lotus

previous page: 1.5 Energy and Momentum Considerations (Special Relativity)
  
page up: Relativity and FTL Travel FAQ
  
next page: 1.5.2 The Energy and Momentum of a Photon (Where m = 0) (Special Relativity)

1.5.1 Rest Mass versus "Observed Mass" (Special Relativity)




Description

This article is from the Relativity and FTL Travel FAQ, by Jason W. Hinson jason@physicsguy.com with numerous contributions by others.

1.5.1 Rest Mass versus "Observed Mass" (Special Relativity)


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

As it turns out, when v is much smaller than c, the equation gamma-1 is
approximately equal to (1/2)*v^2/c^2 such that E_kinetic is approximately
(1/2)*m*v^2 (that's the Newtonian equation for kinetic energy which is
approximately correct for non-relativistic speeds). But with relativistic
velocities, the kinetic energy becomes much larger than we would have
calculated it to be using the Newtonian equations. In that sense, there does
seem to be some "extra energy" which could be considered as extra mass
energy; however, you can't get the correct kinetic energy in relativity by
simply plugging our expression for "observed mass" into the Newtonian
equation for kinetic energy. The observed mass concept doesn't really work
here, and we see that it's better to simply argue that the mass isn't really
increasing, but rather the equations for energy and momentum are different
than expressed by Newtonian physics.

So, "observed mass" has its uses, but physicists today rarely use the
concept in practice. Rather, an object is said to have a rest mass (which
truly is its inherent internal energy) as well as an energy due to its
motion with respect to an observer (kinetic energy) which come together to
produce its total energy, E.

 

Continue to:













TOP
previous page: 1.5 Energy and Momentum Considerations (Special Relativity)
  
page up: Relativity and FTL Travel FAQ
  
next page: 1.5.2 The Energy and Momentum of a Photon (Where m = 0) (Special Relativity)