Cosmo 101 The Schwarzschild metric (math included)

Now I would like to take a moment to talk about the actual Schwarzschild metric. Now there will be a bit of math involved in this but hopefully it will be well worth it. First let’s talk about units again. In general relativity we like for things to be as simple as possible, so we make time and length have the same units. In my book we use meters. Well how do you make time go from seconds to meter? Well you multiply by the speed of light, c. So, now that we have everything in meters things will be pretty easy. We’ve seen the first equation up top (1). It’s the equation for the distance one travels. We’ve also seen the equation right below it (2). It tells us the distance one has traveled in spacetime. But the thing about equation 2 is that it only works for flat spacetime. But now we are dealing with a curved spacetime and we will have to change equation 2. What Schwarzschild discovered is that you can alter this equation to get a distance that works for curved space. But first we have to do some more unit changing. Now we are dealing with gravity and that means we have to deal with masses. Masses normally come in units of kg’s but now we want them in meters as well. So we multiply any mass we get by Newton’s gravitational constant, G, and then dividing it by the speed of light squared. This is in equation 3. Now our new equation for distance will look nice and neat. And there it is, equation 4. There is a lot more to talk about but I think I’ve given you enough already. We’ll talk about all that tomorrow.