In theoretical physics, a Penrose diagram (named after mathematical physicist Roger Penrose) is a two-dimensional diagram capturing the causal relations between different points in spacetime through a conformal treatment of infinity. It is an extension (suitable for the curved spacetimes of e.g. general relativity) of the Minkowski diagram of special relativity where the vertical dimension represents time, and the horizontal dimension represents a space dimension. Using this design, all light rays take a 45° path. ( c = 1 ) {\displaystyle (c=1)} . Locally, the metric on a Penrose diagram is conformally equivalent to the metric of the spacetime depicted. The conformal factor is chosen such that the entire infinite spacetime is transformed into a Penrose diagram of finite size, with infinity on the boundary of the diagram. For spherically symmetric spacetimes, every point in the Penrose diagram corresponds to a 2-dimensional sphere ( θ , ϕ ) {\displaystyle (\theta ,\phi )} . Challenge 3: Keep 3 sets. Represent Set as rectangles with strokeWidth equal to 15. (Hint: you'll also want to set strokeColor to sampleColor(0.5, "rgb") or similar.)

The distortion becomes greater as we move away from the center of the diagram, and becomes infinite near the edges. Because of this infinite distortion, the points i − and i + actually represent 3-spheres. All timelike curves start at i − and end at i +, which are idealized points at infinity, like the vanishing points in perspective drawings. We can think of i + as the “Elephants’ graveyard,” where massive particles go when they die. Similarly, lightlike curves end on \(\mathscr{I}

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Let's say we are making a diagram of things in your house. Then the domain of objects that we are working with includes everything that is in your house. Subsequently, any items that can be found in your house (furniture, plants, utensils, etc.) can be thought of as specific types of objects in your household domain. First, we need to define our domain of objects because Penrose does not know what is in your house or what a chair is. In addition to defining the types of objects in your domain, you will need to describe the possible operations in your domain. For example, you can push a chair, or sit on a chair, which are operations related to a chair. Penrose, Roger (15 January 1963). "Asymptotic properties of fields and space-times". Physical Review Letters. 10 (2): 66–68. Bibcode: 1963PhRvL..10...66P. doi: 10.1103/PhysRevLett.10.66. While Penrose diagrams share the same basic coordinate vector system of other spacetime diagrams for local asymptotically flat spacetime, it introduces a system of representing distant spacetime by shrinking or "crunching" distances that are further away. Straight lines of constant time and straight lines of constant space coordinates therefore become hyperbolae, which appear to converge at points in the corners of the diagram. These points and boundaries represent conformal infinity for spacetime, which was first introduced by Penrose in 1963. [1] The process of creating a Penrose diagram is similar to our intuitive process of analog diagramming. 🎉 ​

For example, we could group the plants in your house based on the number of times they need to be watered on a weekly basis. Then we would have visual clusters of elements.

The coordinates of the Penrose diagram are compactified along the null directions just as in the Minkowski case: Challenge 2: Keep 3 sets. Represent Set as squares with side length equal to 50.0. (Hint: there is no Square type, but you don't need one.) In general, for each diagram, you will have a unique .substance file that contains the specific instances for the diagram, while the .domain and .style files can be applied to a number of different diagrams. For example, we could make several diagrams in the domain of Linear Algebra that each visualize different concepts with different .substance files, but we would preserve a main linearAlgebra.domain file that describes the types and operations that are possible in Linear Algebra, and select from any of several possible linearAlgebra.style files to affect each diagram's appearance. For example, if we want Penrose to know that there are objects of type plant, we would do type Plant or type plant. We normally capitalize type names. ❓ What's the most fundamental type of element in Set Theory? (hint: the name gives it away.) ​

Hawking, Stephen & Ellis, G. F. R. (1973). The Large Scale Structure of Space-Time. Cambridge: Cambridge University Press. ISBN 978-0-521-09906-6. See Chapter 5 for a very clear discussion of Penrose diagrams (the term used by Hawking & Ellis) with many examples.begin {axis }[axis lines=none, xmin=-.1,xmax=1.1,ymin=-1.2,ymax=1.2,width=0.5 \textwidth,height=0.8 \textwidth] For the tensor diagram notation, see Penrose graphical notation. Penrose diagram of an infinite Minkowski universe, horizontal axis u, vertical axis v Two lines drawn at 45° angles should intersect in the diagram only if the corresponding two light rays intersect in the actual spacetime. So, a Penrose diagram can be used as a concise illustration of spacetime regions that are accessible to observation. The diagonal boundary lines of a Penrose diagram correspond to the region called " null infinity," or to singularities where light rays must end. Thus, Penrose diagrams are also useful in the study of asymptotic properties of spacetimes and singularities. An infinite static Minkowski universe, coordinates ( x , t ) {\displaystyle (x,t)} is related to Penrose coordinates ( u , v ) {\displaystyle (u,v)} by: Drawings commonly require explorations and various attempts with colors, sizes, and compositions. The same concept can be visualized in a number of different styles. Recall how you would normally create a diagram of a concept using a pen or pencil. It will most likely involve the following steps:

In this section, we will introduce Penrose's general approach and system, talk about how to approach diagramming, and explain what makes up a Penrose diagram.

Recall that a .domain file defines the possible types of objects in our domain. Essentially, we are teaching Penrose the necessary vocabulary that we use to communicate our concept. For example, recall our example of a house from the introduction. Penrose has no idea that there are objects of type "plant" or "furniture" in a house, but we can describe them to Penrose using the type keyword. d'Inverno, Ray (1992). Introducing Einstein's Relativity. Oxford: Oxford University Press. ISBN 978-0-19-859686-8. See Chapter 17 (and various succeeding sections) for a very readable introduction to the concept of conformal infinity plus examples. It follows naturally that our mathematical domain is Set Theory. Let's take a look at our .domain file. We define the substances in our diagram by declaring their type and variable name in our .substance.