Q: As a consequence of relativity, objects becomes more massive when they’re moving fast. What is it about matter that causes that to happen?

Posted on April 17, 2011 by The Physicist

Physicist: The laws of the universe are relativistic.  That is, Einstein was right and Newton, although accurately and intuitively describing the world around us, was wrong.  When you try to translate the cleaner Einstein laws into a Newtonian form you find that the mass, M, has a \gamma sitting next to it.  Normally it has almost no effect, but at very high speeds it becomes important.  The Newtonian interpretation (where you don’t allow for messed up time and space, but you do allow for alchemy) of that \gamma is that the mass is “exaggerated” by a factor of \gamma.

\gamma is an excellent way to talk about how much relativity is having an impact on distance, time, and mass.  For those of you who’d like to do some back-of-the-envelope relativity \gamma=\frac{1}{\sqrt{1-\beta^2}}, where \beta is the fraction of the speed of light that the object in question is traveling (\beta = \frac{v}{c}).

The source of the \gamma may be obscure, but you’ll find that when you try to write the laws of the universe (which prefer relativity) in a form we’re used to it always seems to show up.

Matter does two big things: it has inertia, and it makes gravity.  Every attempt to measure matter always comes down to one or both of these properties.

Inertial mass means that matter is difficult to push around.  This is summed up by Newton’s second law of motion: F=MA.

(Historical side note: Newton himself originally wanted to call this the “second law of suck it Leibnitz“, but in committee he was talked down to the more demure “Newton’s second law”.  At one time Newton was discovering as many as 9 new Leibnitz-insulting laws a week.)

Here’s a thumbnail sketch of an example of \gamma showing up:

It turns out that, in relativity, the true law is: f=Ma.  (Notice the difference?)
Einstein realized that in order to rewrite Newton’s laws for relativity he needed to be careful about whose time he was talking about.  As it happens, the correct choice is the on-board time of the moving mass (which is what a clock on the object would read).  So here I’m using lower-case to talk about force and acceleration in terms of on-board time, and upper case to talk about them in terms of “world time”, which is the usual time we’re used to talking about.

The ratio of the “on-board time” (\tau) to the “world time” (T) is \gamma.
As a (non-obvious) result, f = \gamma F.  A (hand-waving) way to think about this is “if I push (F) that space ship for a while (T), they’ll think I was pushing (f) longer, because their clock (\tau) is ticking slower”.
At the same time, that acceleration (a) is in terms of on-board time, but if you want A (the acceleration to everyone else) you have to write it in terms of world time.  Acceleration is distance per second per second, so:
a=\frac{d}{\tau^2}=\frac{T^2}{\tau^2}\frac{d}{T^2}=\gamma^2\frac{d}{T^2}=\gamma^2 A
and so
\begin{array}{ll}f=Ma\\\Rightarrow\gamma F=Ma\\\Rightarrow\gamma F=M\gamma^2 A\\\Rightarrow F=\gamma MA\end{array}

So if you look at the forces involved in pushing an object (that’s moving so fast that \gamma is noticeable), you find that it’s harder to push than it “should” be (by a factor of \gamma).  The easiest way to interpret that is as an increase in mass.

You could just as easily say that “force mysteriously gets weaker” by looking at this last equation as: \frac{1}{\gamma}F=MA.  But that wouldn’t be consistent with the interpretation of…

Gravitational mass, which creates gravity and pulls other matter toward it, is the second property of matter.  But matter isn’t really what creates gravity, it’s energy.

For example, if you had a crazy big laser beam (all energy, no matter), that beam would have some gravity.  Similarly, if you have a moving rock (one with kinetic energy) it will have more gravity than a perfectly stationary one.  And the kinetic energy of an object with mass M is given by K=\gamma Mc^2, which is very, very close to Mc^2+\frac{1}{2}Mv^2 (the “rest mass energy” plus the “Newtonian energy“).  Point is: there’s another \gamma M.

These last statements are wrist-breakingly hand-wavy, but they do hold up.  The Frankenstein created by combining gravity and relativity is called “general relativity”, and it’s nasty enough that cute tricks like “multiply by \gamma” don’t come close to telling the whole story, or even enough to be useful.

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12 Responses to Q: As a consequence of relativity, objects becomes more massive when they’re moving fast. What is it about matter that causes that to happen?

  1. Graham Stevenson says:
    April 22, 2011 at 12:17 am

    forgive my ignorance, but is sarcasm implied along with:
    “(Historical side note: Newton himself originally wanted to call this the “second law of suck it Leibnitz“, but in committee he was talked down to the more demure “Newton’s second law”. At one time Newton was discovering as many as 9 new Leibnitz-insulting laws a week.)”
    ?

    is there ANY truth to that other than their dispute? because i cannot find any corroborating sources.

  2. Error: Unable to create directory uploads/2024/12. Is its parent directory writable by the server? The Physicist says:
    April 22, 2011 at 1:11 am

    Aside from Newton really not liking Leibnitz, that was just a hilarious joke.
    Sorry about the confusion.

  3. Johnny says:
    April 22, 2011 at 7:08 pm

    Quote: “Similarly, if you have a moving rock (one with kinetic energy) it will have more gravity than a perfectly stationary one.”

    How would one determine which rock is moving? =P

  4. Error: Unable to create directory uploads/2024/12. Is its parent directory writable by the server? The Physicist says:
    April 22, 2011 at 7:47 pm

    Excellent point!
    The only kind of movement that’s “real” is “relative movement”. Hence the name: “Relativity”.

  5. tonyf says:
    April 24, 2011 at 9:02 pm

    So-called relativistic mass gives you all kinds of problems. The theory becomes much simpler by always treating mass as an invariant (and e.g. E=sqrt(mc^2)^2+(pc)^2) where E is energy of a particle of mass m and momentum p and c is speed of light in vacuum).

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  7. Osimon says:
    December 5, 2012 at 11:45 am

    Hi, you said
    “if I push (F) that space ship for a while (T), they’ll think I was pushing (f) longer, because their clock () is ticking slower”.
    But since the clock of the people on the ship is ticking a slower, shouldn’t they think that you pushed for a shorter amount of time, not a longer one?

  8. goodwill says:
    December 25, 2012 at 11:04 am

    @Osimon: no, i guess. i somehow find its obvious or am i wrong?

  9. Sheldon says:
    September 24, 2013 at 12:41 pm

    So if you fired two rocks toward each other from a ultra-rock-accelerator at relativistic speeds. Sensors mounted on each rock would not detect any change in mass on their rock but would detect the other rock as more massive?

    When talking about speeding up objects to some faction close to c wouldn’t all objects eventually before they get there have an issue with their swartzchild radius or does this require some other factor like density? If so how would that work since from their own frame they wouldn’t be so massive as to have an event horizon but from an observer in a different frame they might?

  10. David Martin says:
    January 20, 2015 at 8:25 am

    Your answer to the question boils down to ‘That’s just the way it is’.

    This is in statements like ‘The laws of the universe are relativistic’ and ‘…when you try to write the laws of the universe (which prefer relativity) in a form we’re used to [relativistic mass] always seems to show up.’ Many people believe the same as you do, that that’s just the way it is, in other words, that there is no further layer of explanation.

    But it doesn’t answer the question, and it may be wrong. It is very bad science not to admit the gaps in our knowledge – it’s an attitude that holds back progress. The correct answer is ‘We don’t know’.

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