Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Why doesn't mass add?

  1. Oct 20, 2006 #1
    Can someone provide a simple and convincing reason why mass dosen''t always add when two system combine but momentum and energy always does?

    Energies and momentum add because of conservation laws but mass is conserved as well isn't it?

    Or is it the case that we are dealing with invariant mass hence is special? This sounds a bit mystical.
  2. jcsd
  3. Oct 20, 2006 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Noether theorem.
  4. Oct 20, 2006 #3

    Meir Achuz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    E and p are components of a 4-vector p^\mu=(E;p).
    m=\sqrt{E^2-p^2}. Why would you think it should add?
  5. Oct 20, 2006 #4
    Please give an example where mass doesn't add, while energy does. If an electron and a positron annihilate into radiation, no rest-mass is left over but energy is radated away. Now, this radiated energy also represents mass (by m=E/c^2) and so the total mass (but not rest-mass) is always conserved.
  6. Oct 20, 2006 #5

    George Jones

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Note that pivoxa15 used the term "invariant mass" in the original post. Also, although Pete disagrees, over the last several decades, the unmodified term "mass" has almost universally come to mean rest mass.
  7. Oct 20, 2006 #6


    User Avatar

    Staff: Mentor

    Invariant mass is the "magnitude" of the energy-momentum four-vector. When you add four-vectors in general, the magnitudes don't usually add, just like the magnitudes of ordinary three-vectors don't add when they're in different directions.
  8. Oct 20, 2006 #7
    This is a nice logical argument for why invariant mass dosen't add.

    I also realised that the conservation law applies for relativistic mass, energy, momentum only. The invariant mass is not relativistic hence not conserved.
  9. Oct 20, 2006 #8
    Once again we need to be precise as to what is meant by mass. If you mean inertial mass (aka relativistic mass) then yes. Inertial mass adds. In fact it is inertial mass that is the time component of 4-momentum. To see that inertial mass is conserved I've worked out a proof. Please see


    Inertial mass, yes. Proper mass, no.

    That is what it sounds like to me too.

    Um ... In order for the OP to use that law they must know what the term "mass" means in order to apply the law. The only result that will yield a conserved quantity from Noether's theorem is "relativistic mass." Is that what you meant?? :biggrin:

    The assertion that 4-vectors add is a correct one and is fundamental to tensor analysis. In curved spacetime the 4-vectors to be added must be located at the same event. In flat spacetime they need not be located at the same event. For details please see


    Best wishes

    Last edited: Oct 20, 2006
  10. Oct 20, 2006 #9


    User Avatar

    Staff: Mentor

    Eh? :confused:

    Can you expand on this? I have the feeling that some other word than "relativistic" might be better here. Invariant quantities are an important part of SR.
  11. Oct 20, 2006 #10
    Noether's Theorem is beuatiful, it gives a hint to the underlying symmertry behind Space and Time before SR!
  12. Oct 20, 2006 #11
    energy and momentum r conserved because they r energetic forms, whereas mass is a form of condensed form of it.mass is not pure energy.it is not quantized
  13. Oct 20, 2006 #12
    I forgot one word, mass. And probably should include a special case as well.

    Here is the complete sentence. 'The invariant mass is not relativistic mass (for nonzero kinetic energy) hence not conserved.'

    Unfortunately, Noether's theorem looks a bit too technical for me.
  14. Oct 21, 2006 #13
    I don't know which are your mathematical bases, otherwise I think I could explain you.
    In simple words it comes out from the Stationary Action Principle that:

    1. Space homogeneity --> linear momentum is conserved
    2. Space isotropy --> angular momentum is conserved
    3. Time homogeneity --> energy is conserved

    Space homogeneity means that the results of experiments inside a laboratory don't change if you perform those experiments with the laboratory which has been put in another place of space.

    Space isotropy means that the results of experiments inside a laboratory don't change if you perform those experiments with the laboratory which has been rotated in space.

    Time homogeneity means that the results of experiments inside a laboratory don't change if you perform those experiments in another moment of time.

    Unfortunately all this is not true in general in curved space-time (if space-time is curved, there cannot be homogeneity or isotropy!).
    So, in general, in a curved space-time, energy, linear momentum and angular momentum are not conserved. They are not even simple to define.
  15. Oct 21, 2006 #14
    If mass would always add, than we would be living in a boring universe. In that case, radiation/ light could never be emitted in a nuclear or chemical reaction and the universe would be completely dark...:frown:
  16. Oct 21, 2006 #15
    Could you give reasons for this claim?
  17. Oct 21, 2006 #16
    I know as much maths as a 2nd year Uni student.

    Given that we live in a curved space-time, Noether's principle doesn't hold in reality? Also what is conserved in reality, that is in a curved space-time?
    Reality as in reality as we know it today offcourse.
  18. Oct 21, 2006 #17
    So, probably you are able to read this document about Noether's Theorem:


    where you only need to know (if you don't already) that the action S is the integral between two instants of time t_1 and t_2 of the Lagrangian L = T-V where T is kinetic energy and V the potential energy, and the variation deltaS is made with fixed values of q_i(t) at the two instants t_1 and t_2.

    About your first questions: far from massive objects (e.g. stars, black holes), the curvature is negligible, so our space-time can be considered flat;

    About your second question: that's a good question! We should ask more expert people about it.
    Think that not even the velocity vector v has a unique global meaning (it depends on how it is transported from one point to another).
    Last edited: Oct 21, 2006
  19. Oct 21, 2006 #18
    The only way to make the mass of the combined system equal to the sum of the separate masses is to ensure that nothing (no energy) escapes during the process of combining. So, it means that no energy/light can be radiated away. I'm not sure how to explain this better.
  20. Oct 21, 2006 #19
    You only need to be clear about what the system is that you're determining the conservation of mass. If the system is closed then the total inertial mass is a constant in time. There was confusion on this issue in a newsgroup once. Someone was making bogus claims and saying that inertial mass (aka relativistic mass) was not conserved. I posted the correction here


    See the part entitled Bogus Claim No. 3 - The inertial mass (aka relativistic mass) of a closed system is not conserved.

    I address a case similar that you mention. I.e. a pion decaying into two photons.

    Best wishes

  21. Oct 23, 2006 #20
    mass addition

    should we mention if the considered systems interact or not?
    sine ira et studio
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook