Minkowski
Minkowski is the unique maximally symmetric flat spacetime.
1. History
Minkowski space traces its origins all the way back to the first investigations in special relativity in the late 19th century, but its status as a geometric entity was not solidified until Hermann Minkowski's paper "Raum und Zeit" (Space and time) of 1908.
2. Topology
Minkowski space has the standard topology $\mathbb{R}^n$, with the basic differentiable structure inherited from it. Variants of Minkowski space include a wide variety of subsets and quotients, most commonly Minkowski space with points removed ($\mathbb{R}^n \setminus S$), the half-Minkowski space($\mathbb R \times \mathbb R^+$), the Minkowski square ($I \times I$), the spacelike cylinder , the timelike cylinder, the torus, the Moebius strip, the Klein bottle, the non-time orientable Minkowski cylinder ($(\mathbb R \times S) / (T \times {0, \pi})$).
3. Metrics and coordinates
3.1. Cartesian
The standard coordinates of Minkowski space are the cartesian coordinates $(t, x_1, x_2, x_3, ...) \in \mathbb R^n$, obtained by using the identity map on the manifold atlas : $(\mathbb{R}^n, \operatorname{Id})$. The metric is
$$ds^2 = -dt^2 + \sum_{i = 1}^n (dx^i)^2$$3.2. Spherical
In four dimension, Minkowski space admits the following metric in spherical coordinates
$$ds^2 = -dt^2 + dr^2 + r^2 (d\theta^2 + \sin^2 \theta d\varphi^2) $$which is singular for $r = 0$ and $\theta = 0$.
3.3. Cylindrical
$$ds^2 = -dt^2 + d\rho^2 + \rho^2 d\varphi^2 + dz^2$$Which is singular for $r = 0$.
Relations to other coordinates
Minkowski :
\begin{eqnarray} x &=& \rho \cos \varphi\\ y &=& \rho \sin \varphi \end{eqnarray}Spherical :
\begin{eqnarray} r &=&\sqrt{\rho^2 + z^2}\\ \theta &=& \arctan (\frac{z}{\rho}) \end{eqnarray}3.4. Rindler
The Rindler coordinates represent the coordinates of an observer accelerating constantly in the same direction. In the case of an observer accelerating in the $x$ direction, this will be
$$ds^2 = -(\alpha x) dt^2 + \sum_{i = 1}^n (dx_i)^2$$Relations to other coordinates
Cartesian :
\begin{eqnarray} t &=& x \sinh(\alpha t)\\ x &=& \sqrt{x^2 - t^2} \end{eqnarray}4. Tensor quantities
4.1. Cartesian coordinates
Determinant
$$g = -1$$Inverse metric
$$g^{\mu\nu} = \operatorname{diag}(-1,1,1,1)$$Christoffel symbols
$${\Gamma^\rho}_{\mu\nu} = 0$$Riemann tensor
$${R^\rho}_{\mu\nu\sigma} = 0$$Ricci tensor
$$R_{\mu\nu} = 0$$Ricci scalar
$$R = 0$$5. Symmetries
Minkowski space is maximally symmetric. The most common way to write its Killing vectors are (in $4$ dimensions)
- Translations :
- $\partial_t$
- $\partial_x$
- $\partial_y$
- $\partial_z$
- Rotations :
- $x \partial_y - y \partial_x$
- $y \partial_z - z \partial_y$
- $z \partial_x - x \partial_z$
- Boosts :
- $t \partial_x - x \partial_t$
- $t \partial_y - y \partial_t$
- $t \partial_z - z \partial_t$
Minkowski space is also invariant under various other transformations, such as space reversal and time reversal.
This corresponds to the full Lorentz group $\operatorname{O}(1, n-1)$.
6. Stress-energy tensor
Minkowski space is a vacuum spacetime, with stress-energy tensor $T_{\mu\nu} = 0$ and $\Lambda = 0$. Assuming matter fields of strictly positive energy, all matter fields are themselves required to be of zero stress-energy tensors.
7. Curves
7.1. In Cartesian coordinates
The geodesic equation simply reduces to
$$\ddot x(\tau) = 0$$With the general solution :
$$x(\tau) = v_0 \tau + x_0$$8. Equations
8.1. In Cartesian coordinates
Wave equation
The wave equation simply reduces to the classical d'Alembert equation
$$-\partial_t^2 \Psi(x) + \sum_i \partial_{i}^2 \Psi(x) = 0 $$For any null 4-vector $k$, there is a solution of the form
$$\Psi(x) = f(g(x, k))$$The type of function $f$ can be is fairly broad but it will usually be sufficient for our purpose to take it to be an $L^2$ function on every spacelike hypersurface.
Harmonics :
$$f_k = e^{ig(k,x)}$$ $$\Psi(x) = \int dp a(p) e^{i(\vec k \cdot \vec x - \omega_k t)}$$9. Causal structure
Minkowski space has the mildest causal structure of all spacetimes. It is globally hyperbolic and has no singularities.
10. Asymptotic structure
10.1. Conformal compactification
The standard conformal compactification of Minkowski space is the following :
With the range $\rho \in \left -\pi, \pi \right]
$$ds^2 = -dT^2 + \frac{1}{4} \left[ d\rho + \sin^{n-2}(\rho) d\Omega \right] $$10.2. Horizons
11. Energy conditions
| NEC | WEC | DEC | SEC |
|---|---|---|---|
| ANEC | AWEC | ADEC | ASEC |
12. Limits and related spacetimes
As Minkowski space is unique, there is no limiting case.
13. Misc.
Minkowski space has a vector space structure, and there is a canonical isomorphism between itself and its tangent spaces.