Let's review the definitions of trigonometric functions. Let's say you have a right triangle, with the small point at the origin. The angle at that point is θ, the horizontal line is x, the vertical line is y, and the diagonal line is r.

cos [theta] = x/r

sin [theta] = y/r

tan [theta] = y/x

Often it is convenient to change the axes of the coordinate system you are using. When you change the axes, the coordinates are different.

In three dimensions, the new coordinates are given by

u cos [alpha] + v cos [beta] + w cos [gamma]

If you use l, m, and n to mean the cosines of the angles between the old and new axes, you have

ul + vm + wn

You can simplify it further by calling the coefficients u_{1}, u_{2}, and
u_{3}, and the cosines of the angles l_{1}, l_{2}, l_{3}

[summation of] u_{i} l_{i} where i = 1, 2, 3

Let's say you have two sets of axes. The first is x_{i}. The second is x_{j}'
and has the same origin as x_{i}, just rotated.

Here I've drawn it as two dimensions but let's say you have the same thing in three dimensions.

How do you change the coordinates from one system to another? Just multiply by the cosine of the angle between them.

cos [theta] = l_{ij}

x_{j}' = (cos [theta]) (x_{i}) = l_{ij}x_{i}

which is the same going back, so

x_{i} = l_{ij}x_{j}'

This formula works not just for position but for displacement, velocity and acceleration. It also works for force as long as the equations of motion are true in all reference frames. You can break this apart into each of the coordinates of the point. In three dimensions, there are three equations, each of which have the following form:

A_{j}' = l_{ij}A_{i} or

A_{i} = l_{ij}A_{j}'

Together, these three equations comprise the components of a vector. This is what defines a vector. The rotation and translation of axes can be drawn like this.

This is why a vector can also be drawn as an arrow.

This is a vector.

x_{j}' = l_{ij} x_{i}

x_{i} = l_{ij} x_{j}'

These are the components of a vector.

A_{j}' = l_{ij} A_{i}

A_{i} = l_{ij} A_{j}'

Let's say you were to have two of each of the variables.

A_{j}' B_{l}' = l_{ij}l_{kl} A_{i}
B_{k}

A_{i}B_{k} = l_{ij}l_{kl} A_{j}'
B_{l}'

These are the individual components, and if you combine them you have this.

k_{jl}' = l_{ij}l_{kl} k_{ik}

k_{ik} = l_{ij}l_{kl} k_{jl}'

This is called a second order tensor. You cannot draw it geometrically like you can a vector but you can deal with it algebraically as easily as a vector. A scalar is a zero order tensor. A vector is a first order tensor. This is a second order tensor, which is also called a dyad. In physics, you frequently use second order tensors. Rarely, you use third and fourth order tensors. You almost never use higher orders, although mathematically you could have any order by just having higher multiples of the number of variables.

If you look at the tensor k_{ik}, you see there are two subscripts, each of which could
have values 1, 2, or 3. Therefore, there are 3^{2} = 3 x 3 = 9 combinations. For this
reason, the tensor can be written as a matrix.

k11 k12 k13 K = k21 k22 k23 k31 k32 k33Even though it's a 3 x 3 matrix, it's a second order tensor, since k has two subscripts.

This is a common notation for tensors. The components k_{11}, k_{22}, and
k_{33} are called the diagonal. Often you have zeroes off the diagonal. The sum of the
components of the diagonal is called the trace or spur.

(K + L)_{ik} = K_{ik} + L_{ik} which is a tensor

K_{jl}' = l_{ik}l_{jl} K_{ik}

l_{ik}l_{jl} = [delta]_{ik} which is the same for any rotation

δ_{ik} is an isotropic tensor which means it is unaltered by rotation. There are no
isotropic tensors of the first order. The only isotropic second order tensors are multiples of
δ_{ik}. The only isotropic third order tensors are multiples of
ε_{ikm}. There are three independent ones of the fourth order which are

[delta]_{ik}[delta]_{mp}

[delta]_{im}[delta]_{kp} + [delta]_{ip}[delta]_{km}

[delta]_{im}[delta]_{kp} - [delta]_{ip}[delta]_{km}

It's possible to multiply a tensor by a vector, and then write it in dyadic notation.

A.K = A_{i}K_{ik} or K.A = K_{ik}A_{k}

If you have a tensor K_{ik} and you exchange rows and columns, you get another
tensor K_{ki}.

If K_{ik} = K_{ki}, the tensor is symmetrical. If
K_{ik} = - K_{ki}, the tensor is antisymmetrical. If K_{ik} is a
tensor, two others are K_{ik} + K_{ki} and K_{ik} -
K_{ki}. The first of these is unaltered if i and k are exchanged and is a symmetrical
tensor. The second has all the components reversed, and is an antisymmetrical tensor. Any tensor
K_{ik} can be written as the sum of a symmetrical and antisymmetrical tensor.

K_{ik} = 1/2(K_{ik} + K_{ki}) + 1/2(K_{ik} -
K_{ki})

The diagonal components of an antisymmetrical tensor K_{ik} must vanish, and since for the others K_{ik} = -K_{ki}, only three independant quantities are needed to specify an antisymmetric tensor, which then takes the form

0 K12 -K31 Kik = -K12 0 K23 K31 -K23 0If K

You now have three values, K_{12}, K_{23}, and K_{31}. However, the vector K_{i}, where i has a value for each of the three axes, x, y, and z, also has three values. Therefore these three values can be used to define a vector. Therefore, an antisymmetrical tensor can be written as a vector.

δ_{ik} is the Kronecker delta, and in Euclidean space, is equal to the metric tensor. ε_{ikm} is the permutation tensor.

Sometimes vectors can be most simply written in tensor notation. For instance, the dot product between two vectors can be simply written as

u . v = u_{i} v_{i}

where repeated indices are summed so that

u_{i} v_{i} = u_{1} v_{1} + u_{2} v_{2} + . . . u_{m} v_{m}

The cross product of two vectors can be written as

u x v = [epsilon]_{ikm} u^{j} v^{k}

A vector is usually portrayed as a 1 x n matrix. If it's portrayed as a m x 1 matrix, it's called a column vector. A two-component complex column vector is called a spinor. It's actually much more complicated than that, but this is sufficient for our present purposes. Here's a typical spinor describing a fermion of arbitrary helicity.

In cosmology, the structure of spacetime is described by a metric. The simplest is flat spacetime which is described by the Minkowski metric, which is as follows.

Minkowski metric, g^{uv}

t - +1 0 0 0 x- 0 -1 0 0 y- 0 0 -1 0 z- 0 0 0 -1This is a second order tensor where the upper left hand value is +1, the other values in the diagonal are -1, and you have zeroes off the diagonal. The Minkowski metric is represented by g

Notice that the time coordinate is +1, and the three spatial coordinates are -1. Some people use an opposite sign convention with the time coordinate -1, and the space coordinates +1. When a tensor used in physics has the form of the first coordinate being the time coordinate, and the remaining three being spatial coordinates, with zeroes off the diagonal, it's called a 4-vector. Technically, it's a second order tensor but you could pretend it's a vector with four components, t, x, y, and z. A 4-vector has the following general form.

A^{u} = (a^{t}, a^{x}, a^{y}, a^{z}) = (a^{0}, a^{1}, a^{2}, a^{3}) = (a^{0}, **a**)

Since the three spatial components are often exactly the same, they are often represented by a single boldface variable a^{1}, a^{2}, a^{3} = **a**. It's also sometimes represented by a variable with an arrow over it.

If you multiply a 4-vector times the Minkowski metric g^{uv}, this has the effect of putting a minus sign on the space part. The result is called a 4-covector, and you switch from superscripts to subscripts.

g^{uv}A^{u} = A_{u} = (a_{t}, -a_{x}, -a_{y}, -a_{z}) = (a_{0}, -a_{1}, -a_{2}, -a_{3}) = (a_{0}, -**a**)

In flat space, the Minkowski metric g^{uv} is the same as the metric tensor, which is actually a function that computes the distance between two points in general space. It's derived from a generalization of the Pythagorean theorem. In curved space, meaning in the presence of a gravitational field, the metric tensor has to be a coordinate dependent field transforming in the right way to keep the proper time invariant.

In Newtonian mechanics, you can change from one inertial frame to another by translations and rotations. This is only within the three spatial coordinates since time is not included. In special relativity, you can change from one inertial frame to another within not just space but spacetime, so all four coordinates are included. You have the translations. You also have a group of transformations called Lorentz transformations, which include rotations and boosts, which are changing from a motionless inertial frame to one moving at a constant velocity. The translations and Lorentz transforms together form the Poincare transforms. If something is unchanged under Lorentz transforms, it's Lorentz invariant. If it's unchanged under Poincare transforms, it's Poincare invariant.

4-vectors are Poincare invariant. That means they are valid in all inertial frames regardless of coordinate system. The components might change from one reference frame to another but the 4-vector is not changed. It's valid for all inertial observers. L^{u}_{v} is the Lorentz transform. It is a transformation tensor that gives relations between different inertial reference frames.

A'^{u} = L^{u}_{v} A^{v}

Lorentz transform, L^{u}_{v}

[gamma] -[gamma]v/c 0 0 [gamma]v/c [gamma] 0 0 0 0 1 0 0 0 0 1where γ is the Lorentz factor, γ = 1/[squareroot of (1 - (v/c)

From this, you get the relation between a given quantity at rest, and then its measured relativistic value.

length L = L_{0}/[gamma]

time t = [gamma]t_{0}

volume V = V_{0}/[gamma]

temperature T = T_{0}/[gamma]

heat Q = Q_{0}/[gamma]

entropy density S = [gamma]S_{0}

Some quantities are unchanged, such as pressure and entropy.

Usually 4-vectors contain only real components. However, it is possible for them to contain imaginary components. An example is the polarization 4-vector.

i = [squareroot of -1] All complex numbers have the form a + bi. For non-imaginary real numbers, b = 0. For imaginary numbers, b is not 0. For pure imaginary numbers a = 0. If you have a number a + bi, the complex conjugate of the number is a - bi. Here is the general form of a 4-vector.

A = (a^{0}, **a**)

If you take into account imaginary numbers, it becomes

A = (a^{0}_{r} + ia^{0}_{i}, **a**_{r} + i**a**_{i})

We use the subscripts r and i to keep track which are real versus imaginary. The complex conjugate of the 4-vector A is as follows.

A = (a^{0}_{r} - ia^{0}_{i}, **a**_{r} - i**a**_{i})

The 4-vector is frequently used in physics. Here is list of some 4-vectors. [h bar] = h/2π where h is Planck's constant. h = 6.626 x 10^{-34} Js. 1 joule = 1 kgm^{2}/s^{2}. c is the speed of light, which is the velocity of a massless particle from the reference frame of a particle with mass. c = 2.99792458 x 10^{8} m/s. m_{0} is the rest mass of a particle, and q_{0} is the charge of a particle. I give the 4-vector, how it's defined, and then its units. These are all Lorentz vectors.

4-vector - definition A = (a_{0}, **a**) - (SI units)

4-position - R = (ct, **r**) - (m)

4-velocity - U = γ(c, **u**) - (m/s)

4-acceleration - A = γ(dγ/dt c, dγ/dt **u** + γ**a**) - (m/s^{2})

4-momentum - P = (E/c, **p**) = γm_{0}(c, **u**) = m_{0}U - (kgm/s)

4-force - F = γ(dE/cdt, f) - (kgm/s^{2})

4-displacement - dR = (cdt, d**r**) - (m)

4-wave vector - K = (w/c, k) = (w/c, wv_{phase}**n**) = 1/[h bar] (E/c, **p**) - (rad/m)

4-current density - J = (cp, **j**) = p_{0}γ(c, **u**) = p_{0}U - (C/m^{2}s)

4-number flux - N = γn_{0}(c, **u**) = n_{0}U - (m/s)

4-polarization - E = (e^{0}, **e**) - (none since it contains imaginary components)

4-gradient - d = d/dx_{u} = (d/cdt, **[nabla]**) - (1/m)

4-vector potential_{EM} - A_{EM} = (V_{EM}/c, **a _{EM}**) - (kgm/Cs)

4-momentum_{EM} - P_{EM} = (E/c + qV_{EM}, **p** + q**a _{EM}**) - (kgm/s, including EM potentials)

4-gradient_{EM} - D = (d/dct + iq/[h bar](V_{EM}c), -**[nabla]** + iq/[h bar]**a _{EM}**) - (1/m, including EM potentials)

If you multiply the 4-gradient by itself you get

dd = (d/cdt, [nabla]).(d/cdt, [nabla]) = (d^{2}/c^{2}dt^{2})-[nabla].[nabla]

This is known as the D'Alambertian operator. It is usually symbolized by a square. It was invented by D'Alambert. Who invented the Fourier series? This might sound like a self-evident question but remember that Monroe was not responsible for the Monroe Doctrine. D'Alambert did work on the vibrating string, and in the process of that he and Euler came up with the Fourier series in 1747. D. Bernoulli got the solution as a sine series in 1753, and what we call the Fourier sine theorem followed from that. Fourier tried to prove it in his work on heat conduction titled, "Analytical Theory of Heat" published in 1822, but the attempt at a proof was inaccurate and almost incoherent. It was actually proven by Dirichlet in 1829.

You use tensors to provide generally valid relations between 4-vectors. Since the components of 4-vectors are altered by change of axes, the components of tensors have to be able to change also, so they can still provide relations between the same 4-vectors. Here is the tensor transformation law.

g'_{[alpha][beta]} = [partial derivative of x^{u} with respect to x'_{[alpha]}] [partial derivative of x^{v} with respect to x'_{[beta]}]g_{uv}

For Cartesian coordinates, it would make no difference if we turn the fractions upside-down.

[partial derivative of x^{u} with respect to x'_{[alpha]}] = [partial derivative of x'^{[alpha]} with respect to x_{u}] = cos [theta]

where θ is the angle of rotation between the two axes. However, this is not in general true.

If you have the 4-vector A_{u}, the components are x_{u} = (ct, -x, -y, -z). These are the covariant components of A_{u}. They transform the same way as a vector. If you multiply by the metric tensor, you get x^{u} = (ct, x, y, z) which are the contravariant components of the 4-vector A^{u}. They transform oppositely of a vector.

A'^{u} = [partial derivative of x'^{u} with respect to x^{v}]A^{v}

A'^{u} = [partial derivative of x^{v} with respect to x'^{u}]A_{v}

It is also possible to contract tensors. If you multiply A^{u} by A_{u}, you get a scalar. The process is called contracting, and the result is called the size or norm of the tensor A^{u}. A^{u} A_{u} is invariant. A^{u} A^{u} would not be a constant. The effects of arbitrary coordinate changes do not cancel unless you have upstairs and downstairs indices.

Let's say you have A^{u} B_{u} = 1, and A^{u} is a 4-vector. Therefore, B_{u} must also be a 4-vector in order for the right hand side to be a scalar. This method of deducing the nature of the quantities in an equation is called manifest covariance.

A coordinate derivative is when a derivative acts on a single coordinate.

The index must be downstairs since the derivative acts on only one coordinate.

[coordinate derivative of u on x^{v}] = [delta] = Diag[1, 1, 1, 1]

Notice we once again see δ, which is an isotropic tensor, meaning that components are the same in all frames. It must exist in order to define the inverse matrix of a tensor. If you have both a partial derivative with upstairs indices and a partial derivative with downstairs indices, you have the D'Alambertian operator.

The scalar product of two 4-vectors is

a_{u} b^{u} = a^{0} b^{0} - a^{1} b^{1} - a^{2} b^{2} - a^{3} b^{3}

The metric tensor is a tool used to raise and lower indices.

A_{u} = g_{uv} A^{v}

For example, in special relativity, the 4-derivatives are

[partial derivative_{u}] = ([partial derivative with respect to ct], **[nabla]**) = ([partial derivative with respect to ct], [partial derivative with respect to x], [partial derivative with respect to y], [partial derivative with respect to z])

[partial derivative^{u}] = ([partial derivative with respect to ct], **-[nabla]**) = ([partial derivative with respect to ct], -[partial derivative with respect to x], -[partial derivative with respect to y], -[partial derivative with respect to z])

so therefore
[partial derivative_{u}]a^{u} = [partial derivative of a^{0} with respect to ct] - = [partial derivative of a^{1} with respect to x] - = [partial derivative of a^{2} with respect to y] - = [partial derivative of a^{3} with respect to z] = [partial derivative of a^{0} with respect to ct] + **[nabla]a**

Therefore conservation laws, such as for the 4-current, can be written as ∂^{u}J_{u} = 0.

Tensor equations with indices in the same relative positions on either side must be generally valid. The equation is called generally covariant. This has nothing to do with the word "covariant" in "covariant components".

The determinant of a tensor is defined as follows.

det(g_{[alpha][beta]}) =

|g11 g12| | | = |g21 g22|g

since in matrices

|a b| | | = ad-bc |c d|

g = -detg_{uv}

Then you have

[integral of] [squareroot of -g]pd^{4}x^{u} = constant

[squareroot of -g]is called the Jacobian. For Minkowski spacetime, g is -1, so the Jacobian is +1 or -1. [squareroot of -g]p is the scalar density. An object formed from a tensor and n powers of the squareroot of -g is called the tensor density of weight n.

If the Jacobian is positive, that is called a proper transformation. If the Jacobian is negative, that is called an improper transformation. In special relativity, a tensor density will transform like a tensor if you restrict yourself to proper transformations. However, with improper transformations, or spatial inversion, tensor densities of odd weights will change sign. These quantities are called pseudotensors. Pseudotensors include pseudovectors and pseudoscalars. The most famous example is the antisymmetric Levi-Civita tensor, ε^{αβγδ}.

[epsilon]_{[alpha][beta][gamma][delta]} = g[epsilon]^{[alpha][beta][gamma][delta]}

It is equal to +1 if the indices are an even permutation of 0123, and -1 if they are an odd permutation. It's 0 for any other value of indices. Here, a "permutation of 0123" is an ordering of the numbers 0, 1, 2, 3 which can be obtained by starting with 0123 and exchanging two of the digits. An even permutation is obtained by an even number of such exchanges, and an odd permutation is obtained by an odd number.

The Christoffel symbol of the first kind is represented in various ways including

These are called components of the affine connection. They are defined by

[ij, k] =

| ij| = | k|g

where g_{uv} is the metric tensor and e_{i} = [partial derivative of r with respect to q^{i}]

Here is the Christoffel symbol expressed in terms of the metric tensor.

The following define the covariant derivative.

DA^{u} = dA^{u} + [Christoffel symbol]^{u}_{[alpha][beta]}A^{[alpha]}dx^{[beta]}

DA_{u} = dA_{u} - [Christoffel symbol]^{[alpha]}_{u[beta]}A_{[alpha]}dx^{[beta]}

You can generalize equations by replacing ordinary derivatives by covariant derivatives. The equation of motion for a particle would become

F^{u} = m(DU^{u}/d[tau])

In order to simplify the notation, partial derivatives are often represented by a comma. Covariant derivatives are often represented by a semicolon.

V^{u},_{v} = [partial derivative of V^{u} with respect to x^{v}] = [partial derivative_{v} V^{u}]

V^{u};_{v} = DV^{u}/[partial derivative]x^{v} = V^{u},_{v} + [Christoffel symbol]^{u}_{[alpha]v}V^{[alpha]},

For instance, for Maxwell's equations, you have F^{uv};_{v} = -u_{0}J^{u}

In physics, you frequently use the energy-momentum tensor T^{uv}. Its value depends on the system. For instance, for a cold fluid with density p_{0} in its rest frame, the only non-zero component is the upper left T^{00} = c^{2}p_{0} In a general frame, you have

T^{UV} = c^{2}p_{0}

[gamma]^2 -[gamma]^2[beta] 0 0 -[gamma]^2[beta] -[gamma]^2[beta]^2 0 0 0 0 0 0 0 0 0 0

where β = v/c and γ = 1/[squareroot of 1 - (v/c)^{2}]. This is similar to the Lorentz transform. You have two powers of γ, one for the change in number density, and one for the relativistic mass increase. For a perfect fluid, the rest-frame T^{uv} is

T^{uv} = Diag[c^{2}[rho], p, p, p]

In a general frame, the energy momentum tensor of a perfect fluid is

T^{uv} = ([rho] + p/c^{2})U^{u}U^{v} - pg^{uv}

In order to explain the fact the Universe is homogeneous and isotropic, we invented inflation which predicted that the universe is flat. Fortunately for us, we observed that the universe is flat since the cosmic microwave background is isotropic. However, then we determined that the amount of matter in the Universe was insufficient to cause the Universe to be flat. We then had to invent some sort of dark energy to flatten the universe. However, this dark energy would also predict that the universe is accelerating. Then Hubble measured the redshift of distant supernovae and determined that the expansion of the Universe really is accelerating, so it's consistent. Primary candidates for the dark energy are the cosmological constant, a rolling scalar field called quintessence, or topological defects.

In 1998, a balloon called Boomerang floated around Antarctica for ten days, and measured the anisotropy of the cosmic microwave background to high precision. The conclusion is that the Universe is very close to flat.

However, on small scales, the Universe is curved. General relativity uses this curvature to explain gravitational effects. The Minkowski metric describes flat spacetime. How do you describe curved spacetime?

In 1884, Erwin A. Abbott wrote a book titled "Flatland" about two dimensional creatures. If a group of such creatures were living on the surface of a large sphere, how would they know it wasn't a plane? Gauss was the first to recognize that you could do this by measuring the angles of a triangle. The angles of a triangle on a plane, or in Euclidean space, always add up to 180^{o} or π radians. The angles of a triangle on a positively curved surface, such as a sphere, add up to more than that. The angles of a triangle on a negatively curved surface, such as a saddle surface, add up to less than that.

This is actually a specific example of a more general case called parallel transport. Imagine that you have a square on a plane. If you move a vector around this square, it's always either parallel or perpendicular to the line it's on. The vector is always facing in the same direction. Let's say you do the same thing with a triangle on a sphere where each of its angles are 90^{o}. You can get the vector pointing in different directions depending on which direction it takes around the triangle.

Take the path integral around the loop. The extent of curvature is the change in the vector which is proportional to both the vector itself, due to rotation, and the distance along the loop, to first order, so that the total change in going around a small loop is given by

[delta]V^{u} = '[R^{u}_{[alpha][gamma][beta]} [path integral ([capital delta]x^{[beta]}dx^{[alpha]} - [capital delta]x^{[alpha]}dx^{[beta]})]]

Since Γ is a function of position in the loop, it can't be taken out of the loop. Make first order expansions of V^{u} and Γ as functions of the total displacement from the starting point Δx^{u}. For Γ, the expansion is a first-order Taylor expansion. For V^{u}, what matters is the first order change in V^{u} due to parallel transport.

There's no first order term because

around the loop. Writing this twice and permutating α and β gives you

where R^{u}_{αγβ} is the Riemann tensor. The Riemann tensor R^{u}_{αγβ} is defined as

which can be written as

The Riemann tensor is a fourth order tensor. However, if you multiply A^{u} by A_{u}, you get a scalar. A^{u} = g_{uv}A_{u}. In this way, you can reduce a fourth order tensor to a second order tensor.

R_{uv} = R^{[gamma]}_{u} R^{u}_{[alpha][gamma][beta]}

R^{uv} is the Ricci tensor. You can do the same thing again and contract it to the curvature scalar R.

R = R^{uv}R_{uv}

The following is the Einstein tensor G^{uv}.

G^{uv} = R^{uv} - (1/2)g^{uv}R

Notice that the Einstein tensor has zero covariant divergence.