Retarded time Lienard Wiechert potential

Question

In a potential which needs to be evaluated at the retarded time, is this the time which represents the actual time the "physics" occurred? So $t_{\text{ret}}=t-\frac{r}{c}$, not just because it may be that you are receiving a signal at light speed but because "causality" spreads out at the maximum speed, $c$, is this correct?

The Lienard-Wiechert 4-potential for some point charge ($q$): $A^\mu=\frac{q u^\mu}{4\pi \epsilon_0 u^\nu r^\nu}$ where $r^\nu$ represents the 4-vector for the distance from the observer. In the rest frame of the charge $A^i$ for $i=1,2,3$ is clearly zero but from what has been said about the retarded time we have that $A^0=\frac{q}{4\pi\epsilon_0c(t-r/c)}$.

Obviously I would like to get $A^0=-\frac{q}{4\pi\epsilon_0 r}$, so where is the misunderstanding of retarded time and instantaneous time? Unless we would like the time since the signal was emitted which is $r/c$? Or if $t$ itself is already $t'-r/c$ and we need to return to the instantaneous time $t$, when the signal was emitted.

Answer 1

Actually since charge is at rest $u_{\nu}r^{\nu} = u_0 r^0 = ct'$ where $t'$ is retarded time, $t'=r/c$, where $r$ is the (constant) distance to the charge.

Answer 2

It's an implicit relation $$t_- = t - \frac{R}{c}, \quad R = |𝐑|, \quad 𝐑 = 𝐫 - 𝒓(t_-),$$ where $t ↦ 𝒓(t)$ is the trajectory of the point source, and $𝒗 = \dot{𝒓}(t_-)$ is its velocity evaluated at time $t_-$.

The three relations between $t_-$, $R$ and $𝐑$ determine them as implicit functions of $\left(𝐫,t\right)$, and a functional of the trajectory $t ↦ 𝒓(t)$. It is well-defined if $|𝒗| < c$.

If you differentiate each relation: $$dt_- = dt - \frac{dR}{c}, \quad dR = \frac{𝐑·d𝐑}{R}, \quad d𝐑 = d𝐫 - 𝒗(t_-)dt_-,$$ you can solve for $dt_-$, $dR$ and $d𝐑$ in closed form to get: $$ dt_- = \frac{Rc dt - 𝐑·d𝐫}{Rc - 𝐑·𝒗}, \quad dR = \frac{𝐑c·(d𝐫 - 𝒗dt)}{Rc - 𝐑·𝒗}, \quad d𝐑 = \frac{Rc(d𝐫 - 𝒗dt) + 𝐑×(𝒗×d𝐫)}{Rc - 𝐑·𝒗}. $$ Thus, for instance, plucking out the differential coefficients of $t_-$, we get: $$ \frac{∂t_-}{∂t} = \frac{Rc}{Rc - 𝐑·𝒗}, \quad ∇t_- = -\frac{𝐑}{Rc - 𝐑·𝒗}. $$

The potential 1-form $A ≡ A_x dx + A_y dy + A_z dz - φ dt = 𝐀·d𝐫 - φ dt$ is given at the space-time point $\left(𝐫,t\right)$ by: $$A\left(𝐫,t\right) = \frac{e}{4πε_0c} \frac{𝒗·d𝐫 - c² dt}{Rc - 𝐑·𝒗}.$$

If the charge is stationary, with $𝒓(t) = 𝟎$, then $𝐑 = 𝐫$, $R = r ≡ |𝐫|$, $t_- = t - r/c$ and $𝒗 = 𝟎$, with the potential 1-form thus reducing to the following: $$A = -\frac{e dt}{4πε_0r},$$ or, component-wise, to $𝐀 = 𝟎$ and $φ = e/{4πε_0r}$, which is independent of $t$, so is also the advanced potential.

The potential is derived (in component form) here:

Liénard–Wiechert potential
https://en.wikipedia.org/wiki/Li%C3%A9nard%E2%80%93Wiechert_potential

and issues concerning the method used with the delta functions (and corrections entailed, when done properly) are described in the reference here:

On the Lienard-Wiechert potentials
https://inis.iaea.org/search/search.aspx?orig_q=RN:22061421