Is there a precise formula giving the increase in mass due an increase in velocity as described by relativity?

Asked by:
Tim M.

Answer

Before I answer your question, I would like to start by giving a more explicit
version of Einstein's famous equation, E = mc^{2}. It is commonly written as follows:

where E is the energy of the object in question, m_{0} is its rest mass, c is the
speed of light, and gamma () is a numerical factor that depends on velocity. The
expression for gamma is

where v is the object's velocity. At very small velocities, gamma is approximately
equal to one. At velocities near the speed of light, gamma becomes larger and
larger (try plugging in some numbers and see). Already you can see why a massive
particle can never travel at the speed of light: it would take an infinite amount
of energy, and no engine in the universe can provide that sort of power!

Now I will answer your question. Einstein interpreted the combination gamma * m_{0} to
be the mass of an object in motion. This is why the equation is usually quoted
simply as E = mc^{2}. The equation relating mass to velocity is

Mass, like gamma, grows with velocity. As the object approaches the speed of light,
the mass approaches infinity!

I should give a word of caution: many physicists simply call m_{0} the mass, dropping
the distinction between the rest mass and Einstein's velocity-dependent mass. So be
careful to note which definition is being used!

This relationship can probably be found in most introductory college physics texts.
A good book for more details, at a slightly more technical level, is Ray
D'Inverno's book Introducing Einstein's Relativity.

A slightly more technical diversion:

It is interesting to look at the behavior of gamma for small velocities. This can
be done using a technique from calculus called Taylor expansion. Consult any
calculus text if you want to see exactly what it is and how it's done. If we Taylor
expand gamma around v = 0, we get

For small velocities, the terms of order 4 and greater are extremely tiny compared
to the 2nd order term, so we can neglect those all together. Plugging this
expression back into Einstein's equation gives

The first term is just the rest mass of the object. The second term you may
recognize to be the kinetic energy of classical Newtonian physics. Thus Einstein's
equation reduces to Newton's equations at small velocities. It took so long to
discover that Newton's laws weren't quite correct because the velocities we
encounter every day are so much smaller than the speed of light. Thanks to
Einstein, we can now study physics at any speed!
Answered by:
Jason Detwiler, B.S., Physics Grad Student, Stanford University, CA

'Physics is mathematical not because we know so much about the physical world, but because we know so little; it is only its mathematical properties that we can discover.'