E2 - p2c2 = m2c4 - Meaning of symbols

[particlemania]

"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...

 

[meopemuk]

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

Eugene.

 

[particlemania]

so m is rest mass, while p is momentum with lorentz factor?

 

[meopemuk]

so m is rest mass, while p is momentum with lorentz factor?

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

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

Eugene.

 

[particlemania]

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


Thanks a lot!

 

[Tomsk]

No E has the Lorentz factor too:
E = \frac{mc^2}{\sqrt{1-v^2/c^2}}
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 - m^2 c^4 - is said to be invariant (same for all observers).

 

[particlemania]

Well the point that E has lorentz factor too was quite obvious if p had and m didnt.

 

[Fredrik]

Note that the formulas in #4 and #6 only hold for massive particles, but the one in #1 holds for massless particles too.

 

[particlemania]

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

then momentum should be

p = \frac{h}{\lambda}

isn't it?

 

[starthaus]

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

then momentum should be

p = \frac{h}{\lambda}

isn't it?

E=hf
p=E/c

 

[utesfan100]

E=hf
p=E/c

c=f\lambda completes the thought

 

[aeon.rs]

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.

 

[JesseM]

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.

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.

 

[aeon.rs]

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.

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.

 

[Phrak]

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

What's an ocean? Is this a physics term?

 

[JesseM]

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

What's an ocean? Is this a physics term?

aeon.rs referred to the Dirac sea, an idea from quantum field theory.

 

[Phrak]

aeon.rs referred to the Dirac sea, an idea from quantum field theory.

hmm.. His oceans seem to be a couple of 3 dimensional hypersurface somewhere else-when.

 

[aeon.rs]

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

What's an ocean? Is this a physics term?

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.

 

[aeon.rs]

aeon.rs referred to the Dirac sea, an idea from quantum field theory.

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).

 

[Vanadium 50]

This sounds highly speculative. Do you have a reference for this "unsplit 4-dimensional ocean of energy"?

 

[aeon.rs]

This sounds highly speculative. Do you have a reference for this "unsplit 4-dimensional ocean of energy"?

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.

 

[Phrak]

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 spatial 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…”

That was well said. Most look at Einstein's equation G_uv = GT_uv 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.

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.

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?

 

[aeon.rs]

That was well said. Most look at Einstein's equation G_uv = GT_uv 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?

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.

 

[Phrak]

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.

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?

 

[Vanadium 50]

Albert Einstein:” Relativity, The Special and The General Theory, Crown Publishers, Inc., New York, 1952.

The words "Dirac sea" appear nowhere in that.

I repeat my question.

 

[JesseM]

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.

Incidentally, I looked over my copy of this book and it seems to me that Penrose does not claim that the equation for relativistic energy alone "creates difficulties because it contains an implicit sign ambiguity". Rather his argument is that when you combine this equation with considerations from quantum mechanics, you get difficulties...without these quantum considerations, he seems to say it is perfectly acceptable to just rule out the negative-energy solutions to the equation (and he also says that even in QM this is acceptable when you are dealing with non-interacting particles). And he also says that in relativistic quantum field theory a way was ultimately found to avoid having any negative-energy particles. Look on p. 615 where he rewrites the equation as a relativistic Hamiltonian, and then says:

There is a more serious difficulty with this square-root expression, because it contains an implicit sign ambiguity. In classical physics, such things might not worry us, because the quantities under consideration are ordinary real-valued functions, and we can imagine that we could keep the positive values separate from the negative ones. However, in quantum mechanics, this is not so easy. Part of the reason for this is that quantum wavefunctions are complex, and the two square roots of a complex-number expression do not tend to separate neatly into 'positive' and 'negative' in a globally consistent way (5.4). This should be considered in relation to the fact that quantum mechanics deals with operators acting on complex functions, and things like square roots can lead to essential ambiguities that are not simply resolved by just saying 'take the positive root'.

There is another way of expressing this difficulty. In quantum mechanics, one has to consider that the various possible things that 'might' happen, in a physical situation, can all contribute to the quantum state, and therefore all these alternatives have an influence on whatever it is that does happen. When there is something like a square root involved, each of the two roots has to be considered as a 'possibility', so even an 'unphysical negative energy' has to be considered as a 'physical possibility'. As soon as there is the potential for such a negative-energy state, then there is opened up the likelihood of a spontaneous transition from positive to negative energy, which can lead to a catastrophic instability. In the case of a non-relativistic free particle, we do not have this problem of the possibility of a negative energy, because the positive-definite quantity p²/2m does not have this awkward square root. However, the relativistic expression (m² + p²)^1/2 is more problematic in that we do not normally have a clear-cut procedure for ruling out negative square roots.

It turns out that in the case of a single free particle (or a system of such non-interacting particles), this does not actually cause a real difficulty, because we can restrict attention to superpositions of positive-energy plane-wave solutions of the free Schrödinger equation, which are just those considered in 21.5, and there are no transitions to negative energy states. However, when interactions are present, this is no longer the case. Even for just a single relativistic charged particle in a fixed electromagnetic background field the wavefunction cannot, in general, maintain the condition that it be of positive frequency. In this case, we begin to perceive the tension between the principles of quantum mechanics and those of relativity.

As we shall be seeing in 24.8, the great physicist Paul Dirac found a way to resolve this particular tension. But as a first step, he put forward an ingenious and deeply insightful proposal—his now famous equation for the electron—which got rid of the troublesome square root in a marvellous and unexpected way. This subsequently led to a highly original point of view in which negative energies are eliminated, their effects being taken over by what was then a startling prediction: the existence of antiparticles.

 

[aeon.rs]

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?

The important thing is that the two continua are immiscible. The explanation to the rest will go beyond this thread.

 

[aeon.rs]

The words "Dirac sea" appear nowhere in that.

I repeat my question.

What I took from that book was about Einstein’s statement on the nature of spacetime to support my argument. Einstein said that spacetime can’t be ascribed as having a separate existence (to the contrary to Minkowski’s statement that it is an independent reality) and can’t be empty. As the special relativity theory has unified and grouped the fundamentals (space, time, energy and matter) into two: spacetime and energy, then spacetime should “contain” energy. This energy continuum which pervades the universe I call metaphorically the ocean of energy. This kind of metaphor is not new. We have Dirac Sea as a metaphor for negative energy states.

 

[aeon.rs]

Incidentally, I looked over my copy of this book and it seems to me that Penrose does not claim that the equation for relativistic energy alone "creates difficulties because it contains an implicit sign ambiguity". Rather his argument is that when you combine this equation with considerations from quantum mechanics, you get difficulties...without these quantum considerations, he seems to say it is perfectly acceptable to just rule out the negative-energy solutions to the equation (and he also says that even in QM this is acceptable when you are dealing with non-interacting particles). And he also says that in relativistic quantum field theory a way was ultimately found to avoid having any negative-energy particles. Look on p. 615 where he rewrites the equation as a relativistic Hamiltonian, and then says:

From that quotation we can draw the conlusion that the solutions of the relativistic energy equation E² = m²c⁴+p²c² are as follows: (1) in classical physics we could keep the positive values separate from the negative ones, (2) in quantum mechanics the positive and negative values do not tend to separate neatly; each of the two roots has to be considered as a possibility, so even an “unphysical negative energy” has to be considered as a physical possibility (3) in relativity we do not have a clear cut procedure for ruling out negative square root.

In all cases the negative energy is not ruled out, whether it is neatly separated from the positive energy or not. Obviously in our discussion we do not take into consideration the case of classical physics or other non-relativistic things. Even Dirac when he took antiparticle to solve the problem, he obliged to introduce the concept of “the ocean of occupied negative energy states” which is now referred to as the Dirac Sea (the same book p. 624-625).

 

[Phrak]

The important thing is that the two continua are immiscible. The explanation to the rest will go beyond this thread.

I don't think the energy-momentum equation is as mysterious as you would like to believe. Both space-time and momentum-energy obey the same rules. Energy is as time. Momentum is as displacement, and both have an invariant length. The rule is the group SO⁺(3,1). You might care to look it up. In the case of momentum-energy it is |mass|. In the case of space-time it is the displacement of one spacetime event from another event.

A temporal interval, positive to one observer, is positive to another observer. If not true, we would observe causally closed loops. The same can be said of energy.

This common temporal direction is not obeyed in QED, by inclusion of unobservable virtual particles with space like paths and negative time-like paths as well. The last has negative energy as relativity requires.

 

[JesseM]

From that quotation we can draw the conlusion that the solutions of the relativistic energy equation E² = m²c⁴+p²c² are as follows: (1) in classical physics we could keep the positive values separate from the negative ones, (2) in quantum mechanics the positive and negative values do not tend to separate neatly; each of the two roots has to be considered as a possibility, so even an “unphysical negative energy” has to be considered as a physical possibility (3) in relativity we do not have a clear cut procedure for ruling out negative square root.

(3) should say that in relativistic quantum mechanics we do not have a clear-cut procedure. In the non-quantum version of relativity we do--that's what he meant by "classical physics" here.

Even Dirac when he took antiparticle to solve the problem, he obliged to introduce the concept of “the ocean of occupied negative energy states” which is now referred to as the Dirac Sea (the same book p. 624-625).

My understanding is that the modern version of quantum field theory basically gets rid of the whole "Dirac Sea" idea, antiparticles are no longer viewed as "holes" but as particles in their own right. See this section of the wikipedia article, or this thread from physicsforums, or the last paragraph before "Free Space Solutions" in this google books result.

 

[aeon.rs]

(3) should say that in relativistic quantum mechanics we do not have a clear-cut procedure. In the non-quantum version of relativity we do--that's what he meant by "classical physics" here.

My understanding is that the modern version of quantum field theory basically gets rid of the whole "Dirac Sea" idea, antiparticles are no longer viewed as "holes" but as particles in their own right. See this section of the wikipedia article, or this thread from physicsforums, or the last paragraph before "Free Space Solutions" in this google books result.

Thank you, you are right. The original text of the quotation was written within the context of quantum mechanics. But the bottom line I would like to express is that we can not take the possibility of the existence of such negative energies very lightly. For me the presence of this symmetry, the pairing of these opposite energies and their interplay will substantially enrich the relativity theory and empower it to resolve the troubles we have both at the cosmic and quantum levels.