Holographic special relativity: observer dependent geometry

Derek Wise, FAU Erlangen
Title: Holographic special relativity: observer space from conformal geometry 
PDF of the talk (600k) Audio [.wav 38MB] Audio [.aif 4MB] 

by Sean Gryb, Radboud University

Introduction

In Roman mythology, Janus was the god of gateways, transitions, and time, whose two distinct faces are depicted peering in opposite directions, as if bridging two different regions (or epochs) of the Universe. The term “Janus-faced” has come to mean a person or thing that simultaneously embodies two polarized features, and the Janus head has come to represent the embodiment of these two distinct features into one.

In this talk (based off the paper [1]), Derek Wise explores the possibility that spacetime itself might be Janus-faced. He explores an intriguing relationship between the structure of expanding spacetime and the scale-invariant description of a sphere. What he finds is a mathematical relationship providing a bridge between these two Janus faces that distinctly represent events in the Universe. This bridge is remarkably similar to the picture of reality proposed by the holographic principle and, in particular, the AdS/CFT correspondence where, on one side, there is the usual spacetime description of events and, on the other, there is a way to imprint these events onto the 3-dimensional boundary of this spacetime.

Aside from providing an alternative to spacetime, Derek's picture may even help illuminate the deeper structures behind a recent formulation of general relativity (GR) called Shape Dynamics, which I will come to at the end of this post. But to begin, I will try to explain Derek's result by first giving a description of the spacetime aspect of the Janus face and then describe how a link can be established to a completely distinct face, which, as we will see, is a description of events in the Universe that is completely free of any notion of scale. The key points of the discussion are summarized beautifully in the depiction of Janus given below, by Marc Ngui, who has provided all the images for this post. The diagram shows how, as I will describe later, events seen by observers in spacetime can be described by information on the boundary. I encourage the reader to revisit this image as its main elements are progressively explained throughout the text.

Relativity, Observers, and Spacetime

In 1908, Hermann Minkowski made a great discovery: Einstein's new theory of Special Relativity could be cast into a beautiful framework, one that Minkowski recognized as a kind of union of space and time. In his own words: “space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.”[2] To understand what Minkowski meant, let's go back to 1904-1905 in order to retrace the discoveries that spawned Minkowski's revolution.

Relativity concerns the way in which different observers organize information about ‘when’ and ‘where’ events take place. Einstein realized that this system of organization should have two properties: i) it should work the same way for each observer, and ii) it should involve a set of rules that allows different observers to consistently compare information about the same events. This means that different observers don't necessary need to agree on when and where a particular event took place, but they do need to agree on how to compare the information gathered by different observers. Einstein expressed this requirement in his principle of relativity, to which he gave primary importance within physical theories. The key point is that relativity is fundamentally a statement about observers and how they collect and compare information about events. Minkowski's conception of spacetime comes afterwards, and it comes about through the specific mathematical properties of the rules used to collect and compare the relevant information.

To try to understand how spacetime works, we will use a slightly more modern version of spacetime than the one used by Minkowski — one with all the same essential properties as the original, but which can accommodate the observed accelerated expansion of space. This kind of spacetime was first studied by Willem de Sitter, and is named de Sitter (dS) spacetime after him. It has the basic shape depicted by the blue grid in the Janus image above. Because this space is curved, it is most convenient to describe it by putting it into a larger dimensional flat space (just like the 2D surface of a sphere depicted in a 3D space). This means that we can label events in this spacetime by 5 numbers: 4 space components, labeled (x, y, z, w) and one time component, t, that obey the relation

x² + y² + z² + w² - t² = ℓ².    (1)
This restriction (which serves as the definition of this spacetime) means that the 4 space components are not all independent. Indeed, the single constraint above removes one independent component, leaving the 3 space directions we know and love. The parameter ℓ is related to the cosmological constant and dictates how fast space is expanding. Adjusting its value changes the shape of the spacetime as illustrated in the figure below.

The middle spacetime in blue depicts a typical dS spacetime. Increasing the parameter ℓ decreases the rate of expansion so that, if ℓ → ∞, the spacetime barely expands at all and looks more like the purple cylinder on the right. The opposite extreme, when ℓ → 0, is the yellow light cone, which is named that way because the space is expanding at its maximum rate: the speed of light. This extreme limit will be very important for our considerations later.

Although this model of spacetime is dramatically simplified, it remarkably describes, to a good approximation, two important phases of our real Universe: i) the period of exponential expansion (or inflation), which we believe took place in the early history of our Universe, and ii) the present and foreseeable future. Different observers compare the labels they attribute to each event by performing transformations that leave the form of (1), and thus the shape of dS spacetime, unchanged. Because of this property, these transformations constitute symmetries of dS spacetime. Since the transformations that real observers must use to compare information about real events just happen to correspond to spacetime symmetries, it is no wonder that the notion of spacetime has had such a profound influence on physicists’ view of reality. However, we will shortly see that these rules can be recast into a completely different form, which tells a different story of what is happening.

Spacetime's Janus Face

We will now see how the symmetries of observers in dS spacetime can be rewritten in terms of symmetries that preserve angles, but not necessary distances, in space. In particular, all information about scale is removed. In mathematics, these are called conformal symmetries. This means that different observers have a choice when analyzing information that they collect about events: either they can imagine that these events have taken place in dS spacetime, and are consequently related by the dS symmetries; or they can imagine that these events are representing information that can be expressed in terms of angles (and not lengths), and are consequently related by conformal symmetries.

To understand how this can be so, consider the very distant future and the very distant past: where dS spacetime and the light cone nearly meet. This extreme region is called the conformal sphere because it is a sphere and also because it is where the dS symmetries correspond to conformal symmetries.

In fact, any cross-section of the light cone formed by cutting it with a spatial plane (as illustrated in the diagram below) is a different representative of the conformal sphere since these different cross-sections will disagree on distances but will agree on angles. Although the intersection looks like a circle (represented in dark green), it is actually a 3-dimensional sphere because we have cut out 2 of the spatial dimensions (which we can't draw on a 2 dimensional page).

To see how events on this 3d sphere can be represented in a scale-invariant way on a 3d plane, we can use a handy technique called a stereographic projection. The stereographic projection is often used for map drawing where the round earth has to be drawn onto a flat map. One of its key properties, namely that it preserves angles, means that maps drawn in this way are useful for navigating since an angle on the map corresponds to the same angle on the Earth. It is precisely this property that will make the stereographic projection useful for us here.

To perform a stereographic projection, imagine picking a point on a sphere, which we can interpret as the location of a particular observer on the sphere (represented by an eye in the diagram below), and call this the South Pole. Now imagine putting a light on the North Pole and letting it shine through the space that the sphere has been drawn in. Suppose our sphere is filled with points. Then, the shadow of these points will form an image on the plane tangentially to the sphere on the South Pole. The picture below illustrates what is going on. Points on the sphere are represented by stars and the yellow rays indicate how their image is formed on the plane.

It is now a relatively straightforward mathematical exercise to show that the symmetries of the light cone represent transformations on the plane that may change the size of the image, but will preserve the angles between the points. Thus, the symmetries of the cone can be understood in terms of the conformal symmetries of this plane.

If we now move our cross-section ever further into the future or the past, then the dS spacetime begins to resemble more and more the light cone. Thus, if we can represent arbitrary events in dS spacetime by information imprinted on two cross-sections in the infinite future and infinite past, then these events can be represented in terms of the images they induce onto our projected planes, and we have obtained our objective.

There is a simple way that this can be done. Imagine taking, as shown in the figure below, an arbitrary event in dS spacetime and drawing all the events in the distant past that could affect things that happen at this point (this region is a finite portion of the spherical cross-sections because no disturbance can travel faster than the speed of light). The result is a 2 dimensional spherical region, called the particle horizon indicated by the red regions in the diagram below, which grows steadily over time. You can think of this region as the proportion of dS spacetime that is visible at any particular place. In fact, you can use the relative size of this region as an indication of the time at which that event occurs. Because this is a notion of time that exists solely in terms of quantities defined in the distant past, it will transform under conformal symmetries. To give an idea of what this looks like, the motion of an observer from some point in the distant past to a new point in the distant future is represented by a series of concentric spheres, starting at the initial point and then spreading out to eventually cover the whole sphere. The diagram below shows how this works. The different regions (a,b,c,d) represent progressively growing regions corresponding to progressively later times.

In this way, you can map information about events in dS spacetime to information on the conformal sphere. In other words, the picture of reality that one gets from Einstein's theory of Special Relativity is a story that can be told in two very different ways. In the first way, there are events which trace out histories in spacetime. ‘Where’ and ‘when’ a particular event takes place depends on who you are, and the information about these events can be transformed from one observer to another via the global symmetries of spacetime. In the new picture, it is the information about angles that is important. ‘Where’ and ‘how big’ things are depends on your point of view and the information about particular events can be transformed from one observer to another using conformal transformations.

From Special to General Relativity

We have just described how to relate two very different views of how observers can collect information about the world. Until now, we have only been considering homogeneous spaces: i.e., those that look the same everywhere. The class of observers we were able to consider was resultantly restrictive. It was Einstein's great insight to recognize that the same mathematical machinery needed to describe events seen by arbitrary observers could also be used to study the properties of gravity. The machinery in question is a generalization of Minkowski's geometry, named after Riemann.

In order to describe Riemannian geometry, it is easiest to first describe a generalization of it (which we will need later anyway), and then show how Riemannian geometry is just a special case. The generalization in question is called Cartan geometry, after the great mathematician Élie Cartan. Cartan had the idea of building general curved geometries by modelling them off homogeneous spaces. The more general spaces are constructed by moving these homogeneous spaces around in specific ways. The geometry itself is defined by the set of rules one needs to use to compare vectors after moving the homogeneous spaces. These rules split into two different kinds: those that change the point of contact between the homogeneous space and the general curved space and those that don't. These different moves are illustrated for the case where the homogeneous space is a 2D sphere in the diagram below.
 The moves that don't change the point of contact (in the case above, this corresponds to spinning about the point of contact without rolling) constitute the local symmetries of the geometry and could, for example, correspond to what different local observers would see (in this case, spinning observers versus stationary ones) when looking at objects in the geometry. Einstein exploited this kind of structure to implement his general principle of relativity described earlier. The moves that change the point of contact (in the case above, this means rolling the ball around without slipping) give you information about the curved geometry of the general space. Einstein used a special case of Cartan geometry, which is just Riemannian geometry, where the homogenous space is Minkowski space. He then exploited the analogue of the structure just described to explain an old phenomenon in a completely new way: gravity. In the process, he produced one of our most radical yet successful theories of physics: General Relativity. The figure below shows how the different kinds of geometry we've discussed are related.

Now, consider what happens when we substitute, as we did in the last section, Minkowski's flat spacetime for de Sitter's curved, but still homogenous, spacetime. We can still describe gravity, but in a way that naturally includes a cosmological constant. However, the conformal sphere is also a homogeneous space. Moreover, as we described earlier, the symmetries of this homogeneous space can be related to the dS symmetries. This suggests that it might be possible to describe gravity in terms of a Cartan geometry modelled off the conformal sphere.

From the Conformal Sphere to Shape Dynamics?

Cartan geometries modelled on the conformal sphere are called conformal geometries because the local symmetries of these geometries preserve angles, and not scale. Although we have laid out a procedure relating the model space of conformal geometries to the model space of spacetimes with a cosmological constant, it is quite another thing to rewrite gravity in terms of conformal geometry. This is, in part, because the laws governing spacetime geometry are complicated and, in part, because our prescription for relating the model spaces is also not straightforward, since it relates local quantities in spacetime to non-local quantities in the infinite future and past. Nevertheless, this exciting possibility provides an interesting future line of research. Furthermore, there are other hints that such a description might be possible.

Using very different methods, it is possible to show that General Relativity is actually dual to a theory of evolving conformal geometry [3]. However, the kind of conformal geometry used in this derivation has not yet been written in terms of Cartan geometry (which makes use of slightly different structures). This new way of describing gravity, called Shape Dynamics, is perhaps making use of the interesting relationship between spacetime symmetries and conformal symmetries described here. Understanding exactly the nature of the conformal geometry in Shape Dynamics and its relation to spacetime could prove valuable in being able to understand this new way of describing gravity. Perhaps it could even be a window into understanding how the quantum theory of gravity should work?

• [1] D. K. Wise, Holographic Special Relativity, arXiv:1305.3258 [hep-th].
• [2] H. Minkowski, The Principle of Relativity: A Collection of Original Memoirs on the Special and General Theory of Relativity, ch. Space and Time, pp. 75–91. New York: Dover, 1952.
• [3] H. Gomes, S. Gryb, and T. Koslowski, Einstein gravity as a 3D conformally invariant theory, Class. Quant. Grav. 28 (2011) 045005, arXiv:1010.2481 [gr-qc].

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