Is $E^2=(mc^2)^2+(pc)^2$ or is $E=mc^2$ the correct one?

Question

I have been having trouble distinguishing these two equations and figuring out which one is correct. I have watched a video that says that $E^2=(mc^2)^2+(pc)^2$ is correct, but I do not know why. It says that $E=mc^2$ is the equation for objects that are not moving and that $E^2=(mc^2)^2+(pc)^2$ is for objects that are moving.

Answer 1

Let me clarify some confusions in the notation that other answers have alluded to but not clearly mentioned.

Historically, physicists liked to talk about two different definitions of mass

• The first is the rest mass of a particle $m_0$. This is the mass of the particle when it is at rest. For example, the rest mass of the electron is $(m_0)_{electron} = 9.1 \times 10^{-31}~Kg$. This is an absolute constant that is independent of the speed of the particle.

• The second is the relativistic mass $m$. This is the apparent mass of the particle when it is moving with speed $v$. It is related to the rest mass via the relation $$ m = \gamma m_0 = \frac{m_0}{\sqrt{1-v^2/c^2}} $$ Note that the relativistic mass is NOT a constant. It depends on $v$.

In this historical notation, Einstein's famous formula that is completely correct in all frames is $$ E = m c^2 $$ However, it turns out via a series of algebraic manipulations that this equation also implies $$ E^2 = ( m_0 c^2)^2 + (pc)^2 $$ Let us prove this. $p$ is the momentum of the particle defined by $p = m v = \gamma m_0 v$. Thus $$ (m_0 c^2)^2 + (pc)^2 = m_0^2 c^4 + \gamma^2 m_0^2 v^2 c^2 = m_0^2 c^4 \left( 1 + \frac{\gamma^2 v^2}{c^2} \right) $$ Now, we have the property $$ 1 + \frac{\gamma^2 v^2}{c^2} = 1 + \frac{\frac{v^2}{c^2}}{\left( 1 - \frac{v^2}{c^2} \right) } = \frac{1}{ \left( 1 - \frac{v^2}{c^2} \right) } = \gamma^2 $$ Thus $$ (m_0 c^2)^2 + (pc)^2 = m_0^2 c^4 \gamma^2 = (\gamma m_0)^2 c^4 = m^2 c^4 = (mc^2)^2 = E^2 $$ Thus, in summary, in the historical notation, we have two completely equivalent formulae $$ \boxed{ E^2 = (m c^2 )^2 = (m_0 c^2)^2 + (pc)^2} $$

In modern day notation, physicists have decided to drop discussion of the relativistic mass $m$ since it is not an absolute constant and depends on the speed of the particle. Nowadays, we only talk about the rest mass, $m_0$. However, in a confusing notational change physicists today decided to use $m$ for the rest mass (which in today's notation is not confusing at all, since we don't talk about relativistic mass, but it is often confusing to students who try to compare Einstein's original papers with books written today).

Following modern day notation then, we only have ONE equation, namely $$ \boxed{ E^2 = (m c^2)^2 + (pc)^2 } $$ where in the above equation $m$ is now the rest mass.

Answer 2

Both are correct, within the domains for which they are correct.

More seriously, the general relation

$$ E^2 = m^2c^4 + p^2c^2$$

holds for all objects, whether they have mass or not, whether they are moving or not.

The special case $E = mc^2$ is for $p = 0$, i.e. objects which do not move, as you said.

The special case $E = pc$ is for objects which have no mass, i.e., photons.

Answer 3

I agree with answer of ACuriousMind, but I think it might also help to think about it like this....

$E^2=m_0^2c^4+p^2c^2 =m^2c^4$

where $m_0$ is the rest mass and $m$ is the relativistic mass (or inertial mass), defined as $m = \gamma m_0 = m_0 / \sqrt{1 - v^2/c^2}$.

The relatavistic mass increases as the momentum of the mass increases. At rest the two are equal to each other. As the speed of an object and its momentum increase the mass of the object increases.

so I think about it as

$E^2=m_0^2c^4+p^2c^2$

and

$E=mc^2$

Answer 4

The equation $$E^2=(mc^2)^2+(pc)^2$$ represents the correct energy-momentum relationship. It gives the total energy $E$ for an object of invariant mass (rest mass) $m$ that is observed to move with momentum $p$. This equation is applicable regardless whether the object is observed to be in motion ($p \ne 0$), or is observed to be at rest ($p = 0$). In the latter case, the energy-momentum equation simplifies into the well-known $E=mc^2$.

As an aside (some might call it nitpicking), when discussing the generics of the energy-momentum equation, it is good form to write the equation such that both sides of the equation are independent of the frame of observation chosen: $$ E^2 - (pc)^2 = (mc^2)^2$$ Same math, different physics. (Note that this relativistically invariant relationship is simply the expression for the square norm of the energy-momentum four-vector.)

Answer 5

All of the other answers are great, and I highly recommend reading them. However, I think something is missing if you don't attempt to get an intuitive understanding with geometry:

This triangle shows that the equation $E^2=(mc^2)^2+(pc)^2$ can be represented via a sort of reverse Pythagorean theorem. The special case of $E=mc^2$ can be found by setting $p$ to zero, and appears as if the $pc$ side of the triangle is zero in size, changing the shape to a line with $E$ on top and $mc^2$ on the bottom. Likewise, for light, we can show the special case of $E=pc$ by setting the rest mass $m$ to zero, which transforms the triangle into a vertical line with $E$ on the left and $pc$ on the right.

Answer 6

Much of this is hard-won insights after a couple of decades of independent study. Look into it, and I think you might find there's something pretty useful here.

Newton's Second Law can be written:

$$\frac{\text{Impulse}}{\text{mass}} = \text{Change in Velocity}.$$

But in relativistic mechanics we have

• Impulse/mass (in ls/s) = $\sinh(w)$,
• Change in Velocity (in ls/s) = $\tanh(w)$
• Time Dilation/Length Contraction Factor (in s/s or ls/ls) = $\cosh(w)$

where $w$ is the rapidity. When rapidity is small, $\sinh(w)= \tanh(w) = w$

You can see part of this in $$E^2 = p^2 c^2 + (mc^2)^2$$

So this equation is essentially the hyperbolic trig equivalent of Pythagoras theorem.

$$(mc^2)^2 \cosh^2(w) = (mc^2)^2\sinh^2(w) + (mc^2)^2 $$

or

$$(mc^2)^2 \gamma^2 = (mc^2)^2 \left(\frac{\text {impulse}}{\text{mass}} \text{(in ls/s)}\right)^2 + (mc^2)^2 $$

You can also get the kinetic energy out of this equation by subtracting 1 from the time-dilation factor, and multiplying the result by $mc^2$. The equation is not terribly useful for that purpose at low velocities, though, since the time-dilation factor, $\gamma$, will be something like 1.00000000004 and it won't fit into your calculator.

Once you confirm that all this is really hyperbolic trig, if you can find a calculator with easy access to hyperbolic trig functions, you'll find it much easier to put things into rapidities.