A Error in Landau-Lifshitz Vol 2, Fig. 20, page 310

[Kostik]

TL;DR Summary

A minor error is noted in Figure 20 of Landau-Lifshitz Vol 2, illustrating the light cones at two points, one inside and one outside the event horizon of a black hole, using Lemaître coordinates.

Landau & Lifshitz introduce what are known as Lemaître coordinates (the coordinates and associated Schwarzschild metric were first used by G. Lemaître in 1932-1935). The figure (Figure 20) on p. 310 is a magnificent illustration of why all timelike and null paths within the black hole region ($r<2m$) go to $r=0$.

However, there is a small error in the figure. At point $a$, located where $r>2m$, a light cone is shown in dotted lines. The paths of the ingoing and outgoing photon are gently curved, and both paths are "concave up". The path of the ingoing photon is concave up, but the path of the outgoing photon should be concave down.

The figure is correct for the light cone at point $a'$ -- both paths are concave up.

The correct concavity is clear upon consideration that the angle of the light cone decreases as $r \rightarrow 2m$.

I have checked this by plotting solutions of the differential equation for the light cone boundaries (found by setting $ds^2=0$ in the Schwarzschild metric, L-L equation (102.3).)

 

[weirdoguy]

Well, last time I read L&L books there were TONS of typos and errors. Which is quite interesting since I had like n-th (polish) edition.

 

[JimWhoKnew]

TL;DR Summary: A minor error is noted in Figure 20 of Landau-Lifshitz Vol 2, illustrating the light cones at two points, one inside and one outside the event horizon of a black hole, using Lemaître coordinates.

Landau & Lifshitz introduce what are known as Lemaître coordinates (the coordinates and associated Schwarzschild metric were first used by G. Lemaître in 1932-1935). The figure (Figure 20) on p. 310 is a magnificent illustration of why all timelike and null paths within the black hole region (r<2m) go to r=0.

However, there is a small error in the figure. At point a, located where r>2m, a light cone is shown in dotted lines. The paths of the ingoing and outgoing photon are gently curved, and both paths are "concave up". The path of the ingoing photon is concave up, but the path of the outgoing photon should be concave down.

The figure is correct for the light cone at point a′ -- both paths are concave up.

The correct concavity is clear upon consideration that the angle of the light cone decreases as r→2m.

I have checked this by plotting solutions of the differential equation for the light cone boundaries (found by setting ds2=0 in the Schwarzschild metric, L-L equation (102.3).)

From equation 102.4 I get
$$\frac{d^2\tau}{dR^2}<0$$
for $R>cτ$ and outgoing null geodesics, so you are probably right.

 

[Kostik]

From equation 102.4 I get
$$\frac{d^2\tau}{dR^2}<0$$
for $R>cτ$ and outgoing null geodesics, so you are probably right.

Yes, I just did the same calculation. It gives the concavity for all four cases (ingoing and outgoing photons, $r>2m$ and $r<m$. It confirms that all photon paths should be concave up except for the outgoing photon with $r>2m$, which should be concave down.

 

[Kostik]

I should have provided a picture from the L-L text:

 

[JimWhoKnew]

Yes, I just did the same calculation. It gives the concavity for all four cases (ingoing and outgoing photons, $r>2m$ and $r<m$. It confirms that all photon paths should be concave up except for the outgoing photon with $r>2m$, which should be concave down.

Are you sure? It seems manifest from (the middle part of) equation 102.4 that $d^2\tau / dR^2$ at any given point has different signs for ingoing and outgoing null geodesics.

 

[Kostik]

Are you sure? It seems manifest from (the middle part of) equation 102.4 that $d^2\tau / dR^2$ at any given point has different signs for ingoing and outgoing null geodesics.

Yes, I am sure. Here is my work. My $\rho$ is Landau's $R$. Also, it's clear because the angle formed by the light cone gets more acute as $r$ decreases. Just check all four cases.

 

[PAllen]

Are you sure? It seems manifest from (the middle part of) equation 102.4 that $d^2\tau / dR^2$ at any given point has different signs for ingoing and outgoing null geodesics.

I don’t have L-L, but am generally familiar with Lemaître coordinates (a personal favorite to use). The outgoing null geodesics exterior to the horizon asymptote to the horizon and bend outwards, thus concave down. At the horizon, these have zero concavity, thus straight lines. Inside, they asymptote to the horizon in the past, then bend upwards to reach the singularity, thus concave up. The horizon is the boundary for concavity change for outgoing null geodesics.

So @Kostik's summary in post #4 is correct.