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Idea of increased mass at relativistic speeds

  1. Aug 13, 2005 #1
    I would like to clarify that this idea that mass increases at relativistic speeds is an explanatory tool only (usually to explain why you cannot exceed the speed of light). It comes from the fact that in some equations, the factor gamma = 1/sqrt(1-v^2/c^2) multiplies the mass. However, I think it is a flawed explanatory tool and I never use it in explanations of special relativity. Not only is it completely unnecessary but I think it creates more confusion than understanding.

    The idea of mass increasing at relativistic velocities leads to the unavoidable conclusion that mass is relative just like velocity, which I find absurd. Reading some of the other post I see that it also leads to the conclusion that mass is different in different directions, which is even more absurd. This idea of mass increasing has forced us, for the sake of clarity, to rename the sensible concept of mass as "rest mass" which is not relative and does not change. In short the absurdites and confusion promoted by this idea of increasing mass warrant giving the whole idea a decent burial.

    The total energy of a mass m at a relative velocity v and thus lorentz contraction factor gamma is given by E = gamma m c^2.
    Well what about kinetic energy? We can extract the classical kinetic energy from a binomial expansion of gamma.
    gamma = 1 + .5(v/c)^2 + .375(v/c)^4 + ...
    when you put this into the above equation you get
    E = m c^2 + .5 m v^2 + .375 m v^4/c^2 + ...
    The first term is the famous mass energy, and the second term is the classical kinetic energy.
    To handle the relativistic correction, we typically write
    E = m c^2 + (gamma-1) m c^2
    and we say that the first term is the mass energy (or rest energy) and the second term here is the relativistic kinetic energy, KE = (gamma-1) m c^2

    In this case we are obviously not thinking that the mass has increased by a factor of gamma at all, because the energy associated with mass has not changed. To say that the mass has increased by a factor of gamma would mean that all of the energy is a part of the mass and there is no energy of motion, and I don't think this helps in understanding special relativity at all.

    I do not even like the idea as an explanation of why you cannot exceed the speed of light because it is too stuck in the thinking of motion as absolute. What I mean is that it puts too much emphasis on one particular relative velocity as if that were special. It is true that increasing the relative velocity with respect to something requires and increasing amount of energy for the same increase in that relative velocity, but I think this misses the point.

    I guess the only way to make what I am saying clear is to look at an example. Suppose you accelerate a big ship to 1/2 the speed of light relative to the earth. If you have a medium ship inside the big ship then you can accelerate that medium ship to 1/2 the speed of light relative to the big ship. Then if you have a small ship inside the medium ship you can accelerate the small ship 1/2 the speed of light relative to the medium ship.

    The energy requirements of all these acceleration depend on the rest masses of these ships (lets call them mbig, mmed, and msmall) in exactly the same way, using the KE shown above KE = (gamma-1) m c^2.
    In each of the three cases gamma = 1/sqrt(1-.25) = 1.1547
    First acceleration: energy required was KE = .1547 mbig c^2
    Second acceleration: KE = .1547 mmed c^2
    Third acceleration: KE = .1547 msmall c^2

    If you want to talk about the resulting velocity with respect to the earth then you need the velocity addition formula v3 = (v1+v2)/(1+ v1 v2/c^2).
    So the velocity of the medium ship with respect to the earth is (c/2+c/2)/(1+.25) = 0.8 c, and the velocity of the small ship with respect to the earth is (.8 c + .5 c)/(1+ .8x.5) = .92857 c. If you look carefully at the velocity addition formula you will see that if both v1 and v2 are less than c then v3 will be less than c, but if either v1 or v2 is equall to c then v3 will also be c.

    The point is there is no increase of mass in this explanation nor should there be. The idea of mass increase promotes a misconception that something changes as you accelerate making an increase of speed more difficult. Absolutely nothing changes. The only limit is on relative velocity at which you see objects receding behind you. It does not even limit how fast you can travel to a destination.

    The speed of light is unreachable because it is like an infinite speed in the sense that if you chase after a light beam your accelertion never reduces the relative velocity between you and the light beam you are chasing, the light continues to speed away from you at 3x10^8 m/s. You cannot catch the light no matter how fast you go, just as if the light were traveling infinitely fast. In fact, we know from the relativity of simultaneity that any travel faster than light would be equivalent to arriving at your destination before you left, leading to the same paradoxes as in time travel. Also if you think of the infinite speed as the limiting case where you get to your destination in no time at all, the speed of light is exactly such a limiting case.

    P.S. Check out my relativistic flight simulator at www.relspace.astahost.com
    Last edited: Aug 13, 2005
  2. jcsd
  3. Aug 14, 2005 #2
    Downloaded the file but couldn't get it to run!

    Is the demo free?
  4. Aug 15, 2005 #3
    Yes, and it is the same program, only a few features are disabled until you register it. What kind of operating system do you have? Did you find the relspace12-2.exe file and try to run it? What happened when you did?
  5. Aug 15, 2005 #4
    It it is never completely unnecessary. That phrase has always been suspect to me. I distrust it because the writer always refuses to back himself up. All you have actually done is to regroup terms and given them different names.

    Its not as easy as you make it out to be. Once mass is defined it follows whether it is or is not a function of speed. Since we humans have never directly percieved such effects we assumed there was none (way back when). However [itex]m = \gamma m_0[/itex] isn't even always correct. Its for that reason you can't claim that rel-mass and energy are the same thing.

    Last edited: Aug 16, 2005
  6. Aug 15, 2005 #5


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    This is the clearest explanation that I have ever read about why relativistic mass can be a source of confusion.

    The substantive point I gather from this is that mass (unlike, say, size) is an invariant quantity and cannot be thought as directional (unlike size, say again). What is more, mass is not energy (and neither is energy mass, or is it?).

    As a non-physicist, the easiest way to rationalize this to myself is to think of the concept of mass as belonging to the pre-Einstein physics. Sorry if this sounds too philosophical or metaphysical, but I say it as I see it.
  7. Aug 15, 2005 #6
    I don't get the point you're trying to make. What's wrong with relativistic mass? It works and it isn't at all confusing for me.

    So are you a teacher? Just curious.

    Which velocity?

    Something does change, it's called inertia.
  8. Aug 15, 2005 #7


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    E.g. velocity relative to an observer, which can be one of many obervers traveling at different velocities, is the OP's point, I believe.
  9. Aug 15, 2005 #8
    And how are we putting emphasis on one observed?
  10. Aug 15, 2005 #9


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    In fact, what most people know as the "relativistic mass", works only for a limited number of purposes. Even as inertia, that is, as a proportionality constant between force and acceleration, it works only if the force applied to an object is perpendicular to the direction of the object's motion. If the force is exerted parallel to the direction of the object's motion, the proportionalty is different. Some people have used the terms "transverse mass" and "longitudinal mass" in reference to this.
  11. Aug 15, 2005 #10
    Relativistic mass is unnecessary, cumbersome and undesirable upon close examination.

    How, physically, does increasing the velocity of an object increase its mass relative to you? Where does the mass confrom? To answer a question like that, we would need a better understanding of mass (i.e. speeding up increases the rate at which higgs bosons are involved some how). The point is, it is better to leave mass (which is ill-understood in some sense) as the mysterious concept that it already is ("rest mass", you would say), when we would be much better of just speaking of energy.

    When we increase the velocity of an object relative to us, its (relativistic kinetic) energy increases. Energy, in special relativity, is unbounded (can be any real number greater then zero). Of course, velocities cannot exceed c. But for any increase in energy, the velocity increases slightly (there is no question of the form of the energy or where it goes, it is in the motion).

    In addition to being kinematically desirable over relativistic mass, relativistic energy is an absolutely necessary part of special relativity (energy-momentum four vector).
  12. Aug 15, 2005 #11
    Well, when something increases it's speed it gains energy, and energy and mass are interchangeable. So therefore, as you gain speed you also gain mass. That's the simplification of my understanding.

    Mass is the property of matter that resists changes in motion and is proportional to the gravitational field created by the object. The faster something is moving the more it "resists change in motion" but I'm unsure if any experiments that show objects' gravitational field is effected by it's kenetic energy but I have heard theories about it. Maybe someone else on the forums know more about it.
  13. Aug 16, 2005 #12
    But I am saying that mass is not so defined, but that this idea of increasing mass was concocted as an explanatory tool, which I claim is a failure at explaining anything. It is not a matter of perception but of interpretation.

    Crosson response is good for me. Thanks Crosson.

    Yes. I have a masters in physics and I teach physics for ITT Tech.

    Yes, thanks for the help. In my example of the three ships I showed how you can change your reference point rather than keeping the same observer every time. This is natural because it is natural and easier to treat acceleration as relative to the frame you are in at the time.

    But inertia is a nearly dead and usless concept. It is replaced by the superior concepts of kinetic energy and momentum. And when you say it is changed, for who has it changed? The ship or the observer? In relativity you cannot say which is really moving. Inertia is relative. Besides, it is a fundamental postulate of relativity that nothing important has changed when you go to a different inertial frame.

    But it is not interchangeable and defining it to be so is not a useful thing to do. That is why we had to invent the term rest mass to recover the original meaning of mass from the confusion created by this flawed explanation. It is rest mass that is used in all the physics I have studied and this relativistic mass has no use at all.

    Your first specification of mass as the property which resists changes in motion is undefined because you do not specify an inertial frame. It would be natural to specify the rest frame but then this would define the rest mass. You cannot define mass by interaction with gravity because the photon is massless and yet it is both affected by gravity and has a gravitational field of its own. I think you are caught in a web of oversimplifications. Of course kinetic energy is affected by gravity. Once you choose an inertial frame to work in, gravity interacts with the stress energy momentum tensor which includes kinetic energy and momentum of the object that would be calculated for that inertial frame.
  14. Aug 16, 2005 #13
    Quite wrong, laddy. There are at least two ways to define "mass" and each has its merits. Defined as m = p/v = mass, can be found in many places. Sayang one is the correct one and the other incorrect is nonsense. And I don't really care if paticle physicist is bothered by it or not.

    Please provide proof that was increasing mass was concocted as an explanatory tool,

    Then prove that
    Physicists Einstein included, never intentionaly fool someone or "concoct" something for a teaching tool.

    I'll await proof for said claims.

    Last edited: Aug 16, 2005
  15. Aug 16, 2005 #14
    This is your basic assumption? You don't like it so it must be wrong huh?
    People who make that claim don't know what they are talking about.

    So. Did you read my entire paper? My website? Give it a whirl.

  16. Aug 16, 2005 #15
    In pre-Einstein physics rods don't contract, volumes don't change and there is non-similaneity is gone. Therefore do you wish to get rid of these thinhgs too?

  17. Aug 16, 2005 #16


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    This is a question for the OP, I take it.
  18. Aug 16, 2005 #17
    You know which is which because atleast one of the objects has been accelerated in its past, in other words it has "felt" a force. This information tells you which object's mass has changed.

    [tex]E = m c^2[/tex]

    It doesn't matter what inertial frame you're in, the mass of an object appears the same to all observers.

    Only when at rest. Photons carry energy and therefore mass except when they are at rest.
  19. Aug 16, 2005 #18


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    Invariant mass is the same for all observers. Relativistic mass depends on your frame (add: it's larger when you are moving then when you are standing still). Which mass were you talking about again?
    Last edited: Aug 16, 2005
  20. Aug 16, 2005 #19
    Its amazing how this subject continues forever.

    Folks. This has been done for prosperity and left at

    I've refered to it many times and guess what? Nobody has read the whole thing I bet. It covers all of these little things in decent detail.

  21. Aug 16, 2005 #20


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    I don't recall seeing this before, though I see from the date at the top that you wrote it last year.

    So far I don't have any major arguments with the first dozen or so pages. (I haven't read the whole thing yet :-)).

    There are a few things I do have to say:

    Mass as it's usually definied in GR (which is usually taught using the 4-vector approach, not the 3+1 approach as you mention) is probably the most closely related to what you call "active gravitational mass". GR actually has multiple concepts of mass, all of which are closely related - the Bondi mass, the ADM mass, and what could be called the Noether mass (though I've never seen the last name used). The detailed discussion of these concepts of mass in GR really goes beyond the scope of the thread - it's probably useful for people to know that they exist, though.

    So far I personally still lean towards teaching students the 4-vector view of relativity and not the 3+1 view, if they are only going to learn one view.
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