There is perhaps only one equation in all
of physics that almost everyone knows- and
that equation is E equals m c squared.
When you get down to it, it’s actually pretty
amazing that this particular equation is so
well known. After all, it makes a pretty weird
assertion, which is that energy is the same
as matter. I mean- that’s what that equal
sign says.
So, given that the equation is so familiar,
it’s probably worth asking the question:
Is it right?
And the answer will probably shock you. It’s
no- or, perhaps more accurately- the equation
is a special case. In order to use it properly
you need to know more.
Now I don’t want my email inbox to explode
because I said that E equals m c squared is
wrong. And you shouldn’t think that I’m
saying something controversial here. Anyone
who has mastered special relativity will agree.
The problem is not with the theory, but rather
simply how the well understood theory of relativity
is portrayed in popular culture.
So, if E equals m c squared is wrong, what's
right? It turns out that there are two more
accurate, but similar equations. Let me tell
you about both of them.
The first one looks pretty similar. It is
E equals gamma times m times c squared. I
made another video in which I talked about
relativistic mass, which introduced this gamma
term, and I made yet another video that talked
specifically about gamma. You can watch those
if you want to learn more about it.
However, I can give you the high points. Gamma
is a term that is ubiquitous in relativity.
It is defined as one over the square root
of the quantity one minus v squared divided
by c squared. V is an object’s velocity
relative to you and c is the speed of light.
If v is zero, then gamma is just one. So that’s
your first insight- E equals m c squared is
only correct if an object isn’t moving with
respect to you.
And, as an object’s velocity increases towards
c, gamma gets higher and higher and the energy
of the object increases. Note that the mass
doesn’t increase, but the energy does.
This is a particularly important to realize,
because I don’t know how many times people
tell me that since energy and mass are the
same, then a highly energetic object is also
a massive one, at least not in the way most
of these people mean. It’s just not true.
So get that misconception out of your head.
There’s another aspect of this equation
that people get at least kinda-sorta wrong.
If you keep the energy constant, but reduce
the mass of the object, then the gamma must
increase. And if you take the mass all the
way down to zero, then the velocity must become
the speed of light, since gamma must become
infinite. This is because you start getting
into mathematical tricks, where infinity times
zero equals a constant.
So that kind of works in calculus, but invoking
infinities in physics is usually a very bad
idea.
Thus this equation doesn’t really apply
for photons, which are massless particles
of light. Indeed- and this is an important
point- this equation only applies for particles
with mass and for speeds below the speed of
light.
Now there is another equation that applies
universally for both particles with and without
mass, and that equation is here. It is E squared
equals p times c, all squared, plus m c squared,
again all squared.
The meaning of the symbols is e is energy,
m is mass- what some people call the rest
mass, but is really the only value for mass-
c is the speed of light, and p is the momentum,
which is a measure of the motion of a particle.
Let’s see what happens in the case of a
particle not moving, which means setting the
momentum to zero. Voila, you get the familiar
equation E equals m c squared.
Now, let’s set the mass of a particle to
zero, which, of course, describes a photon.
We see here that the equation turns into E
equals p times c, which is the correct relationship
between the energy and momentum of a photon.
That’s pretty snazzy too.
Further, this equation demonstrates precisely
why using the E equals m c squared equation
for photons is just simply silly. If you do
that, small children will laugh at you. So
don’t do that.
By the way, there’s one additional point
that I’d like to make. Let’s go back to
the other equation relating energy to mass–
the E equals gamma m c squared one. Yeah-
that one.
Commenters in previous videos asked for physical
meaning for gamma and this is at least the
way I think of it. Remember that m c squared
is the energy of a particle at rest. In contrast,
E is its energy while it's moving. Thus, gamma
is simply the ratio of the total energy of
a particle to its energy when it isn’t moving.
Since particle physics experiments often measure
a particle’s energy, I find this particular
physical implication of gamma to be the most
useful.
So that’s it! Einstein’s most famous equation
is true, but it’s actually a special case
of equations that apply more generally. And
now, you know how to do it the right way.
Relativity is really pretty mind blowing,
no doubt about that. And it’s misused so
often. I hope that this video helped you out.
If it did, please like, subscribe and share
and tell all your friends- we have to get
our numbers up! And feel free to comment.
We want to hear what you have to say. So,
see you next time, and remember- physics is
everything.
