Relativity

Chapter 4: Applications of Derivatives

At the time of this writing, physics is rather annoyingly split into two completely different types of theories. Relativity is a theory about how space and time work, and very basic ideas about how particles must act in space-time. The other half of physics is quantum field theories, which are theories about what types of particles exist and how they interact with each other. One may say that relativity builds a stage, and quantum field theories fill the stage with players and a script.

In 3-space, we're used to putting X,Y,Z into a vector. Then the length of a vector squared is the dot product of the vector with itself, that is, L2 = VV. But now we're working in 4-space, that is space-time, so at the very least we're going to need to have vectors which hold 4 things. Here's the rules we'll use for these 4-vectors:

  1. A vector will have a superscript index, as Vi. We always use superscripts for a vector. Subscripts will mean a different type of object, as we'll see later.
  2. If the superscript is a latin letter like i,j,k, then the vector lives in 3-space and the index runs from 1 to 3. V1, V2, and V3 are respectively X, Y, and Z.
  3. If the superscript is a greek letter like a,b,g, then the vector lives in 4-space and the index runs from 0 to 3. V0 is time, and V1, V2, and V3 are respectively X, Y, and Z. Notice that V2 means the second entry in V, not V squared.
  4. V0, the time coordinate, will be measured with the same units as the space coordinates, so the speed of light is 1.
  5. We can multiply a vector, that is something with a superscript, by something with a subscript. We never multiply two things that both have superscripts. We never multiply two things that both have subscripts.
In 3-space we have the dot product to help us calculate distance. The dot product makes no sense in space-time. A 4 vector dotted with itself gives us X2 + Y2 + Z2 + T2, which has no meaning. Why? Because we are looking for the interval, so we need T2 - X2 - Y2 - Z2. We have seen that T2 - X2 - Y2 - Z2 is zero for a ray of light for all observers. If we calculate X2 + Y2 + Z2 + T2 for a ray of light, we get a number which depends on the observer's velocity, as we'll see in a bit. We need a replacement for the dot product which gets us a minus sign in front of the space terms.

We could do this by remembering that the t term always gets subtracted instead of added, but this is just a recipe for trouble - we'll forget sometimes. We'll handle this with a matrix - a special matrix, called the Metric Tensor. The Metric Tensor, usually just called the Metric, is called h, which is read out loud as "eta." The Metric tensor will be the matrix:

( 1,   0,  0,  0 )
( 0, -1,  0,  0 )
( 0,  0, -1,  0 )
( 0,  0,  0, -1 )

If we multiply the vector V = (V0, V1, V2, V3) by the matrix h, we get (V0, -V1, -V2, -V3). Now, if we multiply our original V by this, we get (V0)2 - (V1)2 - (V2)2 - (V3)2 = T2 - X2 - Y2 - Z2, which is just what we're looking for. So, VV gives us the wrong answer, but VhV gives us the right answer. The purpose of h is to keep track of the minus signs in the space terms for us.

In Euclidean 3-space, the metric tensor is

( 1, 0, 0 )
( 0, 1, 0 )
( 0, 0, 1 )

which just turns a vector into itself, so the metric tensor is always ignored in 3-space. But it's still there, formally, and when we make the move to space-time we can't ignore it any longer. When we move to curved 4-space, we'll call the metric tensor g instead of h. This is because we'll find that the metric tensor g is not a constant, but depends on where we are. We'll find that in most of our universe, so long as we're not moving at close to the speed of light and we're not near any black holes, the metric tensor g is very nearly equal to h, so we can say that g = h + h, where h is a matrix containing only small numbers. We'll find out that h is the gravitational potential. So, this metric tensor stuff is very important, both formally and physically.

We've been cheating here for just a little bit - we've been ignoring our rules above, at least as far as notation goes. Everything we've done is ok, but our notation doesn't make it look ok. So, let's get this problem cleaned up right now.

The V we have been talking about is a vector in space-time, so by our rules it must have a superscript, like Va. The metric tensor h is an object with two subscripts, for example hab. So VhV really means:
    3      3
 S  S  Va * hab * Vb
a=0  b=0

Now we can see explicitly that we followed our rules. There's two superscripts, and two subscripts, and we are always multiplying something with a superscript by something with a subscript. Things with superscripts are called vectors, or contravarients. Things with subscripts are called forms, or covarients. The metric tensor is called a 2-form, because it has two subscripts.

Einstein published a lot of papers, and one day the guy who did his typesetting said to Einstein, "Every time you have an index repeated, you have one of these sum symbols. Why bother?" So, Einstein invented the rule that whenever an index is repeated, it means multiply the terms containing that index, and sum from 0 to 3. Even though it was the typesetter who thought this up, Einstein gets the credit. We add to this the rule that if an index is repeated, one of them must be "up" (superscript) and one must be "down" (subscript). This saves us from writing a bunch of "*" and "S" characters. This is called the Einstein Convention.

    3      3
 S  S  hab * Va * Vb º  habVaVb.
a=0  b=0

Next, we know that the speed of light is the same for all observers. That means that hab must be the same for all observers. So, if the transformation matrix which gets us from one person's frame of reference to another is L, then this must be true:

hab = Lag Lbd hgd

There are a couple different forms of notation for what we're learning here, so I'm going to take this chance to talk briefly about them. Some people use 4-vectors where the index runs from 1 to 4, and the 4th component is i times T, where i is the square root of -1. This notation was invented by Minkowski, one of Einstein's professors when Einstein was a college student. Einstein always said he hated this notation, but in spite of that he sometimes used it. I will never use it. If you use Minkowski notation, you don't need the metric tensor, which somehow make us feel like space-time is more life space, but also makes the transition to General Relativity much harder. Some people use g for the metric tensor, but I will reserve that letter for the metric tensor in curved space-time. The g will remind us that there are gravity fields around. The notation I'm using is the modern notation, but not everyone has gotten with the program yet. Just so you know.

Problems

4.1: Find the minimums and maximums of
    a) 4X - 12
    b) 3X2 + 5X - 64
    c) X3 - 3X2 + 5X/3 -27
    d) 3X5/5 + 2X4 - X3

answer: none, -5/6, (5/3, 1/3), (0, 1/3, -3)
 

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Copyright © 2002 Mark Lawrence. All rights reserved. Reproduction is strictly prohibited.