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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.

Bibliography

1. Hermann Minkowski, Raum und Zeit, 1909