Beginning with a metric with small perturbations \begin{eqnarray} g_{00} &=& 1-2\frac{U}{c^2} + \mathcal{O}(c^{-4}) \\ g_{0i} &=& \mathcal{O}(c^{-3}) \\ g_{ij} &=& -\delta_{ij}\left(1+2\frac{U}{c^2} + \mathcal{O}(c^{-4})\right) \end{eqnarray}

I'm trying to show that the components of the 4-velocity vector $u^0$ and $u^i$ can be written as \begin{eqnarray} u^0 &=& 1 + \frac{1}{2}\frac{v^2}{c^2} + \frac{U}{c^2} + \mathcal{O}(c^{-4}) \\ u^i &=& \frac{v^i}{c}\left(1 + \frac{1}{2}\frac{v^2}{c^2} + \frac{U}{c^2}\right) + \mathcal{O}(c^{-5}) \end{eqnarray}

I want to explicitly keep powers of $c$ in the calculation to help learn about different orders of smallness in PPN. The method I am attempting to use is to write $u^\mu u_\mu = u^0 u_0 + u^i u_i = +1$. This may be rearranged as \begin{eqnarray} u^0 u^0 g_{00} &=& 1 - u^i u^j g_{ij} - 2u^0 u^i g_{0i} \\ \left(u^0\right)^2 \left[ 1-2\frac{U}{c^2}\right] &=& 1 + \left(\frac{dx^i}{ds}\right)^2\left[ 1+2\frac{U}{c^2} \right] \end{eqnarray} I dropped the $g_{0i}$ term because it is $\mathcal{O}(c^{-3})$. I found that if I write $\frac{dx^i}{ds} = \frac{1}{c}\frac{dx^i}{dt}\frac{dt}{ds} = \frac{v}{c}$, solve the equation above for $u^0 = \sqrt\frac{...}{...}$, and then taylor expand about the small terms $\left(\frac{v}{c}\right)^2$ and $\frac{U}{c^2}$, I get the correct form of $u^0$.

My questions are:

  1. Is this the best approach to solve for $u^0$ and $u^i$, or is there a better/more highly recommended method to use?

  2. I don't understand if or why it is correct to write $\left(\frac{dx^i}{ds}\right)^2 = \left(\frac{v}{c}\right)^2$. The term $\frac{dx^i}{ds}$ should have no units (like $\frac{v}{c}$), but when I use the chain rule to expand $\frac{dx^i}{ds}$, shouldn't there be a factor of c next to dt in the numerator above as well? Could it have something to do with thinking of ds as proper time, and then writing $dt\approx ds$ for slow moving objects?

  3. Also, I read that metric components of $g_{0i}$ will have odd powers of $c^{-1}$, while metric components of $g_{00}$ and $g_{ij}$ will have even powers of $c^{-1}$, but have never seen a justification for that claim.

  4. Ultimately my goal is to understand the PPN formalism. Are there any textbooks or papers where the author works through expanding a perturbed metric while keeping factors of $c^{-1}$ explicitly in the calculation to show different orders of the small terms (e.g. does not set $c=1$)? So far, the report I linked below is the only paper I have ever seen that does not work in units where $c=1$.

My question is about how to arrive at eq. (15) in this link (the link will automatically download a pdf of the paper).

If you don't want to click on a link that automatically downloads a paper, the title of the paper is "The post-Newtonian formalism" by Rene Michelsen.

  • $\begingroup$ Is it helpful to express the four velocity as $u^\beta = g^{\beta \alpha} u_{\alpha}$? $\endgroup$ – Rumplestillskin Jul 1 '18 at 22:43
  • $\begingroup$ I have been using that relation to make all 4-velocities contravariant (e.g. they all have upper indices). I haven't yet tried making 4-velocities covariant (have lower indices), which I may attempt next, but it's not obvious to me whether that will help or not. Somewhat related, this method hasn't yet worked for $u^i$, since Taylor expanding a term $\propto\sqrt\frac{v}{c}$ gives infinity at lowest (non-zero) order. $\endgroup$ – Bob Jul 2 '18 at 3:52

Express four-velocities $u^\alpha$ and coordinate velocities $v^i$ according to

$$ u^\alpha = \frac{dx^\alpha}{d\tau}, \quad v^i = \frac{dx^i}{dt}, $$

where proper time and coordinate time is given by $\tau$ and $t$ respectively. Hereafter, Latin ($i,j = 1,2,3$) and Greek ($\alpha,\beta = 0,1,2,3$) indices account for spatial and spacetime variables respectively.

I will answer your question $(1)$ in terms of arbitrary metric tensor components, where I will outline the procedure you need to follow in order to get the desired results.

A timelike particle obeys the normalisation condition given by

$$ g_{\alpha \beta} u^\alpha u^\beta = -c^2, \tag{1} \label{eqn1}$$

where $c$ is the speed of light. Eq. (\ref{eqn1}) is given explicitly by

$$ \left( u^0 \right)^2 \left( g_{00} + 2 g_{0i} \frac{u^i}{u^0} + g_{ij} \frac{u^i}{u^0} \frac{u^j}{u^0} \right) = -c^2. \tag{2} \label{eqn2} $$

Now, we know $x^0 = ct$. Hence $d x^0 = cdt. $ Using this we can express $u^i/u^0$ as the following

$$ \frac{u^i}{u^0} = \frac{dx^i}{d\tau} \left(\frac{d x^0}{d\tau} \right)^{-1} = \frac{dx^i}{dt} \frac{dt}{d\tau} \left( \frac{dx^0}{dt} \frac{dt}{d\tau} \right)^{-1} = \frac{v^i}{c}. \tag{3} \label{eqn3}$$

(Note: I hope that addresses your confusion regarding question $2$. ) Returning to Eq. (\ref{eqn2}) we have

$$ u^0 = c \left[ - g_{00} - 2g_{0i} \left( \frac{v^i}{c} \right) - g_{ij} \left(\frac{v^i}{c}\right) \left( \frac{v^j}{c} \right) \right]^{-1/2}.$$

Substituting in the appropriate metric tensor components and expanding the square root to PN order will give the desired $u^0$. Consulting Eq. (\ref{eqn3}) gives the appropriate expression for the spatial part of the four-velocity namely

$$ u^i = \frac{v^i}{c} u^0. $$

So to answer your question $(1)$, I would consult the normalisation condition to easily obtain the components of $u^\alpha$. By following the method outlined above you will obtain the expressions you have quoted for $u^0$ and $u^i$.

I should note that there is a big difference between the post-Newtonian and parametrised post-Newtonian formalisms which I would advise you familiarise yourself with. I assume you mean you want to understand the PN formalism in your question $(4)$. Works by Chandrasekhar on the PN formalism generally retain the factor of $1/c$ for convenience. However, you should know that this is simply a matter of correctly determining the appropriate dimensions and reinstating. Being able to do this is more beneficial in the longrun.

  • $\begingroup$ Thanks for the reply. In regard to question (4), I mean I want to work with the PPN formalism. I have a equation of motion that reduces to the trace-free EFE at lowest order and has a scalar field, but want to know if it is ruled out at first order in PPN parameters. $\endgroup$ – Bob Jul 13 '18 at 14:22
  • $\begingroup$ Should your eq. (1) read $g_{\mu\nu}u^\mu u^\nu = +1$ since (i) the spacetime metric signature is (+,-,-,-) and (ii) we are normalising the 4-velocities? $\endgroup$ – Bob Jul 13 '18 at 20:15
  • $\begingroup$ @Bob if you are concerned with the PPN formalism, shouldn’t there be parameters $\alpha,\beta$ in your metric components? Again I stress the PN and PPN formalisms are different animals. $\endgroup$ – Rumplestillskin Jul 13 '18 at 23:56
  • $\begingroup$ @Bob the sign will change according to your signature. However, I thought you wanted to retain $c$ explicitly? $\endgroup$ – Rumplestillskin Jul 13 '18 at 23:58

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