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E2 - p2c2 = m2c4 - Meaning of symbols

  1. Apr 13, 2010 #1
    "E2 - p2c2 = m2c4" - Meaning of symbols

    What do the symbols mean in the equation-

    E2 - p2c2 = m2c4


    I know this is so basic, but I am really confused about what all are rest parameters here, and what all involve Lorentz factor...
     
  2. jcsd
  3. Apr 13, 2010 #2
    Re: "E2 - p2c2 = m2c4" - Meaning of symbols

    c = speed of light
    m = rest mass of the object
    E = object's total energy
    p = object's momentum

    Eugene.
     
  4. Apr 13, 2010 #3
    Re: "E2 - p2c2 = m2c4" - Meaning of symbols

    so m is rest mass, while p is momentum with lorentz factor?
     
  5. Apr 13, 2010 #4
    Re: "E2 - p2c2 = m2c4" - Meaning of symbols

    That's right. Relativistic momentum is related to the velocity via Lorentz factor

    [tex] p = \frac{mv}{\sqrt{1- v^2/c^2}} [/tex]

    Eugene.
     
  6. Apr 13, 2010 #5
    Re: "E2 - p2c2 = m2c4" - Meaning of symbols

    That means among all variables there (except E) only p has a lorentz factor.


    Thanks a lot!
     
  7. Apr 13, 2010 #6
    Re: "E2 - p2c2 = m2c4" - Meaning of symbols

    No E has the Lorentz factor too:
    [tex]E = \frac{mc^2}{\sqrt{1-v^2/c^2}}[/tex]
    So you can see that even though E and p depend on the velocity, when you take the difference of their squares, the result is independent of the velocity. The result - [itex]m^2 c^4[/itex] - is said to be invariant (same for all observers).
     
  8. Apr 13, 2010 #7
    Re: "E2 - p2c2 = m2c4" - Meaning of symbols

    Well the point that E has lorentz factor too was quite obvious if p had and m didnt.
     
  9. Apr 13, 2010 #8

    Fredrik

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    Re: "E2 - p2c2 = m2c4" - Meaning of symbols

    Note that the formulas in #4 and #6 only hold for massive particles, but the one in #1 holds for massless particles too.
     
  10. Apr 13, 2010 #9
    Re: "E2 - p2c2 = m2c4" - Meaning of symbols

    @Fredrik : So for massless particles, if my third term becomes 0

    then momentum should be

    [tex]p = \frac{h}{\lambda}[/tex]

    isn't it?
     
  11. Apr 13, 2010 #10
    Re: "E2 - p2c2 = m2c4" - Meaning of symbols

    E=hf
    p=E/c
     
  12. Apr 13, 2010 #11
    Re: "E2 - p2c2 = m2c4" - Meaning of symbols

    c=f[tex]\lambda[/tex] completes the thought
     
  13. Apr 17, 2010 #12

    aeon.rs

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    Re: "E2 - p2c2 = m2c4" - Meaning of symbols

    As the meaning of symbols has been exhaustively elucidated, I wish to respond to your remark about the importance of this relativist energy equation. You are absolutely right, the equation E2 = m2c4 + p2c4 is one of the most basics in physics. It demonstrates that energy as a whole consists of positive and negative energies.

    Everything in nature exists in pairing, the energy is no exception. We may say that this pairing of energy is the prerequisite of the creation (and annihilation). As I stated in my reply to ZirkMan’s “What is spacetime made of?” energy is the only independent reality in nature (the spacetime is merely its structural quality) from which everything else is derived. How?

    Allow me to explain a little bit further. Energy as a whole tends to break its symmetry. Eventually it splits into its two opposite elements: the positive and negative energies. As such, the spacetime becomes polarized. When it happens, a hypersurface or more precisely hyper-interface is raising between the two, just like the interface of oil-water system. The 3-space or, in a more technical term, the 3-brane is thus created. I think the study of the brane should be done in this direction.
     
  14. Apr 17, 2010 #13

    JesseM

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    Re: "E2 - p2c2 = m2c4" - Meaning of symbols

    How do you figure? Negative energy would only be present with exotic matter, this may be possible in some real-world situations like the Casimir effect but it's an unusual phenomenon, not really implied by that equation.
     
  15. Apr 18, 2010 #14

    aeon.rs

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    Re: "E2 - p2c2 = m2c4" - Meaning of symbols

    Roger Penrose in his book [“The Road to Reality”, P.614, Vintage Book, London, 2005] elaborates the difficulties with this relativistic energy equation quiet thoroughly. Let me quote and summarize his view. The square root of the expression (m2c4+p2c2)1/2 indeed creates difficulties because it contains an implicit sign ambiguity.

    In quantum mechanics the two square roots are complex number and therefore do not tend to separate neatly into positive and negative in a consistent way. In quantum mechanics, however, each of two roots has to be considered as a possibility, so even an unphysical negative energy has to be considered as a physical possibility.

    The relativistic expression (m2c4+p2c2)1/2 is, however, more problematic in that we do not normally have a clear-cut procedure for ruling out negative square root.

    Paul Dirac found a way to resolve this problem. When he convinced that the negative frequency solutions could not be mathematically eliminated, he put forward an ingenious proposal which got rid of the negative energies, their effect being taken over by introducing the idea of antiparticles and what is called “Dirac sea” of negative energy.

    Now, Dirac sea of negative energy represents only the half of the reality. The give the whole picture, as I presented previously, there should be both oceans of positive and negative energies where the 3-interface, our 3-space, is arising in between.
     
  16. Apr 18, 2010 #15
    Re: "E2 - p2c2 = m2c4" - Meaning of symbols

    "...there should be both oceans of positive and negative energies..."

    What's an ocean? Is this a physics term?
     
  17. Apr 18, 2010 #16

    JesseM

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    Re: "E2 - p2c2 = m2c4" - Meaning of symbols

    aeon.rs referred to the Dirac sea, an idea from quantum field theory.
     
  18. Apr 18, 2010 #17
    Re: "E2 - p2c2 = m2c4" - Meaning of symbols

    hmm.. His oceans seem to be a couple of 3 dimensional hypersurface somewhere else-when.
     
  19. Apr 18, 2010 #18

    aeon.rs

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    Re: "E2 - p2c2 = m2c4" - Meaning of symbols

    Don’t forget that we are talking about four-dimensional ocean of energies. As we have discussed in the foregoing, the spacetime is just the structural quality of energy. The number of dimensions of the spacetime is the reflection of the potency or the degree of freedom of the underlying energy.

    Relativist energy is four-dimensional. What we know about energy in our [3D] daily life is only superficial. We perceive energy as a mere abstraction that has no existence apart from its relationship to other variables.

    The interface between these two 4-dimensional oceans of energy is naturally 3-dimensional: our 3-dimensional space.
     
  20. Apr 18, 2010 #19

    aeon.rs

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    Re: "E2 - p2c2 = m2c4" - Meaning of symbols

    Dirac Sea is most likely 3-dimensional(?) and could contain particle/antiparticle. What I mean by the oceans of energies here are 4-dimensional oceans of “pure” energies. These oceans of positive and negative energies are originated from the split of the original 4-dimensional spacetime (unsplit 4-dimensional ocean of energy).
     
  21. Apr 18, 2010 #20

    Vanadium 50

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    Re: "E2 - p2c2 = m2c4" - Meaning of symbols

    This sounds highly speculative. Do you have a reference for this "unsplit 4-dimensional ocean of energy"?
     
  22. Apr 18, 2010 #21

    aeon.rs

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    Re: "E2 - p2c2 = m2c4" - Meaning of symbols

    Albert Einstein:” Relativity, The Special and The General Theory, Crown Publishers, Inc., New York, 1952. His Note to the Fifteenth Edition, page vi: “… I wished to show that space-time is not necessarily something to which one can ascribe a separate existence, independently of the actual objects of physical reality. Physical objects are not in space, but these objects are spatially extended. In this way the concept empty space loses its meaning".

    And based on that Einstein’s statement I interpreted as in my reply (#19) to: “What is Spacetime made of.” originally posted by ZirkMan:
    … “The [special relativity] theory concerns about the fundamental: the space, time, energy, and matter. The special relativity has unified those fundamentals into two distinct entities: the [four-dimensional] spacetime as the unification of space and time and the energy from which matter is derived.
    The spacetime and energy are not two separate entities as we think. The spacetime is not like a sort of container and energy something that fills the container. On the contrary, they are inextricable just like water substance and its spherical form in a drop of water…”

    This energy as a whole together with its related 4-dimensional spacetime I refer to as the 4-dimensional “ocean” of energy, or the [original] 4-dimensional unsplit ocean of energy, analogous to Dirac Sea of energy.
     
  23. Apr 18, 2010 #22
    Re: "E2 - p2c2 = m2c4" - Meaning of symbols

    That was well said. Most look at Einstein's equation Guv = GTuv and say "energy curves spacetime", rather than "energy is spatial-temporal curvature."

    Of course it's not really energy per se. "Energy" is shorthand for the genre of terms that stand on the left hand side of the equation.

    In fact, there may be no reason, other than tradition to group them under general banner of energy. I could change units, dividing by the action, and now say "frequency is spatial curvature." But I digress.
    In your scenario, why should we be confined to this 3 dimensional region? Or what make our three dimensional part of this 4-dimensional spacetime any different than the rest?
     
  24. Apr 19, 2010 #23

    aeon.rs

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    Re: "E2 - p2c2 = m2c4" - Meaning of symbols

    I have posted already in #12, 14, 18,19, 21 the subject related to your inquiry. Let me summarize. The 4-dimensional [ocean of] energy potentially consists of positive and negative energies (E2 = m2c4 + p2c4). This energy, in its originality, is highly unstable and eventually splits into two 4-dimensional opposite [oceans of] energies.

    This phenomenon of splitting is analogous to the separation of oil and water which creates an interface in between. Under a 4-dimensional [relativity] framework, such an interface should naturally have 3 dimensions. [We can extend this to an n-dimensional framework where the dimensions of the hyper-interface would be (n-1)]. This 3-interface is nothing but our 3-dimensional material world. Beyond this there is nothing but [4-dimensional oceans of] energies.
     
  25. Apr 19, 2010 #24
    Re: "E2 - p2c2 = m2c4" - Meaning of symbols

    What makes the stuff on one side different than the stuff other such that there is an interface? How to you reconcile the observed asymmetry of energy; that all material has positive energy rather than a distribution, both positive and negative?
     
  26. Apr 19, 2010 #25

    Vanadium 50

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    Re: "E2 - p2c2 = m2c4" - Meaning of symbols

    The words "Dirac sea" appear nowhere in that.

    I repeat my question.
     
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