I was wandering around the web looking for a high precision Schwarzchild radius of the Earth. There are a lot of pages that list it, but it’s usually described with pretty low precision, something like “8.87 millimeters”. Is there nothing more precise?

But What is the Schwarzchild radius? Well, if you smashed the Earth into a tiny ball, so that it became a black hole, it would be very small. How small? Well, it would be a ball with a radius figured out by Karl Schwarzchild back in the early 1900s, when he managed to find a solution to Einstein’s Field Equations of General Relativity.

How do you calculate it?

Schwarzchild Radius = 2 * G * M / square of lightspeed

G is the Newton’s Universal Gravitational Constant

M is the Mass of the Earth

Now. Lightspeed (c) of course is 299792458 meters per second. Exactly. Why? Because meters have been redefined in terms of the speed of light, with the caveat that we are speaking of lightspeed in a vacuum of course.

What about G and M? Well the funny thing is that it turns out that we don’t know G nor M very well on their own.

For G, according to Wikipedia, the “CODATA” international scientific committee recommends the following value:

G = 6.67384(80) x 10^-11 meters^3 / kilogram seconds^2

The parenthesis around the 80 is for something called “standard uncertainty” which means there is a good chance of G being 6.67384 etc etc give or take 80.

That surprised me, though. 6 decimal digits of precision and that big two digit error? We have known about G since the 1600s, that’s over 400 years, and we haven’t figured out a better value for it yet? I guess not!

Then there is M. . . . the Mass of the Earth. According to NASA, via Wikipedia, this is about 5.9722(6)×10^{24} kg.

OK. That’s not very good either. 5 digits of precision, give or take a digit? We have been looking at this question for thousands of years… and that’s all we got. OK.

But here is the tricky bit. Remember the equation for Schwarzchild Radius

Schwarzchild Radius = 2 * G * M / square of lightspeed

It just so happens that we know the quantity G*M to a lot higher precision than we know either G or M. Why? Well, I am not sure. But often in nature it is easier to measure the ratio of things rather than something ‘absolute’. So by measuring things like the orbits of Earth-Moon or Earth-Sun, we can calculate various relative quantities.

G*M in fact has it’s own special name. The Standard Gravitational Parameter = G*M.

For Earth, it has an even more special name. By Wikipedia’s article:

“The value for the Earth is called the

geocentric gravitational constantand equals 398600.4418±0.0008 km^{3}s^{−2}.”

That is 10 digits of precision! Not too shabby, especially compared to 5 or 6, which is what we have for G or M by themselves.

So,

Schwarzchild Radius = 2 * 398600.4418±0.0008 km^{3}s^{−2 }/ square of 299792458 m/s

With a bit of python computer language code:

# calculate Schwarzchild Radius of Earth, return low and hi estimate def schwarzchild_radius_of_earth(): # ggc = geocentric gravitational constant = # Universal Gravitational Constant * Mass of Earth ggc_lo = Fraction( 398600.4418 - 0.0008 ) # km^3/s^2 ggc_hi = Fraction( 398600.4418 + 0.0008 ) # km^3/s^2 lightspeed = 299792458 # m/s lightspeed_km = Fraction( lightspeed, 1000 ) lightspeed_sq = lightspeed_km*lightspeed_km schwarzchild_radius_lo = Fraction( 2 * ggc_lo , lightspeed_sq ) schwarzchild_radius_hi = Fraction( 2 * ggc_hi , lightspeed_sq ) return [schwarzchild_radius_lo * 1000, schwarzchild_radius_hi * 1000]

We get these results, in meters. First, as fractions:

lo = 6687405696352255859375/753930488239309834944512

hi = 3343702861597900390625/376965244119654917472256

Then, as an easier to read Decimal approximations of the fractions:

lo = 0.008870056060432941

hi = 0.008870056096037741

In other words, roughly 8.8700560 millimeters, give or take. That seems pretty good. I haven’t a nice way to express the uncertainty but you can kind of see the low value and hi value are the same for the first 8 decimal digits.

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So where might one use this?

Let’s say you wanted to calculate Time Dilation for an object due to General Relativity and Gravity, assuming the Schwarzchild solution of Einstein’s equations. The “normal” way to write this uses Gravitational Constant G and mass M,

sqr( time_dilated / time_undilated ) = 1 – 2GM/object’s_altitude*c^2

but you can also write it using Schwarzchild Radius –

sqr( time_dilated / time_undilated ) = 1 – Schwarzchild radius / objects_altitude

This is nice for a few reasons. It feels nice, at least subjectively to me. But also, you can kind of get a ‘ballpark idea’ of the dilation involved.

The radius of Earth is 6371 kilometers. The radius of a low earth orbit satellite or space station is about 160 kilometers above that. Either way, if you compare those to 8.87 millimeters, you are going to get a tiny number. 8.87 / 7 000 000 for example. Now, subtract this tiny number from 1. That’s the time dilation. In other words, for normal orbit objects, like satellites, the dilated time is going to be very, very close to 1 times the “normal” undiluted time.

But near a black hole, it can also help understand. If the Schwarzchild radius, also known as the “event horizon”, is 1 kilometer, well, we can imagine that if you get down close to say, 1.2 kilometers, then the ration is going to be like 1/1.2, which is pretty close to one. That will make 1-1/1.2 pretty small, which means the ratio of the times will be pretty significant.

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See Also

http://www.sussex.ac.uk/Users/waa22/relativity/What_is_gravitational_time_dilation.html (William Astill’s pages on Relativity)

http://en.wikipedia.org/wiki/Gravitational_redshift

http://en.wikipedia.org/wiki/Gravitational_time_dilation

http://en.wikipedia.org/wiki/Standard_gravitational_parameter

http://en.wikipedia.org/wiki/Earth_mass

http://en.wikipedia.org/wiki/Jet_Propulsion_Laboratory_Development_Ephemeris

http://en.wikipedia.org/wiki/Planetary_mass

http://en.wikipedia.org/wiki/Standard_gravitational_parameter