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I thought I'd present some plots for the Fermi-normal coordinates (only in the r-t plane) for someone falling into a black hole "from infinity".

Fermi-normal coordinates radiate a set of space-like geodesics from some point on the worldine of an object - in this case, the worldline of an observer falling into a black hole. The set of points lying on space-like geodesics that pass through the observer's worldline at a proper time of tau represent the set of points with a time coordinate of tau. The distance coordinate is given directly by the affine parameter of the geodesic, which is equivalent to measuring the distance to the point along the arc-length of the geodesic curve connecting the observer to the observee.

In our diagram, the blue dashed line represents our infalling observer. The red lines are the space-like geodesics - they are "lines of simultaneity" for the fermi-normal coordinate system. Boxes are placed at unit intervals along the space-like geodesics to display the affine parameter or the "distance away from the observer".

We assume for simplicity that m=1/2, i.e the line element is

ds^2 = -(1-1/r)dt^2 + dr^2 / (1-1/r)Then, for the infalling observer the defining equations are:

[tex]

\frac{dt}{d\tau} = \frac{1}{ \left( 1 - 1/r \right) }}

[/tex]

[tex]

\frac{dr}{d\tau} = \frac{1}{\sqrt{r}}

[/tex]

for the space-like geodesics, passing through some time t0, r0 the defining equations are:

[tex]

\frac{dt}{ds} = \frac{1}{\sqrt{r0} \, \left(1 - 1/r \right) }

[/tex]

[tex]

\frac{dr}{ds} = \sqrt{1 + \frac{1}{r0} - \frac{1}{r} }

[/tex]

These equations were derived first by noting that (dt/ds) / (1-1/r) must be constant, just as (dt/dtau) / (1-1/r) is constant.

dr/ds is found by solving g_00 (dt/ds)^2 + g_11 (dr/ds)^2 = 1, similarly to the way dr/dtau is found by setting g_00 (dt/dtau)^2 + g_11 (dr/dtau)^2 = -1.

On a highly technical note, [itex]u^{a} \nabla_a u^{b}[/itex] was computed and confirmed to be zero to confirm that the vector fields above are the tangent fields of geodesics.

Fermi-normal coordinates radiate a set of space-like geodesics from some point on the worldine of an object - in this case, the worldline of an observer falling into a black hole. The set of points lying on space-like geodesics that pass through the observer's worldline at a proper time of tau represent the set of points with a time coordinate of tau. The distance coordinate is given directly by the affine parameter of the geodesic, which is equivalent to measuring the distance to the point along the arc-length of the geodesic curve connecting the observer to the observee.

In our diagram, the blue dashed line represents our infalling observer. The red lines are the space-like geodesics - they are "lines of simultaneity" for the fermi-normal coordinate system. Boxes are placed at unit intervals along the space-like geodesics to display the affine parameter or the "distance away from the observer".

We assume for simplicity that m=1/2, i.e the line element is

ds^2 = -(1-1/r)dt^2 + dr^2 / (1-1/r)Then, for the infalling observer the defining equations are:

[tex]

\frac{dt}{d\tau} = \frac{1}{ \left( 1 - 1/r \right) }}

[/tex]

[tex]

\frac{dr}{d\tau} = \frac{1}{\sqrt{r}}

[/tex]

for the space-like geodesics, passing through some time t0, r0 the defining equations are:

[tex]

\frac{dt}{ds} = \frac{1}{\sqrt{r0} \, \left(1 - 1/r \right) }

[/tex]

[tex]

\frac{dr}{ds} = \sqrt{1 + \frac{1}{r0} - \frac{1}{r} }

[/tex]

These equations were derived first by noting that (dt/ds) / (1-1/r) must be constant, just as (dt/dtau) / (1-1/r) is constant.

dr/ds is found by solving g_00 (dt/ds)^2 + g_11 (dr/ds)^2 = 1, similarly to the way dr/dtau is found by setting g_00 (dt/dtau)^2 + g_11 (dr/dtau)^2 = -1.

On a highly technical note, [itex]u^{a} \nabla_a u^{b}[/itex] was computed and confirmed to be zero to confirm that the vector fields above are the tangent fields of geodesics.

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