Schwarzschild spacetime in Kruskal coordinates

[PeterDonis]

From which region is the free-falling Painleve observer traced backward ?

From the right exterior region, region I, using ingoing Painleve coordinates, which cover only regions I and II. In the case I described, you are looking at a timelike geodesic that emerges from the "white hole" region, the bottom wedge of the Kruskal diagram, but ingoing Painleve coordinates do not cover that region, so all you have in those coordinates is the worldline approaching the past horizon (the white hole horizon) asymptotically as $T \to - \infty$.

 

[PeterDonis]

I believe its path should be in the past light cone centered at the event its worldline is traced backward.

This makes no sense; a light cone is not the same thing as a worldline.

 

[cianfa72]

In the case I described, you are looking at a timelike geodesic that emerges from the "white hole" region, the bottom wedge of the Kruskal diagram, but ingoing Painleve coordinates do not cover that region, so all you have in those coordinates is the worldline approaching the past horizon (the white hole horizon) asymptotically as $T \to - \infty$.

Ok, so we are looking at a timelike geodesic starting from region IV. Its path is inside the local light cones along the curve up to the past horizon. Then it passes the past horizon entering in region I; the ingoing Painleve coordinates assigns to a such event/point the value $T= - \infty$.

Edit: the timelike worldlines of ingoing or outgoing Painleve free-falling observers are orthogonal to the hypersurfaces of constant Painleve coordinate time $T$. Using $T$ as path parameter it turns out that the Painleve radially free-falling worldlines are of type $r=f(T), T=T$, right ?

 

[PeterDonis]

we are looking at a timelike geodesic starting from region IV. Its path is inside the local light cones along the curve up to the past horizon. Then it passes the past horizon entering in region I; the ingoing Painleve coordinates assigns to a such event/point the value $T= - \infty$.

Yes.

the timelike worldlines of ingoing or outgoing Painleve free-falling observers are orthogonal to the hypersurfaces of constant Painleve coordinate time $T$.

Yes.

Using $T$ as path parameter it turns out that the Painleve radially free-falling worldlines are of type $r=f(T), T=T$, right ?

I believe that you can use the Painleve $T$ as an affine parameter along Painleve free-falling worldlines, yes.

 

[cianfa72]

So, the ingoing and outgoing Painleve charts overlap in region I. It makes sense since generally in an atlas charts may overlap.

 

[PeterDonis]

the ingoing and outgoing Painleve charts overlap in region I.

Yes. And also, if you read the Insights article you referenced, you will see that I say that there should also be another pair of Painleve charts that overlap in region III (the left exterior region). Then the two "ingoing" charts will overlap in the black hole (region II) and the two "outgoing" charts will overlap in the white hole (region IV).

 

[Orodruin]

So, the ingoing and outgoing Painleve charts overlap in region I. It makes sense since generally in an atlas charts may overlap.

Not only are charts allowed to overlap, it is essential that there are chart overlaps. Otherwise you would simply not describe how different parts of the manifold are stitched together.

 

[cianfa72]

I'm still confused about geodesic affine parameterization. For a timelike geodesic any affine parameter $\lambda$ is related to the proper time $\tau$ along the curve via $\lambda =a\tau + b$. So I believe the Schwarzschild coordinate time $t$ (since is not related via an affine map to the proper time of free-falling observers) cannot be used as affine parameter for timelike geodesics in Schwarzschild spacetime.

 

[PeterDonis]

I believe the Schwarzschild coordinate time (since is not related via an affine map to the proper time of free-falling observers) cannot be used as affine parameter for a timelike geodesic in Schwarzschild spacetime.

You are correct. One easy way to see it is to note that Schwarzschild coordinate time along an ingoing timelike geodesic increases without bound as the horizon is approached, where of course proper time along the geodesic is finite.

 

[cianfa72]

The same should apply for geodesic spacelike curves. In the sense that an affine parameterization has to be related by an affine map to the proper lenght along the spacelike geodesic ?

 

[PeterDonis]

The same should apply for geodesic spacelike curves. In the sense that an affine parameterization has to be related by an affine map to the proper lenght along the spacelike geodesic ?

Yes.

 

[cianfa72]

So when we write down the geodesic equation as $$ \frac {D} {d\lambda} \frac {dx^{\mu}} {d\lambda} =0$$
$\lambda$ is implicitly an affine parameter.

 

[Orodruin]

So when we write down the geodesic equation as $$ \frac {D} {d\lambda} \frac {dx^{\mu}} {d\lambda} =0$$
$\lambda$ is implicitly an affine parameter.

Yes

Edit: For a non-affine parameter, the RHS would be proportional to $dx^\mu/d\lambda$

Edit 2: … and in that case you can use the chain rule to obtain an ODE expressing the relationship between $\lambda$ and proper time.

 

[cianfa72]

So a solution of the equation $$\frac {D} {d\lambda} \frac {dx^{\mu}} {d\lambda} = K \frac {dx^{\mu}} {d{\lambda}}, K \neq 0$$ gives a geodesic $x^{\mu} ({\lambda})$ implicitly parametrized by a non-affine parameter $\lambda$.