Theory of Relativity

Space time:

Space time interval between two events is:

\begin{align*} (\Delta s)^2 = (-c \Delta t)^2 + (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2 \end{align*}

This interval is invariant under change in intertial frame of references (i.e. shift in location or change in velocity of observer).

Using the metric tensor \(\eta\) we can express this more succintly as:

\begin{align*} (\Delta s)^2 = \eta_{\mu, \nu}(\Delta x)^\mu (\Delta x)^\nu \end{align*}

This value is negative for time like separated events (slower than light travel), zero for light-like / null spearated events and positive for spacelike separated events.

Proper time:

We defined proper time (\(\tau\)) between two events to be the time elasped as seen by an observer moving on a straight path between the events:

\begin{align*} (\Delta \tau)^2 = - (\Delta s)^2 = - \eta_{\mu, \nu}(\Delta x)^\mu (\Delta x)^\nu \end{align*}

In space the shortest distance between two points is a straight line, in spacetime the longest proper time between two points is a straight line.

Time along curves:

If we consider a curved path through spacetime \(x^\mu(\lambda)\), parameterized by \(\lambda\) then for timelike paths, the proper time is:

\begin{align*} \Delta \tau = \int \sqrt{ - \eta_{\mu, \nu} \frac {d \Delta x^\mu}{d\lambda} \frac {d \Delta x^\nu}{d \lambda}} d\lambda \end{align*}

These are coordinate transforms from one inertial frame (one coordinate system) to another inertial frame. Such transformation must leave the spacetime distance unchanged.

An Example:

E.g. Translation from coordinates \(\mu\) to \(\mu'\):

\begin{align*} x^\mu \rightarrow x^{\mu'} = \delta_\mu^{\mu'} (x^\mu + a^\mu) \end{align*}

where \(a^\mu\) adds the offsets in \(\mu\) coordinates and \(\delta_\mu^{\mu'}\) is Kronecker delta.

Note that the prime is in the index \(\mu\) not on \(x\) because the underlying object (the spacetime position \(x\)) is not changed. Rather only its representation is changed from one coordinate system to another.

General Form:

Other transformations can be rotation in space coordinates, travel by constant velocity (called boost). In general, if \(\Lambda\) denotes the transfom:

\begin{align*} x^{\mu'} = \Lambda^{\mu'}_\mu x^{\mu} \end{align*}

Where the matrix (tensor) \(\Lambda\) has to keep the spacetime distance unchanged:

\begin{align*} (\Delta s)^2 &= \eta_{\mu, \nu}(\Delta x)^\mu (\Delta x)^\nu \\ &= \eta_{\mu', \nu'}\Lambda^{\mu'}_\mu (\Delta x)^\mu \Lambda^{\nu'}_\nu (\Delta x)^\nu \end{align*}

And thus:

\begin{align*} \eta_{\mu, \nu} = \Lambda^{\mu'}_\mu \Lambda^{\nu'}_\nu \eta_{\mu', \nu'} \end{align*}

The matrices that satisfy above relation are called Lorentz transformations. The set of them forms a group under matrix multiplication, known as Lorentz group. Also denoted as \(O(3,1)\) ¹. This group is nonabelian.

Lorentz group includes rotation and boost transformation and can be specified by 6 parameters. If we also consider translations, then we have the Poincare Group specified with 10 parameters.

At each point \(p\) in spacetime, we associate a set of all possible vectors located at that point. This space is called Tangent space at \(p\), or \(T_p\).

Thus in constrast to flat spacetime, once we consider curvatures we have to stop thinking of vectors as streching from one point to another. Also, now we can't move vectors around the manifold.

• Vector space: Set of vectors
• Vector field: If we associate a vector at each point in spacetime, we have a vector field.
• Tangent bundle: Set of all tangent spaces of a \(n\) dimensional manifold \(M\) can be assembled into a \(2n\) dimensional manifold called the tangent bundle, \(T(M)\).

Tangent vector to a curve:

The tangent vector \(V(\lambda)\) along the curve \(x^\mu(\lambda)\) has the coordinates:

\begin{align*} V^\mu = \frac {d x^\mu} {d\lambda} \end{align*}

And the entire vector is \(V = V^\mu \hat e_{(\mu)}\) where \(\hat e_{(\mu)}\) are the basis vector along the coordinate direction.

Under lorentz transform the coordinates change:

\begin{align*} V^\mu \rightarrow V^{\mu'} = \Lambda^{\mu'}_{\mu} V^\mu \end{align*}

But the vector itself doesn't change:

\begin{align*} V = V^\mu \hat e_{(\mu)} = V^{\mu'} \hat e_{(\mu')} \\ \implies \hat e_{(\mu)} = \Lambda^{\mu'}_\mu \hat e_{(\mu')} \\ \implies \hat e_{(\mu')} = \Lambda^{\mu}_{\mu'} \hat e_{(\mu)} \end{align*}

So, the new basis vectors are obtained from old basis vectors by inverse lorentz transformation. We denote the inverse transform by the same symbol \(\Lambda\) but with the prime and non prime symbols switched.

Dual space:

Space of all linear maps from a vector space to reals also form a vector space called the dual vector space.

We can construct a set of basis vectors \(\hat \theta^{(\nu)}\) in dual space by demanding:

\begin{align*} \hat \theta^{(\nu)} \hat e_{(\mu)} = \delta^\nu_\mu \end{align*}

Dual vectors are called covariant vectors, or also one-forms. The original vectors are called contravariant.

The basis of dual vectors transform under the lorentz transformation (thus called covariant) while the basis of vector space transform under inverse of lorentz transformation (thus called contravariant).

Example: In spacetime, gradient of a scalar function is a dual vector.

Just as a vector is a linear map from a dual vector to real.

A tensor \(T\) of rank \((k,l)\) is a multilinear map from a colelction of \(k\) dual vectors and \(l\) vectors to \(\mathbb R\).

We can construct the basis for a tensor using tensor product and the basis of the (dual) vector spaces.

Examples:

1. Metric tensor

   The metric tensor \(\eta_{\mu\nu}\) is a (0,2) tensor. And we call such tensors inner product because they take two vectors and give a scalar.

   Once we have inner product, we can defined orthogonality and norm.

2. Kronecker delta

   Can be viewed as a map from vectors to vectors. It is a rank (1,1) tensor.

These two are rather special tensors in that for metric tensor the coordinates are unchanged under any coordinate system in flat spacetime, and for Kronecker delta the coordiantes are unchanged in even curved spacetime.

Raising and Lowering Indices:

The metric and inverse metric can be used to raise and lower indices on tensors. i.e.

\begin{align*} V_\mu = \eta_{\mu\nu}V^\nu \\ \omega^\mu = \eta^{\mu\nu}\omega_\nu \\ T^{\alpha \beta \mu} = \eta^{\mu\nu} T^{\alpha \beta}_{\nu} \end{align*}