I’m enjoying a little break from lectures now that it’s Summer, which gives me time to learn more about subjects that interest me. Today, we’ll be having a quick look at the most basic black hole physics we have: Schwarzschild black holes!
The Schwarzschild metric is used for black holes with NO charge and NO angular momentum. Whilst this most likely isn’t ever the case, the metric provides a very good approximation for things with nearly no charge and angular momentum, such as our Sun and Earth.
The Schwarzschild radius (where the event horizon is) of a black hole is calculated using this formula:
This metric shows the singularity of a black hole to be in the very centre, and stable circular orbits can only be made 3 times away from the Schwarzschild radius. Once you get closer to the black hole, about 1.5 times the Schwarzschild radius, you’re screwed.
Don’t listen to Interstellar, listen to the physics!
I’m no black hole expert, so instead a short derivation of the Schwarzschild metric can be viewed here.
The equation you’ll find first looks like this:
This essentially shows you how space changes as you’re near a black hole, the lower equation being a simplified version where I’ve simply subbed in the Schwarzschild radius.
If you have ever covered coordinate geometry in maths or physics, you most likely used cartesian (x, y, z) coordinates to do things like find equation for lines and circles, but you may have also come across spherical polar coordinates. Cartesian isn’t the most ideal metric for spheres, but spherical polar coordinates are great! When you want to find a point in spherical polar you basically create a sphere in which the point lies on the surface. When it comes to black holes, a naturally spherical object, this metric will alleviate a lot of mathematic headaches that come with cartesian coordinates, simply because spheres are already built into the spherical polar metric.
Below is the general coordinate equation accompanied by a diagram of how this coordinate system works. Along with them is the Schwarzschild metric at the bottom, and hopefully you can see the similarities in red.
As you may have noticed, there’s an extra coordinate, the time coordinate (t). The change in time squared multiplied by the speed of light squared results in another distance component that we need to take into account.
The presence of a singularity is shown when this metric blows up. This happens with two values of r: the true singularity, and the event horizon.
If we look at one part of the metric…
If we sub in the true singularity (r=0), we are dividing the Schwarzschild radius by zero straight away, and the metric blows up. At the event horizon (defined to be the Schwarzschild radius) the amount in the brackets totals to zero, which we then divide by and blow the metric up again.
Of course, the singularity at the event horizon is just an illusion. There is only one singularity at the centre of the black hole.
Freddie should have put that in the chorus in “One Vision”.
One Star, one (gigantic) mass, One Siiiingularity.
And now, a pretty picture!
Back to the metric! There’s nothing attached to theta and phi, so the gravitational field at radius r is the same regardless of theta and phi.
In fact, there’s this beautiful shape that describes the spacial curvature defined by the Schwarzschild metric of a body called Flamm’s Paraboloid. It’s a pretty visualisation of the Schwarzschild metric, but it doesn’t include time. It is NOT the same as a gravity well, which doesn’t look the same.
This specific image below was created by Allen McC using Mathematica. Enjoy!
Featured Image: GALEX image of NGC 6744 by NASA/JPL