Does Einstein's mass-energy equation (E = mc^{2}) imply that the mass of an object increases when we give it enough energy, be it kinetic or otherwise?

Asked by: Marc

Answer

No. First, we should define the mass. Mass is defined
with respect to a frame where the velocity of this object
is zero, and often called as 'rest mass'. In this
equation 'm' is not the rest mass. If we denote rest mass
with m_{0}, then the equation should be written as:

Where gamma is equal to:

And, this equation gives the energy of the particle with mass m,
with respect to a frame whose travelling with speed
Answered by: v

.

So, the kinetic energy can be calculated, by finding the
energy difference between each case; when the particle is
at rest w.r.t. frame that gamma = 1, and when it travels
with some speed.

Note that gamma is always greater then 1. And, when v is much smaller than c,
the K.E. equation can be approximated with a great precision
by the non-relativistic K.E. equation:

As a result, 'm' in this equation is actually 'm_{0} * gamma',
where m_{0} is the mass of the object, gamma is a unitless
number which depends on the velocity of the object, w.r.t.
the frame where the measurements are done.
Answered by: Taylan Akdogan, Physics, Ph.D. candidate, M.I.T.

I respectfully, disagree with the answer to the above question.

Mass m in the equation E = mc^{2} is equal to the rest mass
(m_{0}) times the factor gamma. Gamma has the value 1 when v = 0, and
increases toward infinity as v approaches c. That is, the mass in E =
mc^{2}, which is clearly the mass the questioner is concerned about, does
indeed increase as an object acquires energy due to motion = kinetic
energy. E = mc^{2} clearly implies that, as total relativistic energy E
increases, the total (rest and relativistic kinetic) mass increases
correspondingly!

It is this _total_ mass that has to be reckoned with when interacting
with the object. For example, the mass of the electrons in the electron
beam of a CRT increases due to the acceleration of the electrons toward
the face of the tube. This factor has to be taken into account in
designing the deflection system that scans the beam across the tube
face. Likewise, the considerable increase of the mass of particles in
particle accelerators has to be taken into account in the design of
accelerators.

Furthermore the answer to the question is not complete. There was also
a 'be it kinetic, or otherwise?' The questioner should be informed
that, yes indeed, it doesn't make any difference what form of energy is
acquired by the object. Kinetic energy is only one possibility.
Thermal energy and potential energy (of various kinds) also matter. For
example, the binding potential energy between the constituents of the
nucleus contributes significantly to the mass of the nucleus. It is
precisely the liberation of this mass-due-to-binding-energy that is the
basis of nuclear fission reactors and the atomic bomb!
Answered by: Warren F. Davis, Ph.D., President, Davis Associates, Inc., Newton, MA USA

'There is no inductive method which could lead to the fundamental concepts of physics. Failure to understand this fact constituted the basic philosophical error of so many investigators of the nineteenth century.'