The Schwarzschild Metric is very often used to describe nonrotating, uncharged, black holes (as well as other gravitational bodies) and the effect that they have on the spacetime around them.
Hey guys, I'm back with a video that I kinda threw together, because I'm on holiday at the moment lol. But I wanted to make a (serious) video about black holes, and the equation most commonly used to describe them. In this video, I try to explain what the Schwarzschild Metric means, and where it comes from.
So we can build all the information we need in order to understand the equation that gives the proper time interval between two events in 4 dimensional spacetime (jargon alert lol) just by starting from Pythagoras' theorem. We can see how it can be used to find the distance between two points on a 2 dimensional grid, and we can even extend this theorem into 3d and 4d.
It's the 4d version that's most useful for us - after all, according to relativity, we live in a 4 dimensional spacetime. However, the equation we find by using Pythagoras' theorem and extending it to 4d only applies if the spacetime itself is flat, or non-warped, or non-wibbly-wobbly-timey-wimey'd. So we need to modify it further, using the equations of general relativity (specifically "Einstein's Field Equations") which tell us just how mass, and therefore gravity, ends up warping spacetime.
One such solution of the Einstein Field Equations is the Schwarzschild Metric - which is the equation that Brian Greene, legendary physicist, posted on his FaceBook page as a response to the announcement that humans had finally imaged a black hole. However, I reckon Brian got something slightly wrong... stick around until the end to find out what lol.
By the way, yes I've been a bit inconsistent in this video regarding a couple of things - firstly, I've said "point" instead of "event" and "distance" instead of "interval" a few too many times (but hopefully not when it actually matters). Secondly, in some of the equations I've left in powers of the speed of light, c, whereas in others I've used the form of the equation when we set c = 1. Hopefully this isn't a massive issue though.
Also, some of the stuff in this video is quite complicated and takes a while to wrap your head around (or it did for me at least), so if you have any questions then let me know down in the comments. As well as this, if you spot a mistake I've made, then let me know too.
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