# Tortoise Coordinate

1. Apr 24, 2006

### yenchin

Hi. I have been reading the forums for quite sometime now, but this is the first time I decided to join in the fun. I am wondering why the tortoise coordinate is called tortoise coordinate (why not turtle or hippo... :tongue:) . I have tried searching for the answer online but couldn't find any, and my lecturers are not sure too. Any idea?

2. Apr 24, 2006

### Garth

Hi yenchin!

The ''tortoise coordinate'' $r^\star$ is defined by:

$$r^\star = r + 2GM\ln\left|\frac{r}{2GM} - 1\right|$$.

The tortoise coordinate $r^\star$ approaches $- \infty$ as ''r'' approaches the Schwarzschild radius ''r'' = 2''GM''. It satisfies

$$\frac{dr^\star}{dr} = \left(1-\frac{2GM}{r}\right)^{-1}$$.

Watch the object fall towards the Schwarzschild radius at a constant

$$\frac{dr^\star}{dt}$$

it 'slows right up', hence $r^\star$ is called the 'tortoise coordinate'.

Garth

Last edited: Apr 24, 2006
3. Apr 24, 2006

### yenchin

Oh... Thanks.

4. Apr 29, 2006

### Cexy

This thread has just reminded me of something! Apparently in Arnol'd's book on GR, he refers to the Lie derivative as "the Angler's derivative". Does anybody have any idea why that might be?!

5. Apr 29, 2006

### George Jones

Staff Emeritus
Because the Lie derivative is defined using a flow (of a vector field), like the flow of a stream in which the angler angles!

Regards,
George

6. Aug 14, 2011

### yangbin

I don't think that
"it 'slows right up', hence r⋆ is called the 'tortoise coordinate'."
That coordinate was named tortoise coordinate due to the story of Achilles and the Tortoise, we know that the Tortoise think that Achilles wouldn't catch up itself forever, this case just like the observer of outside of blackhole who never saw anything fall into the blackhole, but in fact for a free falling observer who fall into the blackhole in a finite time. What saw outer of blackhole observer is just because he used the "Tortoise coordinate", that it means!

7. Aug 14, 2011

### yenchin

Wow. Thanks for resurrecting my 5-year-old thread ;-)