Can calculus find the momentum of light?

[lugita15]

The relativistic momentum formula is p = \frac{mv}{\sqrt{1-v/c^{2}}}. Since photons are massless and travel the speed of light, this formula doesn't apply to them directly, but it seems logical that we could find the momentum of a photon by calculating lim_{(m,v) \rightarrow (0^{+},c^{-})} \frac{mv}{\sqrt{1-v/c^{2}}}. Unfortunately, this double limit does not exist, but we can still calculate the limit along specific "path" in the "m-v plane". If we choose a "curve of constant energy", namely \frac{mc^{2}}{\sqrt{1-v/c^{2}}} = E, then we find the that the limit along this path is \frac{E}{c}, as expected.

But what a priori reason do we have for choosing such a path, as opposed to any other path? There doesn't seem to be any physical rationale; is there any situation where a physical object gets lighter and lighter as it approaches the speed of light, all the while keeping its total energy constant?

Any help would be greatly appreciated.

Thank You in Advance.

 

[atyy]

An equation for a massive photon is found in Eq. 3 of http://arxiv.org/abs/0809.1003. When m is taken to zero, we get a massless photon.

Footnote 4 is interesting! "In the period 1862-1867 the Dane Ludwig Lorenz independently derived the “Maxwell equations” of 1865, but received relatively little credit for this work."

 

[pervect]

We know that for all particles, E^2 - (pc)^2 = (mc^2)^2

So, we can write directly that E = sqrt(p^2 c^2 + m^2 c^4), and take the limit as m->0 rather than worry about v,( which becomes irrelevant), and find that in the limit as m->0, E= pc

 

[lugita15]

We know that for all particles, E^2 - (pc)^2 = (mc^2)^2

So, we can write directly that E = sqrt(p^2 c^2 + m^2 c^4), and take the limit as m->0 rather than worry about v,( which becomes irrelevant), and find that in the limit as m->0, E= pc

Yes, I'm already aware of this simple proof. I'm just trying to see whether other method can achieve the same result.

 

[atyy]

Yes, I'm already aware of this simple proof. I'm just trying to see whether other method can achieve the same result.

I don't think so. It's really through wave-particle duality that light and massive particles can both have mass. The 0/0 argument is a great heuristic, but I think it remains that at best.

 

[matphysik]

The relativistic momentum formula is p = \frac{mv}{\sqrt{1-v/c^{2}}}. Since photons are massless and travel the speed of light, this formula doesn't apply to them directly, but it seems logical that we could find the momentum of a photon by calculating lim_{(m,v) \rightarrow (0^{+},c^{-})} \frac{mv}{\sqrt{1-v/c^{2}}}. Unfortunately, this double limit does not exist, but we can still calculate the limit along specific "path" in the "m-v plane". If we choose a "curve of constant energy", namely \frac{mc^{2}}{\sqrt{1-v/c^{2}}} = E, then we find the that the limit along this path is \frac{E}{c}, as expected.

But what a priori reason do we have for choosing such a path, as opposed to any other path? There doesn't seem to be any physical rationale; is there any situation where a physical object gets lighter and lighter as it approaches the speed of light, all the while keeping its total energy constant?

Any help would be greatly appreciated.

Thank You in Advance.

You say that, "..this double limit does not exist..":

For any ε:0<ε<<1, let 0<m₀≤√(ε) and 0<v/c≤√(1-ε). Then,
p=mv= m₀v / √(1 - v²/c²)= m₀(v/c)c² / c√(1 - v²/c²)≤√(ε)√(1-ε)c/√(1-(1-ε))= c√(1-ε)→c, as ε→0+.
Hence, p=O(1) (ε→0+).

 

[matphysik]

We know that for all particles, E^2 - (pc)^2 = (mc^2)^2

So, we can write directly that E = sqrt(p^2 c^2 + m^2 c^4), and take the limit as m->0 rather than worry about v,( which becomes irrelevant), and find that in the limit as m->0, E= pc

`v` is very relevant, since light propagates with speed `c` (in vacuum). We must approach `c` from the left Recall that, v∈[0,c). Furthermore, in the expression E^2 - (pc)^2 = (m₀c^2)^2 (for a free relativistic particle), `p` is understood to be equal to `m₀v / √(1 - v²/c²)`. We cannot simply let m₀→0+ for we would have, E^2=0.

 

[Dickfore]

@op:

because the energy and momentum for a free particle must be constant.

 

[bcrowell]

I don't think so. It's really through wave-particle duality that light and massive particles can both have mass. The 0/0 argument is a great heuristic, but I think it remains that at best.

No, it has nothing to do with wave-particle duality. It's a purely classical fact.

There is nothing nonrigorous about taking the limit of the expression p=m\gamma v. Nobody has proposed substitution as a way of evaluating that limit, so your remarks about 0/0 are a straw-man argument.

 

[atyy]

No, it has nothing to do with wave-particle duality. It's a purely classical fact.

There is nothing nonrigorous about taking the limit of the expression p=m\gamma v. Nobody has proposed substitution as a way of evaluating that limit, so your remarks about 0/0 are a straw-man argument.

How do you get the energy of a photon? (Or a massless classical particle?)

 

[ctxyz]

You say that, "..this double limit does not exist..":

For any ε:0<ε<<1, let 0<m₀≤√(ε) and 0<v/c≤√(1-ε). Then,
p=mv= m₀v / √(1 - v²/c²)= m₀(v/c)c² / c√(1 - v²/c²)≤√(ε)√(1-ε)c/√(1-(1-ε))= c√(1-ε)→c, as ε→0+.
Hence, p=O(1) (ε→0+).

This is a very nice proof. What do you think about this alternative:

By definition

E=\gamma m_0 c^2
p=\gamma m_0 v

so

\frac{E}{p}=\frac{c^2}{v}

When v-&gt;c \frac{E}{p}-&gt;c ?

 

[Dickfore]

This is a very nice proof. What do you think about this alternative:

By definition

E=\gamma m_0 c^2
p=\gamma m_0 v

so

\frac{E}{p}=\frac{c^2}{v}

When v-&gt;c \frac{E}{p}-&gt;c ?

By dividing the expressions for the energy and momentum, you eliminated the mass and essentially expressed the speed as a function of E and p. You can go back and express the mass as:
<br /> m = \frac{E}{c \gamma}<br />
So, when taking the limit v \rightarrow c^{-}, you are actually approaching the point (v, m) = (c, 0) along a particular path.

 

[ctxyz]

By dividing the expressions for the energy and momentum, you eliminated the mass and essentially expressed the speed as a function of E and p. You can go back and express the mass as:
<br /> m = \frac{E}{c \gamma}<br />

You must mean m = \frac{E}{c^2 \gamma}, surely.

So, when taking the limit v \rightarrow c^{-}, you are actually approaching the point (v, m) = (c, 0) along a particular path.

I am not so sure, it is clear from the definition of E and p that \frac{E}{p} is not a function of m_0. As an aside, it is not a function of \gamma either.

 

[bcrowell]

How do you get the energy of a photon? (Or a massless classical particle?)

If the question is how you get the energy of a massless classical particle, I don't really understand the question. Are you asking how you get the energy if you know its momentum? E=p.

 

[atyy]

If the question is how you get the energy of a massless classical particle, I don't really understand the question. Are you asking how you get the energy if you know its momentum? E=p.

How do I calculate the energy or momentum of a massless classical particle? More generally, what is a massless classical particle?

How do I use Newton's second law on a massless classical particle? Could I, say, formulate a consistent theory, of a massless charged classical particle? Is there any consistent theory of massless classical particles interacting with massive classical particles or classical fields?

 

[atyy]

OK, I guess we can have a consistent theory of point particles having definite position and momentum traveling at the speed of light. By definition, they interact by enforcement of energy and momentum conservation. But are these really related to massive particles? ie. are they things with m=0, or things for which the notion of mass does not apply - ie. it isn't really fair to use Newton's second law on them, or to define p=γmv.

 

[matphysik]

This is a very nice proof. What do you think about this alternative:

By definition

E=\gamma m_0 c^2
p=\gamma m_0 v

so

\frac{E}{p}=\frac{c^2}{v}

When v-&gt;c \frac{E}{p}-&gt;c ?

Thank you.

You are using the expressions for a relativistic free MASSIVE particle. E=E(m₀,v), p=p(m₀,v) so that: E(m₀,v)/p(m₀,v)=c²/v and, E(m₀,v)/p(m₀,v)→E(m₀,c)/p(m₀,c)=c as v→c-. What about m₀>0?

 

[ctxyz]

Thank you.

You are using the expressions for a relativistic free MASSIVE particle. E=E(m₀,v), p=p(m₀,v) so that: E(m₀,v)/p(m₀,v)=c²/v and, E(m₀,v)/p(m₀,v)→E(m₀,c)/p(m₀,c)=c as v→c-. What about m₀>0?

True, the starting point is the expressions for massive particles, yet the limit can be taken for the case when m_0 tends to 0. Why do my posts appear so funny, on multiple lines?

 

[matphysik]

This is a very nice proof. What do you think about this alternative:

By definition

E=\gamma m_0 c^2
p=\gamma m_0 v

so

\frac{E}{p}=\frac{c^2}{v}

When v-&gt;c \frac{E}{p}-&gt;c ?

Hello.
I`ve been thinking about your result carefully, and i haven`t found anything wrong with it. The quotient of two simple algebraic expressions with the rest mass canceling out. Excellent!

 

[ctxyz]

Hello.
I`ve been thinking about your result carefully, and i haven`t found anything wrong with it. The quotient of two simple algebraic expressions with the rest mass canceling out. Excellent!

thank you, we have TWO excellent methods, I really liked yours. Do you know why my posts are appearing so funny, I am not hitting any "return" key...

 

[matphysik]

thank you, we have TWO excellent methods, I really liked yours. Do you know why my posts are appearing so funny, I am not hitting any "return" key...

They look fine to me. Probably you`re using TeX, and this site can`t handle it :)

 

[George Jones]

Why do my posts appear so funny, on multiple lines?

If you want mathematics in line with your text, use itex and /itex instead of tex and /tex. Use tex and /tex when you mathematics to appear on a separate line.

 

[ctxyz]

If you want mathematics in line with your text, use itex and /itex instead of tex and /tex. Use tex and /tex when you mathematics to appear on a separate line.

Thank you, this is what I was doing wrong. Test1: p=\gamma m v.Test2 : p=\gamma m v

You two are absolutely right, thank you!

 

[lugita15]

By dividing the expressions for the energy and momentum, you eliminated the mass and essentially expressed the speed as a function of E and p. You can go back and express the mass as:
<br /> m = \frac{E}{c \gamma}<br />
So, when taking the limit v \rightarrow c^{-}, you are actually approaching the point (v, m) = (c, 0) along a particular path.

What particular path would that in the m-v plane would that be?

 

[lugita15]

You say that, "..this double limit does not exist..":

For any ε:0<ε<<1, let 0<m₀≤√(ε) and 0<v/c≤√(1-ε). Then,
p=mv= m₀v / √(1 - v²/c²)= m₀(v/c)c² / c√(1 - v²/c²)≤√(ε)√(1-ε)c/√(1-(1-ε))= c√(1-ε)→c, as ε→0+.
Hence, p=O(1) (ε→0+).

Could you please explain or elaborate on what you wrote? Having taken Real Anaysis, I'm familiar with epsilon-delta proofs for both single and double limits, but I'm not too clear on the details of your proof.

 

[Dickfore]

How do I calculate the energy or momentum of a massless classical particle? More generally, what is a massless classical particle?

How do I use Newton's second law on a massless classical particle? Could I, say, formulate a consistent theory, of a massless charged classical particle? Is there any consistent theory of massless classical particles interacting with massive classical particles or classical fields?

See my derivation:

https://www.physicsforums.com/showpost.php?p=3333233&postcount=52"

 

[ctxyz]

Could you please explain or elaborate on what you wrote? Having taken Real Anaysis, I'm familiar with epsilon-delta proofs for both single and double limits, but I'm not too clear on the details of your proof.

He gave an extremely clever proof that the \gamma m_0 v -&gt;c when m_0-&gt;0 and v-&gt;c by using epsilon-delta in two dimensions.
By aplying the same algorithm to E=\gamma m_0 c^2 rewritten as E=\gamma m_0 v \frac{c^2}{v} you get E-&gt;c^2 so, E/p-&gt;c.

 

[Dickfore]

What particular path would that in the m-v plane would that be?

The path determined by:

<br /> m_{0} = \frac{E}{c^{2}} \, \sqrt{1 - \frac{v^{2}}{c^{2}}}<br />

 

[lugita15]

The path determined by:

<br /> m_{0} = \frac{E}{c^{2}} \, \sqrt{1 - \frac{v^{2}}{c^{2}}}<br />

. OK, so it's just a curve of constant energy again, in which case my question remains why that's the right path to take the limit over.

 

[lugita15]

@op:

because the energy and momentum for a free particle must be constant.

But we're not talking about what happens to energy and momentum as we vary time, which is where conservation laws would matter. We're varying mass and speed, and moreover we're already doing it in a way that makes the momentum change. So there's no obvious reason why energy should remain constant.

 

[Dickfore]

But we're not talking about what happens to energy and momentum as we vary time, which is where conservation laws would matter. We're varying mass and speed, and moreover we're already doing it in a way that makes the momentum change. So there's no obvious reason why energy should remain constant.

How are we varying mass and velocity exactly?

Energy and momentum depend on velocity and mass. You can think of it as a mapping from the (v, m) -> (p, E) The Jacobian of this mapping is:

<br /> \frac{\partial(p, E)}{\partial(v, m)} = \left|\begin{array}{cc}<br /> m \, \gamma^{3} &amp; \gamma \, v \\<br /> <br /> m \, v \, \gamma^{3} &amp; c^{2} \, \gamma<br /> \end{array}\right| = m \, c^{2} \, \gamma^{4} - m \, v^{2} \, \gamma^{4} = m \, c^{2} \, \gamma^{2}<br />

The Jacobian is zero if m = 0 or \gamma = 0 \Leftrightarrow v = c and these are singular paths of the mapping.

 

[ctxyz]

\gamma = 0 \Leftrightarrow v = c

This isn't right, \gamma=\infty for v=c, so the Jacobian is not zero for v=c.

 

[atyy]

OK, I guess we can have a consistent theory of point particles having definite position and momentum traveling at the speed of light. By definition, they interact by enforcement of energy and momentum conservation. But are these really related to massive particles? ie. are they things with m=0, or things for which the notion of mass does not apply - ie. it isn't really fair to use Newton's second law on them, or to define p=γmv.

OK, I am convinced by matphysik's and ctxyz's approaches to p=γmv. I notionally add a reference non-zero mass M to matphysik's limit to get the correct dimensions. I had never seen either of those before. Very nice - thanks!

I think we still cannot use Newton's second law, is that right? We have to go directly by energy-momentum conservation.

 

[lugita15]

You say that, "..this double limit does not exist..":

For any ε:0<ε<<1, let 0<m₀≤√(ε) and 0<v/c≤√(1-ε). Then,
p=mv= m₀v / √(1 - v²/c²)= m₀(v/c)c² / c√(1 - v²/c²)≤√(ε)√(1-ε)c/√(1-(1-ε))= c√(1-ε)→c, as ε→0+.
Hence, p=O(1) (ε→0+).

As I mentioned before, I don't understand the details of this proof. This is what I would consider to be an epsilon delta proof in this situation: we would have to find a number L, such that for any \epsilon &gt;0 there exists a \delta &gt;0 such that \left|\frac{mv}{\sqrt{1-v^{2}/c^{2}}} -L \right| &lt; \epsilon whenever m&gt;0, v&lt;c, and m^{2}+(v-c)^{2} &lt; \delta^{2}. So what are L and \delta in this case?

 

[PAllen]

I think all of this is unnecessarily complicated. I see nothing invalid about the idea that 'if a particle without rest mass exists' you want it to follow as much of SR as possible. Thus relationships independent of rest mass allow you state, if such a thing exists, it must behave as follows. Similar to what is done with tachyons.

To me, it is completely sufficient to say:

E^2 = E0^2 + p^2 c^2

implies that a particle with only kinetic energy, no rest energy, must have momentum:

E/c

Similarly, it is easy to derive that:

sqrt( 1 - v^2/c^2) = 1 - KE / (KE + E0)

From which you can directly state that v=c must be true for a particle with nonzero KE but zero E0.

 

[atyy]

For any ε:0<ε<<1, let 0<m₀≤√(ε) and 0<v/c≤√(1-ε). Then,
p=mv= m₀v / √(1 - v²/c²)= m₀(v/c)c² / c√(1 - v²/c²)≤√(ε)√(1-ε)c/√(1-(1-ε))= c√(1-ε)→c, as ε→0+.
Hence, p=O(1) (ε→0+).

If I take the equalities m₀=√(ε) and v/c=√(1-ε), then I seem to get m₀ as a function of v. But shouldn't m₀ be independent of v?

 

[matphysik]

If I take the equalities m₀=√(ε) and v/c=√(1-ε), then I seem to get m₀ as a function of v. But shouldn't m₀ be independent of v?

The variable `ε` has been introduced as a means of proving that the limit of the relativistic momentum exists, and is bounded as both m₀→0+ and v/c→1- together.

 

[atyy]

The variable `ε` has been introduced as a means of proving that the limit of the relativistic momentum exists, and is bounded as both m₀→0+ and v/c→1- together.

But doesn't this seem contrary to the physical meaning of rest mass?

 

[matphysik]

But doesn't this seem contrary to the physical meaning of rest mass?

NO. Given a free relativistic test particle. We are letting both the rest mass `m₀` approach zero arbitrarily close from the right, and `v/c` approach unity arbitrarily close from the left.

 

[atyy]

NO. Given a free relativistic test particle. We are letting both the rest mass `m₀` approach zero arbitrarily close from the right, and `v/c` approach unity arbitrarily close from the left.

So the use of the inequality, rather than the equality, is key?

 

[matphysik]

So the use of the inequality, rather than the equality, is key?

Yes.

 

[matphysik]

Could you please explain or elaborate on what you wrote? Having taken Real Anaysis, I'm familiar with epsilon-delta proofs for both single and double limits, but I'm not too clear on the details of your proof.

What i wrote was not an ε-δ proof.

 

[matphysik]

As I mentioned before, I don't understand the details of this proof. This is what I would consider to be an epsilon delta proof in this situation: we would have to find a number L, such that for any \epsilon &gt;0 there exists a \delta &gt;0 such that \left|\frac{mv}{\sqrt{1-v^{2}/c^{2}}} -L \right| &lt; \epsilon whenever m&gt;0, v&lt;c, and m^{2}+(v-c)^{2} &lt; \delta^{2}. So what are L and \delta in this case?

No. It`s not an ε-δ proof.

 

[matphysik]

You say that, "..this double limit does not exist..":

For any ε:0<ε<<1, let 0<m₀≤√(ε) and 0<v/c≤√(1-ε). Then,
p=mv= m₀v / √(1 - v²/c²)= m₀(v/c)c² / c√(1 - v²/c²)≤√(ε)√(1-ε)c/√(1-(1-ε))= c√(1-ε)→c, as ε→0+.
Hence, p=O(1) (ε→0+).

So once the existence of the limit has been proved we may write:

p= m₀v / √(1 - v²/c²)= m₀(v/c)c² / c√(1 - v²/c²) → E`/c=: p`, as both v→c- and m₀→0+. Where the prime indicates of light/photon(s). Therefore, for light the equation E²=m₀²c⁴ + p²c² reduces to E²=E`².

 

[atyy]

For any ε:0<ε<<1, let 0<m₀≤√(ε) and 0<v/c≤√(1-ε). Then,
p=mv= m₀v / √(1 - v²/c²)= m₀(v/c)c² / c√(1 - v²/c²)≤√(ε)√(1-ε)c/√(1-(1-ε))= c√(1-ε)→c, as ε→0+.
Hence, p=O(1) (ε→0+).

Is the limit a Lorentz invariant? How do we Lorentz transform it?

 

[ctxyz]

Is the limit a Lorentz invariant? How do we Lorentz transform it?

He proved that the limit is c, so it is invariant. See post 27 for a detailed explanation.

 

[atyy]

He proved that the limit is c, so it is invariant. See post 27 for a detailed explanation.

How can E and p be Lorentz invariant? Do E and p of a massless classical particle transform differently from the non-Lorentz invariant E and p of a photon?

 

[ctxyz]

How can E and p be Lorentz invariant?

Not E and p, but their limits when v-&gt;c and m_0-&gt;0. It is the limits that are Lorentz invariant, as explained in post 27.

 

[ctxyz]

I think all of this is unnecessarily complicated. I see nothing invalid about the idea that 'if a particle without rest mass exists' you want it to follow as much of SR as possible. Thus relationships independent of rest mass allow you state, if such a thing exists, it must behave as follows. Similar to what is done with tachyons.

To me, it is completely sufficient to say:

E^2 = E0^2 + p^2 c^2

I don't think that you can do that (pervect tried to use the same argument eralier in the thread).

E^2-(pc)^2=(m_0c^2)^2 is a consequence of E=\gamma m_0 c^2 and p=\gamma m_0 v that results when one does the calculation of the norm of the energy-momentum four-vector. Indeed:

(\gamma m_0 c^2)^2-(\gamma m_0 vc)^2=(m_0c^2)^2

The evaluation of the LHS requires that \gamma is defined properly (i.e. is not infinite) .

 

[atyy]

Not E and p, but their limits when v-&gt;c and m_0-&gt;0. It is the limits that are Lorentz invariant, as explained in post 27.

Why is it ok for the limits to be Lorentz invariant? I thought the limits were supposed to be the energy of a massless particle?