We kicked off the countdown to Newton’s birthday with his second law of motion, which is almost but not quite everything you need to understand and predict the motion of objects. The missing piece is today’s equation:


This is the full and correct definition of momentum, good for any speed all the way up to the speed of light. Newton’s second law tells us how the momentum changes in response to a force, but in order to use that to predict the future, you need to know what momentum is, and that’s where this equation comes in.

(Wouldn’t it make more sense to do this first, and the second law afterwards? Yes, but it’s more thematically appropriate to start with one of Newton’s laws. And, anyway, holidays don’t need to make sense.)

So, why is this important? Mostly, the reason I just gave you– that you need a definition of momentum before you can use the second law to predict the future– but also because this equation brings together the two greatest titans of physics: Isaac Newton and Albert Einstein.

Newton, of course, is the founder of physics as a mathematical science, and was the first person to recognize that momentum was an important quantity. But, through no fault of his own, Newton did not have the complete story: he thought that the momentum of an object was just its mass multiplied by its velocity.

The full definition, shown above, is a little more complicated, and includes that square root factor involving the speed of light. We have Einstein to thank for this version– not because he invented this out of whole cloth, because the relevant factor had previously been identified by Hendrik Lorentz, and is often referred to as the “Lorentz factor.” Einstein was the one responsible for making a really convincing argument that this had to be the correct expression, though, and thus getting it accepted by the wider world of physics.

It’s important to stress that Newton wasn’t wrong, here. Newton’s definition of momentum as mass times velocity is perfectly good for speeds that are slow compared to the speed of light. The Lorentz factor increases very slowly at low velocity– you need to be moving at something like 14% the speed of light (a bit more than 42,000,000 m/s, several thousand times the speed of the fastest man-made object) before the correct momentum differs from Newton’s definition by more than 1%. In Newton’s day, there was absolutely no way to work with objects at such high speeds, so there’s no reason why he ever would’ve seen his error.

The advance of physics and technology over the couple of centuries between Newton and Einstein, and particularly the development of Maxwell’s theory of electromagnetism, forced physicists to think more carefully about the motion of objects. This process led to Einstein’s theory of special relativity, and the third equation of our advent calendar.

This expression for momentum has been confirmed countless times, both in experiments that look for it directly– we sometimes do a lab in our junior-level lab course where students look at beta decay and measure a clear difference between the Newton and Einstein versions of momentum– and in experiments that involve it more indirectly. The Large Hadron Collider accelerates beams of protons up to 0.999999991 times the speed of light, and if they didn’t use the relativistic expression above, they wouldn’t be able to correctly predict the motion of their proton beams to collide them together. So we know that this is the right version.

So, as we continue counting down the days to Newton’s birthday, remember that while Newton kicked things off, Einstein’s relativity brought it to completion. This is the equation where the two most clearly come together.

And come back tomorrow to see the next equation of the season.


  1. #1 Michael Woods
    December 3, 2011

    I absolutely love this series of posts, Chad! Thumbs up for a great idea!

  2. #2 Paul Orwin
    December 3, 2011

    So do neutrinos that travel FTL have imaginary momentum?

  3. #3 Otto Krog
    December 3, 2011

    What if the speed of light isn’t constant?

    What if the speed of light varies through time and space?


    Antimatter is the mind and consciousness of all living entities.

    You are your own universe.


    Interested? Then read my crankpot philosophical multiverse theory.

    Google crestroyer theory, and find it instantly.


  4. #4 Nikolas
    December 4, 2011

    Hey isn’t the Lorentz factor meant to have the magnitude of velocity squared, not the vector squared?

  5. #5 RM
    December 4, 2011

    Nikolas@4 – Hey isn’t the Lorentz factor meant to have the magnitude of velocity squared, not the vector squared?

    Depending on your definition of “vector squared”, the two are equivalent.

    Squaring usually means multiplying something by itself, and although there are two ways to multiply one vector by another vector, the “default” is usually taken as the dot product. (Besides which, the cross product of a vector with itself is somewhat pointless, as it is always a zero length vector.)

    The dot product of 3-vector with itself is x*x + y*y + z*z. The square of the magnitude is (sqrt(x^2 + y^2 + z^2))^2, which (if we assume we have no imaginary values for our vector components) works out to the same thing.

  6. #6 CCPhysicist
    December 9, 2011

    I think this expression is much clearer if the square root factor is put under v rather than out in front like the afterthought it was in 1905.

    Einstein only changed the v dependence of momentum, and did so in a way that the leading (low velocity limit) term in the expression was unchanged. The correction should be at the end, not the beginning.

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