This is from "Exploring Black Holes" by Taylor and Wheeler. It's a very good book but I struggle not with the math, but the explanations (sometimes)(adsbygoogle = window.adsbygoogle || []).push({});

On page B-13 is a frame called "Metric for the Rain Frame", which is a transformation of the Schwarzschild Metric from "bookkeeper coordinates" to shell coordinates to "rain coordinates"

All well and good.

Here is the final equation for the rain frame:

[tex]d \tau^2 = (1-\frac {2M}{r}) dt^2 - 2( \frac{2M}{r})^\frac{1}{2} dt_{rain} dr -dr^2 -r^2 d \phi ^2 [/tex]

The text continues:

Now I have a few observations about this: This metric can be used anywhere around a non-rotating black hole, not just inside the horizon. Our ability to write the metric in a form without infinities at [tex] r = 2M[/tex] is an indication that the plunger feels no jerk or jolt as she passes through the horizon

1. Yes, the metric has got rid of the infinity at [tex]r=2M[/tex] but

when r < 2M all of the parts of the right hand side must be negative because [tex]1 - \frac {2M}{r} < 0 [/tex] this means that [tex]d \tau^2 < 0 [/tex] which means that [tex]d \tau [/tex]is imaginary

2. When r is slightly greater than 2M, then in order for the wristwatch of the plunger to record real time then

[tex](1-\frac {2M}{r}) dt^2 > 2( \frac{2M}{r})^\frac{1}{2} dt_{rain} dr -dr^2 -r^2 d \phi ^2[/tex]

Now I interpret part (2) to mean that regardless of initial conditions, any infalling object or person must cross normally to the event horizon.

But (1) puzzles me. There may be no "jolt" but how does a wristwatch measure imaginary time?

Perhaps the hands bend at [tex]90^ \circ [/tex] to the plane of the watchface...

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# Does a wristwatch measure imaginary time?

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