# What is Wald talking about here in his book "General Relativity"?

I'm reading General Relativity by Robert Wald and his notation is confusing me.

On page $$61$$ of the book, Wald claims that by setting $$c=1$$ we can derive $$E=mc^2$$ from the following information.

(1) $$u^au_a=-1$$ where $$u^a$$ is a tangent vector to a timelike curve.

(2) $$p^a=mu^a$$ where $$p^a$$ is an energy-momentum $$4$$-vector of the rest mass $$m$$.

(3) The energy of a particle as measured by an observer, present at the site of the particle, whose $$4$$-velocity is $$v^a$$ is written as $$E=-p_av^a$$.

(4) The particle is at rest with respect to the observer, so $$u^a=v^a$$ is assumed.

As far as I am aware, it is impossible to algebraically derive $$E=mc^2$$ from this as written unless I am completely misunderstanding Wald's notation. However, if we set $$E=-p^av_a$$ then we have $$E=m$$, but this is extremely silly since we can also have $$E=mc^{100}$$ or something if $$c=1$$. Is there something I am missing here? Is there a typo, or is Wald saying something different?

A similar issue arises with more notation introduced on the next page. He writes $$\partial^a$$, which was never used prior to page 62 (unless Wald means $$\partial^a=\partial_a^{-1}$$...?). What is he talking about here?

Up until this point, the book has been a pretty nice read and I've understood most of the content since the first three chapters are primarily math review and understanding Wald's new notational conventions. I just need help with this notational discrepancy in Chapter 4. Thanks for the help.

• If you understand the difference between $u^a$ and $u_a$, the difference between $\partial^a$ and $\partial_a$ is similar. Commented Oct 23, 2023 at 3:46
• $E=mc^{100}$ wouldn’t be dimensionally correct when using SI units. Commented Oct 23, 2023 at 3:48
• $E=m$ is not “extremely silly”. Is this the first time that you have encountered the use of natural units such as $c=1$? Natural units are generally first encountered when learning SR, not GR. Likewise with upper vs. lower spacetime indices. Commented Oct 23, 2023 at 3:55
• Commented Oct 23, 2023 at 6:13
• I don't really see how setting $c=1$ simplifies the argument here. Maybe clearer to leave $c$ as $c$ and then get an explicit factor of $c^2$ at the end. Commented Oct 23, 2023 at 9:03

First, note that

$$p^a = mu^a \rightarrow \eta_{ab}p^a = m\eta_{ab}u^a \rightarrow p_b = mu_b$$

Then, relabel the dummy indice to get $$p_a = mu_a$$. We can use this to make use of your (4) relations. We start with energy (3)

$$E=-p_av^a$$

Then, insert (2)

$$E=-mu_av^a$$ Then, insert (4)

$$E=-mu_au^a$$

Finally, use (1)

$$E = m.$$

As Ghoster said, this is not ridiculous at all when using natural units. Wald does use the notation that $$\partial^a = \partial_a^{-1}$$. This is common notation in GR.

Finally, to emphasize Ghoster's correct point, $$E = mc^{100}$$ wouldn’t be dimensionally correct when using SI units. It is, indeed, a power of 2 that is the correct input of $$c$$ despite $$c$$ being $$1$$ in natural units.

• "it being $1$" referring of course to $c$, not its exponent, which in natural units is taken as $0$.
– J.G.
Commented Oct 23, 2023 at 6:47
• yes, let me rephrase @J.G. Commented Oct 23, 2023 at 6:54