Cosmo 101 The Schwarzschild metric

So we’ve recently been looking at one metric in particular, the Schwarzschild metric. There is an interesting history to this one. So Einstein published some of his finding concerning general relativity in 1915. This same year a man by the name of Karl Schwarzschild was serving in the German army and thinking about these same sorts of things. He came up with some equations that are now the Schwarzschild metric. And then he died only a couple of month later. Einstein didn’t think it was even possible to come up with these sorts of equations. It makes you wonder what he would have come up if his life hadn’t been cut short. Anyway, these equations, like any metric, describe a world and the way physics works around this world. Formally we say these equations describe the spacetime geometry for a certain situation. Well the Schwarzschild metric describes the world around a spherical non-rotating mass. The most famous examples are black holes! Yes, that’s right, we get to talk about black holes! Perhaps I should do a bit of an introduction. Most black holes arise from a star finally giving in to gravity and collapsing in on its self. At this point all of that matter is compresses so much that it sort of tears spacetime and creates the mysterious black hole. But anyway if you were to stand right outside a black hole your spacetime and hence the laws of physics will be governed by the Schwarzschild metric. It’s a very twisted spacetime that I don’t really understand completely but I think I shall have more information for you tomorrow.

GOD! ?

Oh my! I had forgotten how hard homework was and how long it took! All we did today was math. So today I think I’m going to put a God post up.

Cosmo 101

Forgive the lack of a post last night I was at a wedding and it took much longer than initially anticipated. But I have come to a decision. With this class and all of the other things I am doing. I think I’m going to only write a post once during the weekend (Friday- Sunday). Ok if figured out how we do velocity. If we are just trying to find the velocity of a thing we us normal (or coordinate) time. So now I think it’s time to discuss some interesting properties of gravity. Ok so let’s say we are characters in a sci-fi movie and we are going to visit a black hole. To our horror one of our androids slip over the edge of the observation deck of our spaceship and falls towards the black hole. As much as that android was useful, helpful and part of the team, this mishap provides us with a scientific opportunity. We watch the poor little android as he is sucked further towards the back hole and see a confirmation of general relativity. You remember the experiment with the gobstoppers when we feel towards the earth. The same thing happens to the android. As he falls gravity pulls on his feet more than on his head and he gets stretched. Also gravity pushes him inward and his arms get compressed. This goes on until he gets stretched into a thin string and finally sucked beyond the event horizon. This event has one of the coolest names in science; we say the android underwent spaghettification. Also, and I won’t provide an example for this just take my word for it, the curvature of spacetime messes with the lengths of things and the wavelengths of light. These effects are not due to special relativity, they are not associated with time dilation or length dilation, they are solely due to the curvature of spacetime. But I don’t know why. Perhaps I will have more information on Monday.

Cosmo 101 (a bit of math included)

Well…I was gonna talk about tidal effects and the Schwarzschild metric but I think I would like to talk about our homework more. I know that sounds weird but…it is, my homework, I mean. We are currently working with special relativity and it is just really weird. I’ve always known, well since 1687 (Newton’s year), which feels like forever, I’ve always known that gravity is a force and that v=d/t and that t is the same for everybody that hasn’t even been a concern, but now it is and I don’t really know how to handle it. Now all of the equations that I’ve always fallen back on are not valid anymore and I don’t know which ones are and aren’t. For example there are two (or possible more) types of time in relativity. There is measured time which we represent with a normal t and there is wristwatch (or proper) time, which is designated with the greek letter tau. Now which of those do you use if you want to find the velocity? And for example if you have people traveling at different speeds they will see different times for different events. In other words not everyone agrees on the time separation, delta t, between two events. And not everyone agrees on the spacing either. But there is hope (haha I sound like a public service announcement) there are some values that are the same for everybody, tau is the same no matter who you are or how fast you are going. The equation to calculate tau is tao^2=t^2-s^2. Here s is the displacement in space. It’s the same as the r we used a couple weeks ago.

Cosmo 101

Ok so hop back into your glass elevator we’re gonna need it again. So let’s get some more goobstopers and do a little experiment. The experiment is to toss one lightly up into the air and see what happens. First let’s start out on earth. You’re just sitting there in the chocolate factory and you throw the goobstoper up into the air. What does it do? It goes up and slows down, then comes to a stop and falls back down. Ok after you record the results you take off into space and find a nice little spot away from any gravitational fields. You cut the power and drift along at a nice constant velocity. Now you try the experiment again. You throw the goobstoper up but this time it just keeps going until it hits the ceiling. Hmm…interesting. Now you turn your power back on and set the controls to accelerate you at 9.8 meters per second per second (the ‘acceleration of gravity’ felt on earth) and you do the experiment again. This time the goobstoper goes up and comes back now just the way it would on earth. Very peculiar. Well now you go back towards earth and let yourself free fall down towards the earth. Now we repeat the experiment one more time. This time when you throw the goobstoper up it keeps going just as it did when you at constant velocity in deep space. This set of thought experiments is called the Equivalence Principle and, more succinctly, look like this:

A lab on earth = an accelerating lab in deep space

And a non-accelerating lab in deep space = a feel-fall lab on/near earth

Now there is one distinction that must be made. When I say earth what I really mean is a uniform gravitational field (remember the experiment from last night), but on earth’s surface it’s very close to uniform. You know what, I think I’ve been spelling goobstoper wrong…by George I have! It’s spelled gobstopper. Oh well, you knew what I meant.

Cosmo 101

Ok time for some more really cool introductory concepts. Proper time. I and the author of the book we will be reading like to call it wristwatch time and it has to do with moving clocks…or watches as the case may be. So let’s go back to wolverine and deadpool in their epic battle. Let’s say that in the computer chips inside deadpool’s head he has a clock. Now this clock measures the time when the battle starts, the first event and let’s say that deadpool himself is present when the call to the army is made, event two. If that were the case then the clock inside deadpool’s head would measure wristwatch time, or proper time. Right now it’s just a vocab term but latter it will mean something more. I also wanted to tell you about the reading we did for tomorrow. It was all about gravity. So you know in high school physics how they say that gravity exerts a force on you when you fall…well all the time…but it hurts you when you fall. Well turns out that there is really no force at all…we are accelerating towards the earth due to the fact that the space around the earth is curved by it’s gravity. Here’s an experiment to illustrate. Let’s say you have working replica of Willy Wonka’s glass elevator. Now first you take it way into deep space where there are no planets or anything with a very strong gravitational pull. Now you have 4 everlasting goobstopers and you set one near the ceiling, 2 in the center one on the left and one on the right, and one near the bottom of the elevator. As you sit there in space your goobstoper don’t move, not one iota. Now let’s say you come back to earth and let yourself free fall through the atmosphere and plummet towards the earth. Now you are experiencing the same sort of weightless feeling you did in deep space so you decide to try the goobstoper experiment again. You put the goobstopers in the exact same places and watch them. The funny thing is that they do move but only very slightly. The top one inches toward the ceiling the center ones move more towards the center and the bottom one goes towards the floor. Again this effect is very small but if you were to fall for quite a long time (and not die in a giant fireball upon impact with the earth’s surface) you would see the goobstopers move! And they are moving because of gravity. Cool experiment right? Now all I need is a glass elevator and some goobstopers…

Cosmo 101

So I went to my first cosmology class today and it was amazing. I am very much looking forward to walking through all of this with you. And guess what! All of the things we’ve been talking about up to now have been almost a perfect introduction to where we started off in class today. But there are some things I didn’t mention, so I would like to take today and explain what an inertial reference frame is. Ok so whenever I think of this topic I always picture people in bubbles floating around in spacetime and it always makes me think of this commercial. But anyway inertial means that the bubbles aren’t accelerating, they’re just floating along at a constant velocity looking at events that take place. The other thing I wanted to cover today way the units that I couldn’t remember from before. So when dealing with spacetime in any sort of relativistic way you have to make a choice. Both time and distance either have to be in meters or in seconds. The book we are using picks meters. So whenever you have a time 12 seconds for example you multiply by the speed of light, 3x10^8 m/s and you get an answer in meters. The units of everything else change too, for example velocity is unitless, it’s just a number.