- #1

George Keeling

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- TL;DR Summary
- I want to check my formula for spaghettification and then apply it to the gas cloud G2 which approached Sagittarius A* in 2014 with disappointing results.

I did an exercise about beacons falling radially into black holes from Carroll's book and got a formula for the proper velocity$$

\frac{dr}{d\tau}=-\sqrt{\frac{R_S}{r_\ast}}\sqrt{\frac{r_\ast-r}{r}}

$$It's in natural units (##c=1)##, ##r_\ast## is where the beacon is dropped from and ##R_s=2GM## is the Schwarzschild radius. By inverting the equation and integrating it, one can calculate the proper time between two radii and it gives the correct answer for time from event horizon to the centre (##\pi GM## or 28 hours in M87*).

If I differentiate that I can get the proper acceleration ##d^2r/d\tau^2=-R_S/2r^2## and that turns out to be ##9.8\ ms^{-2}## at the surface of the earth. So that seems to work.

I can now differentiate that with respect to the ##r## to get the radial rate of change of the acceleration which will give me a measure of spaghettification. Call that ##S## and$$

S=\frac{2GM}{r^3}\ \ s^{-2}

$$The units are a bit odd, but make sense if you think of them as ##\left(ms^{-2}\right)m^{-1}## or even Newtons per kilogram per metre. Wikipedia gives the equivalent tensile force on a rod with mass ##m## length ##l## as $$

F=\frac{GMlm}{4r^3}

$$Using my formula, I calculate that for a male human at ##70 kg## & ##2 m##, ##S=0.1## would give 14 Newtons, that's how much 1.4 kg weighs. That sounds bearable - the opposite of putting on a bit of weight. Women and children would have an advantage!

Now we come to the gas cloud known as G2, which was discovered heading towards Sagittarius A*, the black hole at the centre of our galaxy, in 2011 by some folk at the Max Planck Institute. G2 was destined to come closest to Sagittarius A* in Spring 2014 "with a predicted closest approach of only 3000 times the radius of the event horizon". There was great excitement because spaghettification and great fireworks were expected. However nothing much happened and G2 continues on its way, orbiting Sagittarius A*.

I re-read about it in a 2014 paper from the American Astronomical Society.

One thing continues to puzzle me: At ##r=## 3000 times the radius of the event horizon, that's ##r=4\times{10}^{13}\ m## the ##g## force is only ##0.4\ ms^{-2}## and the ##S## is a very feeble or even undetectable ##2.2\times{10}^{-14}##.

It might be because they estimate the size of G2 as ##r_{G2}=3au\approx5\times{10}^{11}\ m## which is pretty big. But at closest approach where ##S=2.2\times{10}^{-14}\ ms^{-2}m^{-1}## g-forces would change across the diameter by ##r_{G2}\times S=0.001\ ms^{-2}##. Still tiny compared to ##0.4##.

\frac{dr}{d\tau}=-\sqrt{\frac{R_S}{r_\ast}}\sqrt{\frac{r_\ast-r}{r}}

$$It's in natural units (##c=1)##, ##r_\ast## is where the beacon is dropped from and ##R_s=2GM## is the Schwarzschild radius. By inverting the equation and integrating it, one can calculate the proper time between two radii and it gives the correct answer for time from event horizon to the centre (##\pi GM## or 28 hours in M87*).

If I differentiate that I can get the proper acceleration ##d^2r/d\tau^2=-R_S/2r^2## and that turns out to be ##9.8\ ms^{-2}## at the surface of the earth. So that seems to work.

I can now differentiate that with respect to the ##r## to get the radial rate of change of the acceleration which will give me a measure of spaghettification. Call that ##S## and$$

S=\frac{2GM}{r^3}\ \ s^{-2}

$$The units are a bit odd, but make sense if you think of them as ##\left(ms^{-2}\right)m^{-1}## or even Newtons per kilogram per metre. Wikipedia gives the equivalent tensile force on a rod with mass ##m## length ##l## as $$

F=\frac{GMlm}{4r^3}

$$Using my formula, I calculate that for a male human at ##70 kg## & ##2 m##, ##S=0.1## would give 14 Newtons, that's how much 1.4 kg weighs. That sounds bearable - the opposite of putting on a bit of weight. Women and children would have an advantage!

**Is that all correct? Would a brave black hole explorer be able to rely on me?**Now we come to the gas cloud known as G2, which was discovered heading towards Sagittarius A*, the black hole at the centre of our galaxy, in 2011 by some folk at the Max Planck Institute. G2 was destined to come closest to Sagittarius A* in Spring 2014 "with a predicted closest approach of only 3000 times the radius of the event horizon". There was great excitement because spaghettification and great fireworks were expected. However nothing much happened and G2 continues on its way, orbiting Sagittarius A*.

I re-read about it in a 2014 paper from the American Astronomical Society.

One thing continues to puzzle me: At ##r=## 3000 times the radius of the event horizon, that's ##r=4\times{10}^{13}\ m## the ##g## force is only ##0.4\ ms^{-2}## and the ##S## is a very feeble or even undetectable ##2.2\times{10}^{-14}##.

**Why was there such great excitement or expectation of fireworks?**It might be because they estimate the size of G2 as ##r_{G2}=3au\approx5\times{10}^{11}\ m## which is pretty big. But at closest approach where ##S=2.2\times{10}^{-14}\ ms^{-2}m^{-1}## g-forces would change across the diameter by ##r_{G2}\times S=0.001\ ms^{-2}##. Still tiny compared to ##0.4##.