Penrose diagram

Fig. 1: Penrose diagram as a conformal mapping of the (asymptotically) flat Minkowski spacetime (see Fig. 2). Both diagrams represent the identical causal relationships between events, whereby in both diagrams the world lines of the light rays are with inclined straight lines. The amber-colored curve shows a time-like world line which in the Penrose diagram, as in the Minkowski diagram, always runs within the purple-colored forward light cone. ${\displaystyle \textstyle \pm 45^{\circ }}$

In theoretical physics , a Penrose diagram (named after the mathematician and physicist Roger Penrose ) is a two-dimensional diagram that shows the causal relationship between different points in space-time (see Fig. 1). It is an extension of the Minkowski diagram , wherein the horizontal of the space , the vertical and time are registered, and a light cone indicates the causal relationship between different events in space-time. The metric to be represented in the Penrose diagram is compacted by means of conformal transformation , so that an infinite time and an infinite space coordinate are represented as a two-dimensional finite subspace. This diagram can be used to graph the global structure of the solutions of general relativity (such as black holes and other singularities , event horizons, asymptotic flatness ).

properties

Penrose diagrams or, more correctly, Carter-Penrose diagrams or Penrose-Carter diagrams , since they were developed independently of Roger Penrose at the same time by the physicist Brandon Carter , expand the idea of ​​Minkowski diagrams for any space-time metrics. Here, a two-dimensional physical sub-space with a time and a spatial coordinate and the line element using conformal transformation so in a "unphysical" subspace with displayed that the original finite or infinite intervals of the coordinates are mapped to finite intervals of new coordinates, wherein the representation of the Zero geodesics (rays of light) than with inclined straight lines is preserved. As in the Minkowski diagram, time-like world lines in the Penrose diagram run within the forward light cone (the tangents of the world lines form an angle smaller than one with the vertical axis ). The mapping of finite or infinite intervals to finite intervals is called compactification and allows the analysis of the causal relationships of a metric even in borderline cases . ${\displaystyle \textstyle {\widetilde {\mathcal {M}}}}$ ${\displaystyle x^{0},\,x^{1}}$ ${\displaystyle \mathrm {d} {\widetilde {s}}}$ ${\displaystyle \Omega }$ ${\displaystyle \textstyle {\mathcal {M}}}$ ${\displaystyle \mathrm {d} s=\Omega \,\mathrm {d} {\widetilde {s}}}$ ${\displaystyle \pm 45^{\circ }}$ ${\displaystyle \pm 45^{\circ }}$ ${\displaystyle \lim _{x^{0},\,x^{1}\rightarrow \pm \infty }g(x^{0},x^{1})}$

Construction and description

Fig. 2: Minkowski diagram as the basis for the construction of the corresponding Penrose diagram (see Fig. 1).

The construction of the Penrose diagram has the following scheme (explained using the example of the Minkowski space with Cartesian coordinates, see Fig. 2):

1. The time and space coordinates are selected, which are then transformed:

   ${\displaystyle -\infty <t<\infty ,\quad -\infty <x<\infty .}$

   With is the line element ${\displaystyle \mathrm {d} y=\mathrm {d} z=0}$

   ${\displaystyle \mathrm {d} {\widetilde {s}}^{2}=-\mathrm {d} t^{2}+\mathrm {d} x^{2}.}$

2. These coordinates are transformed into zero coordinates. Zero coordinates have tangent vectors of their coordinate lines with zero length.

   If the original coordinates are the zero coordinates, then applies with ${\displaystyle x^{0},\,x^{1}}$ ${\displaystyle \xi ^{0},\,\xi ^{1}}$

   ${\displaystyle x^{0}=x^{0}(\xi ^{0},\xi ^{1}),\quad x^{1}=x^{1}(\xi ^{0},\xi ^{1}),}$

   that

   ${\displaystyle g_{ij}{\frac {\partial x^{i}}{\partial \xi ^{m}}}{\frac {\partial x^{j}}{\partial \xi ^{m}}}=0\quad \left(i,j,m=0,1\right).}$

   There will be for example

   ${\displaystyle u=t-x,\quad v=t+x}$

   With

   ${\displaystyle -\infty <u<\infty ,\quad -\infty <v<\infty }$

   chosen as zero coordinates and the result is the new line element

   ${\displaystyle \mathrm {d} {\widetilde {s}}^{2}=-\mathrm {d} u\,\mathrm {d} v.}$

3. The new zero coordinates are compacted by mapping their unrestricted intervals to restricted intervals with another coordinate transformation:

   ${\displaystyle p=\arctan v,\quad q=\arctan u}$

   With

   ${\displaystyle -{\frac {\pi }{2}}<p<{\frac {\pi }{2}},\quad -{\frac {\pi }{2}}<q<{\frac {\pi }{2}}}$

   and

   ${\displaystyle \mathrm {d} {\widetilde {s}}^{2}=-{\frac {\mathrm {d} p\,\mathrm {d} q}{\cos ^{2}p\,\cos ^{2}q}}.}$

4. The compactized zero coordinates are transformed back to "unphysical" time and space coordinates (reversal of step 2):

   ${\displaystyle T=p+q,\quad X=p-q}$

   With

   ${\displaystyle -\pi <T+X<\pi ,\quad -\pi <T-X<\pi ,}$

   ${\displaystyle \mathrm {d} {\widetilde {s}}^{2}=-{\frac {1}{\cos ^{2}\left({\frac {T+X}{2}}\right)\,\cos ^{2}\left({\frac {T-X}{2}}\right)}}\left(\mathrm {d} T^{2}-\mathrm {d} X^{2}\right)}$

   and

   ${\displaystyle \mathrm {d} {s}^{2}=-\left(\mathrm {d} T^{2}-\mathrm {d} X^{2}\right)=\left(\cos ^{2}\left({\frac {T+X}{2}}\right)\,\cos ^{2}\left({\frac {T-X}{2}}\right)\right)\mathrm {d} {\widetilde {s}}^{2}=\Omega (T,X)\,\mathrm {d} {\widetilde {s}}^{2}.}$

   Inside of is true and on the edge of is . ${\displaystyle {\mathcal {M}}}$ ${\displaystyle \Omega >0}$ ${\displaystyle {\mathcal {M}}}$ ${\displaystyle \Omega =0}$ ${\displaystyle \left({\text{d. h., }}T+X=\pm \pi {\text{ und }}T-X=\pm \pi \right)}$

The meaning of and in Fig. 1 can be seen from the consideration of the conformal mapping of the straight line from Fig. 2 in the Penrose diagram Fig. 1. Time-like world lines (in Figs. 1 and 2 the pale green coordinate lines with and the amber world line a uniform moving observer) begin in . The space coordinate is finite there and the time coordinate approaches minus infinity. Hence, past time-like infinity is called. The time-like world lines end at the point . There the space coordinate is finite and the time coordinate approaches plus infinity. This point is called future time-like infinity . All time-like world lines run within the pale purple cones of light. Space-like world lines (in Fig. 1 and 2 the pale blue coordinate lines with ) begin and end at the point . There the time coordinate is finite and the space coordinate approaches infinity. This point is called space-like infinity . Light-like world lines, so-called zero geodesics (the paths of light rays or photons, in Figs. 1 and 2 the black with inclined straight lines) begin at the edges and end at the edges . Hence, past zero infinity and future zero infinity are called. ${\displaystyle \textstyle {i}^{-},\,{i}^{+},\,{i}^{0},\,{\mathcal {I}}^{-}}$ ${\displaystyle \textstyle {\mathcal {I}}^{+}}$ ${\displaystyle \textstyle x=\mathrm {const.} }$ ${\displaystyle i^{-}}$ ${\displaystyle i^{-}}$ ${\displaystyle i^{+}}$ ${\displaystyle \textstyle t=\mathrm {const.} }$ ${\displaystyle i^{0}}$ ${\displaystyle \pm 45^{\circ }}$ ${\displaystyle \textstyle {\mathcal {I}}^{-}}$ ${\displaystyle \textstyle {\mathcal {I}}^{+}}$ ${\displaystyle \textstyle {\mathcal {I}}^{-}}$ ${\displaystyle \textstyle {\mathcal {I}}^{+}}$

Application example of the Schwarzschild metric

Fig. 3: Penrose diagram of a static black hole . The basis is the Schwarzschild metric with Kruskal-Szekeres coordinates as the maximum analytical extension of the Schwarzschild solution.

The advantage of the Penrose diagram becomes apparent when analyzing the causal relationships for the Schwarzschild metric. Fig. 3 shows the Penrose diagram of the maximum analytical expansion of the Schwarzschild solution with Kruskal-Szekeres coordinates. Section I is the area outside the event horizon, Section II the area within the event horizon. The horizontal blue line above with is the singularity in the center of the black hole. ${\displaystyle \textstyle r=0}$

For example, the amber world line in Section I describes a spaceship orbiting the black hole in a constant radius orbit . Two probes are deployed in quick succession, heading for the black hole (the two amber-colored lines that branch off to the left). The two probes alternately send radio signals to stay in contact (the red dashed lines), which is possible even after crossing the event horizon . However, signals from the probes do not reach the spaceship after they have crossed the event horizon. However, the spaceship can still send signals to the probes. All world lines with events in Section II within the event horizon end in the singularity. ${\displaystyle \left(r=1{,}05\right)}$ ${\displaystyle \textstyle {\mathcal {H}}^{+}}$

Individual evidence

1. ^ ^{A b} Roger Penrose: Republication of: Conformal treatment of infinity . In: General relativity and gravitation . tape 43 . Springer, 2011, ISSN  1572-9532 , p. 901-922 (English).
2. ↑ Brandon Carter: Complete Analytic Extension of the Symmetry Axis of Kerr's Solution of Einstein's Equations . In: Phys. Rev. Band 141 , no. 4 , 1966, pp. 1242–1247 , doi : 10.1103 / PhysRev.141.1242 , bibcode : 1966PhRv..141.1242C (English).
3. ^ PK Townsend: Black Holes . Lecture notes for a 'Part III' course 'Black Holes' given in DAMTP. Cambridge July 4, 1997, arxiv : gr-qc / 9707012 (English, the course covers some of the developments in Black Hole physics of the 1960s and 1970s).
4. ↑ Frederic P. Schuller: Lecture 23: Penrose Diagrams (International Winter School on Gravity and Light 2015). The WE-Heraeus International Winter School on Gravity and Light 2015, accessed on January 6, 2018 .  
5. ^ Andreas Müller: Penrose diagram. In: Spektrum.de. Retrieved January 6, 2018 .