Introduction to Special Relativity

[VorpalNeko]

[Warning--work in progress]

The purpose of this article is to introduce the spacetime of special relativity rigorously with no more than high-school level mathematics. The first part will be mainly review of such; it is not intended to be comprehensive, but there merely to point out to the reader what he or she should be aware of, and possibly what should be studied in more detail in class or looked up in school textbooks for more in-depth coverage. Under the United States system, it requires competence in Algebra II and prior mathematical subjects; the exercises should be fairly straightforward but not trivial to those who have just completed this.

Prerequisites
Essential: prior familiarity with elementary algebra, trigonometry, complex numbers, and the exponential/logarithmic functions e^x and ln(x). A review of complex numbers will be provided, but those encountering them for the first time will find it a bit diffucult to get accustomed to.
Bonus: conic sections, matrices, and Euler's formula. Mastery of matrix multiplication is not essential (indeed, completely optional), but it will make things easier.

So far, only the preliminary review of mathematics has been fully written. Nevertheless, I would appreciate some feedback on this. Is the math too hard? Too easy? I feel that every single one of the tools mentioned below is very important to truly understanding the mathematical structure of spacetime in special relativity, but should something be explained in more detail? Would diagrams be helpful? If so, where? Please post feedback here.

Contents
I. Preliminaries
II. Spacetime
III. Lorentz transform


Note on notation: the ^ indicates exponent or superscript, e.g., 2^{x+y} means 2 raised to the power (x+y).  

[VorpalNeko]

Preliminaries
Before delving into relativity, let's first briefly review some concepts from coordinate geometry. Consider the standard (x,y) coordinate plane, which is governed by the Euclidean metric ds² = dx²+dy². This may look fancy to those without familiarity with calculus, but it is really just the Pythagorean theorem. The distance s between two points (x,y) and (a,b) is the length of the line segment between them, which can be thought of as the hypotenuse of a right triangle with sides |x-a| and |y-b|, leading to the familiar distance formula s² = (x-a)² + (y-b)², as per the Pythagorean theorem.

Exercise 1: [Geom] In three dimensions, show that the distance formula between two points (x,y,z) and (a,b,c) extends naturally to s² = (x-a)² + (y-b)² + (z-c)². In differential form, this is written as ds² = dx²+dy²+dz².

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The locus of all points a certain distance r from the origin is simply the circle r² = x²+y²; in parametric form, this is {x = r cos t, y = r sin t}. Rotations through the origin leave this distance fixed, so that points stay on the same circle. If a point (x,y) is rotated counterclockwise by an angle θ, the new coordinates (x',y') are found by
x' = (cos θ)x - (sin θ)y
y' = (sin θ)x + (cos θ)y
Or, in matrix notation, [x';y'] = [cos θ, -sin θ; sin θ, cos θ ][x;y], where a semicolon indicates a new row (see image). Note that rotating every point counterclockwise is equivalent to rotating the coordinate axes themselves in the clockwise direction.

Exercise 2: [Trig] Prove that the above rotation formula works.

Rotations in three dimensions are combinations of three basic rotations: around the x-axis by α, which leaves the x-coordinate the same rotates only in the (y,z) plane; around the y-axis by β, which occurs in the (x,z) plane and leaves y invariant; and finally around the z-axis by γ, which occur in the (x,y) plane. Thus, we can simply "embed" the two-dimensional rotational matrix in a 3x3 that one leaves the corresponding coordinate invariant. In matrix form, rotation about the x-axis takes the form: R_x = [ 1, 0, 0; 0, cos α, -sin α; 0 sin α, cos α ]. All three-dimensional rotations can be decomposed into a sequence of these basic rotations.

Exercise 3: [Matr] Find the three-dimensional rotation matrices R_y and R_z.

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A complex number is defined as a pair of real numbers (x,y), subject to the the following rules of arithmetic:
(x,y)+(a,b) = (x+a,y+b)
(x,y)·(a,b) = (xa-yb,xb+ya)
Complex numbers are commutative (w·z = z·w, w+z = z+w), associative ((w·z)·c = w·(z·c), (w+z)+c = w+(z+c)), and distributive (w·(z+c) = w·z+w·c)), where w = (u,v), z = (x,y) and c = (a,b) are arbitrary complex numbers. Complex numbers can be interpreted as points in the "complex plane"--the pair of numbers (x,y) being the standard Cartesian coordinates.

Exercise 4: [Alg] Assuming the corresponding properties of real numbers (commutativity, associativity, distributivity), verify that complex numbers satisfy them also.

Defining i = (0,1) and identifying every real number x with the complex number (x,0), a complex number z = (x,y) can be written simply as z = (x,y) = (x,0) + (0,1)·(y,0) = x+iy. Since i² = (-1,0) = -1, this form allows for easy computations with complex numbers. For example, (3+2i)+(-1+4i) = 2+6i, while (3+2i)·(-1+4i) = -3+12i-2i+8i² = -11+10i.

[work in progress]

The final tool in our arsenal is complex exponentiation, Euler's formula to be precise: e^{it} = exp(it) = cos t + i sin t. This is a very powerful tool that can be used to prove numerous trigonometric identities. For example, cos(s+t) + i sin(s+t) = e^{is+it} = e^{is}e^{it} = (cos s + i sin t)(cos s + i sin t). Multiplying this out and setting the respective real and imaginary components equal to each other, we have cos(s+t) = cos s cos t - sin s sin t and sin(s+t) = cos s sin t + sin s cos t, which are the familiar angle addition identities.

Exercise 5: [Alg] Show that cos t = [e^{it} + e^{-it}]/2 and sin t = [e^{it}-e^{-it}]/2i.

Following the above form, the hyperbolic trigonometric functions are defined as cosh t = [e^t + e^{-t}]/2 and sinh t = [e^t - e^{-t}]/2. Analogously to cos²t + sin²t = 1, we have cosh²t - sinh²t = 1.

Exercise 6: [Alg] Show that cos(it) = cosh(t), sin(it) = i sinh(t). Conclude that cosh²t - sinh²t = 1.

Just as the parametric equations {x = a cos t, y = b sin t} describe an ellipse, the equations {x = a cosh t, y = sinh t} describe a hyperbola, or rather, one branch of a hyperbola. The other branch is {x = -a cosh t, y = b sinh t}.

Exercise 7: [Alg] Let A be the point (-sqrt(2),0) and B be the point (sqrt(2),0). Show if C is the point (cosh t, sinh t), t any real number, then AC - BC = 2. Conclude that all points (cosh t, sinh t) lie on the same hyperbola with foci A and B.  

[VorpalNeko]

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