as
(Eq 1:11)
E = (observed mass) * c^2
This is like the energy of an object at rest (E = m*c^2) with the "observed
mass" substituted in for the "rest mass."
Thus, one way to interpret relativity's effect on our view of momentum
and energy is to say that because of relativity, an observer sees an
object's mass increase as the object approaches the speed of light in that
observer's frame of reference. The mass (m) in our equations is thus the
mass as measured when the object is at rest in our frame of reference (the
rest mass), not the "observed mass" we have defined.
However, this concept of observed mass doesn't really work for
gravitational mass. In a relativistic setting, you can't figure out the
gravitational effects of an object that is moving (in your frame) by simply
figuring out what gravitational effects its mass would have at rest and
replacing its mass with the observed mass in your frame of reference. For
example, as the velocity of an object with respect to you approaches c, its
"observed mass" approaches infinity. However, this does not mean that the
object will eventually look like a black hole predicted by general
relativity (as it would if the same object really did have a huge mass
sitting at rest).
Also, let's look at kinetic energy in relation to mass. Kinetic energy
is energy of motion--it's the total energy of a free object minus the amount
of that energy that is internal to the object:
(Eq 1:12) E_kinetic = E_total - E_internal = gamma*m*c^2 - m*c^2 = (gamma-1) *m*c^2
 
Continue to: