# Proof that $v=\frac{dE}{dp}$ at relativistic speeds?

This is a formula I have seen stated in two different places in a relativistic context, but I do not know the proof and seem unable to derive it on my own. $$E$$ is the energy of a free particle, $$v$$ is its speed and $$p$$ its momentum.

The fundamental energy-momentum relation in special relativity is $$E^2=p^2+m^2$$ (let’s ignore factors of $$c$$, since they will factor out in the end — that is, let’s work in natural units).

Taking the derivative with respect to $$p$$ on both sides gives

$$2E\frac{\partial E}{\partial p}=2p,$$

since $$m$$ is a constant. Thus,

$$\frac{\partial E}{\partial p}=\frac{p}{E}.$$

Now, noting that $$E=m\gamma$$ and $$p=mv\gamma$$, where $$\gamma=1/\sqrt{1-v^2}$$, the result is proven.

In the last step, we could have instead used the fact that the four-momentum is given by

$$p^{\mu}=m\frac{\mathrm{d}x^{\mu}}{\mathrm{d}\tau},$$

So that $$E=m\,\mathrm{d}t/\mathrm{d}\tau$$ and $$p=m|\mathrm{d}\vec{x}/\mathrm{d}\tau|$$ (this only works in the massive case of course, but proving your identity is trivial in the massless case, since $$E=p$$). Thus,

$$\frac{\partial E}{\partial p}=\frac{p}{E}=\frac{|\mathrm{d}\vec{x}/\mathrm{d}\tau|}{\mathrm{d}t/\mathrm{d}\tau}=\left|\frac{\mathrm{d}\vec{x}}{\mathrm{d}t}\right|,$$

which is the definition of the speed. Note that in the second step we used the chain rule.

I hope this helps!

• Doing these manipulations using $\gamma$ and normalized velocity vectors is needlessly complicated and, as you note, forces you to handle the massless case separately. It's both simpler and more general if you just prove $v=p/E$ from the fact that the energy-momentum vector has to be parallel to the velocity vector (which can be unnormalized). (If it weren't parallel, then we would have a violation of rotational symmetry.) – user4552 Oct 14 '19 at 23:05

We have $$E=\gamma mc^2$$ (relativistic total energy) and $$p=\gamma mv$$, where $$\gamma=1/\sqrt{1-v^2/c^2}$$. Therefore, we can write:

$$E=\frac{pc^2}{v} \space,$$

differentiating both sides WRT $$p$$ implies:

$$\frac{dE}{dp}=c^2\left(\frac{v-p\frac{dv}{dp}}{v^2}\right)=\frac{c^2}{v^2}\left(v-\frac{p}{\frac{dp}{dv}}\right) \space. \tag{*}$$

Moreover, we have:

$$\frac{dp}{dv}=\frac{d(\gamma mv)}{dv}=m\frac{d}{dv}\left(\frac{v}{\sqrt{1-\frac{v^2}{c^2}}}\right)=\frac{m}{\left(1-\frac{v^2}{c^2}\right)^\frac{3}{2}}={\gamma^3}m \space. \tag{**}$$

Substituting both $$p=\gamma mv$$ and its derivative [Eq. $$(**)$$] into Eq. $$(*)$$ infers:

$$\frac{dE}{dp}=\frac{c^2}{v^2}\left(v-\frac{\gamma mv}{\gamma^3 m}\right)=\frac{c^2}{v}\left(1-\frac{1}{\gamma^2}\right)=\frac{c^2}{v}\left(1-1+\frac{v^2}{c^2}\right)=v \space.$$

• This is much more complicated than necessary, and only works in the case where $m>0$. – user4552 Oct 14 '19 at 23:09

I use the signature convention $$(+,-)$$.

Any spacetime-displacement vector can be written as $$\tilde T=a\hat t+ \tilde s,$$ where $$\hat t$$ is unit and future-timelike and $$\tilde s\cdot\hat t=0$$ (that is, $$\tilde s$$ is purely spacelike to the observer with 4-velocity $$\hat t$$). Assume $$a \gt 0$$. Since $$a$$ is the "temporal displacement" and $$\tilde s$$ is the "spatial displacement", then we can define the "spatial velocity according to the $$\hat t$$-observer" as $$\tilde v\equiv \displaystyle\frac{\tilde s}{a}.$$ So, $$\tilde T=a(\ \hat t+\tilde v\ )$$. For non-spacelike vectors ($$\tilde T\cdot\tilde T \geq 0$$), we have $$1=\left\|\hat t\right\|^2=\hat t\cdot \hat t\ \ \geq\ -\tilde v \cdot \tilde v=\left\|\vphantom{\hat t}\tilde v\right\|^2=v^2$$.

Similarly, any [future-pointing non-spacelike] energy-momentum vector can similarly be decomposed into $$\tilde P=E\hat t+ \tilde p,$$ where $$E$$ is the relativistic-energy and $$\tilde p$$ is the relativistic momentum. When $$\tilde T$$ is chosen to be parallel to $$\tilde P$$ (so the particle with this constant momentum $$\tilde P$$ will traverse this displacement $$\tilde T$$), we have $$\tilde v =\displaystyle\frac{\tilde p}{E},$$ as suggested by @Ben Crowell .

Now, as @Bob Knighton does, \begin{align} m^2 \equiv \tilde P\cdot \tilde P &=E^2(\ \hat t\cdot \hat t\ )+(\ \tilde p\cdot \tilde p\ )\\ &=E^2 - p^2 \end{align} and, by differentiation, \begin{align} 0 &= 2 E \frac{dE}{dp} - 2p, \end{align} so that \begin{align} \frac{dE}{dp} = \frac{p}{E} =v, \quad \mbox{where the last equality uses the relation above}. \end{align} This should be in line with @Ben Crowell , who hints that a good proof should be able to cover both the timelike ($$m^2>0$$) and lightlike ($$m^2=0$$) cases.

• Sweet coordinate-independent presentation. At the point where you say $\tilde{v}=\tilde{p}/E$, it seems to me that you ought to invoke the fact that the energy-momentum is parallel to the displacement. Otherwise it's not obvious why this $\tilde{v}$ is the same as the earlier $\tilde{v}$. – user4552 Oct 15 '19 at 14:41
• You start off with a displacement that can be in any direction regardless of light-cone, then make some remarks about norms in the non-spacelike case, then make an argument about the $\tilde{P}$ vector that's restricted to future non-spacelike $\tilde{P}$. It's not clear to me where you ever used the facts about the norms (maybe you didn't?), or whether the restriction to future non-spacelike $\tilde{P}$ was ever necessary. Is there something in the argument that fails for tachyons? – user4552 Oct 15 '19 at 14:49
• I agree that it should be specified that $\tilde P$ is parallel to $\tilde T$. Thanks... I'll make the change. I restricted the displacement's temporal component to avoid the $a=0$ case (infinite velocity in $\hat t$'s frame) and to keep it pointing forward in time (in $\hat t$'s frame). I also wanted $\tilde P$ to be future-pointing causal. – robphy Oct 15 '19 at 16:56