Visualizing Four Dimensions in Specialand General Relativity

Magdalena Kersting

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Mathematics of Space and Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Four-Dimensional Spacetime and the Special Theory of Relativity . . . . . . . . . . . . . . . . . . . . 4Gravity, Geometry, and the General Theory of Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Black Holes and Numerical Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Revealing Spacetime Through Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Imagination and Artistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Analogies and Metaphors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Spacetime Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Relativistic Ray Tracing and First-Person Visualizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Gravitational Lensing and Astrophysical Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Numerical Simulations of Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Virtual, Augmented, and Mixed Reality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Abstract

Modern physics unfolds on the stage of four-dimensional spacetime. Grap-pling with century-old ideas of space and time, Albert Einstein revolutionizedour understanding of the cosmos by merging space and time into a four-dimensional entity that takes an active role in shaping the laws of physics.While experiments have repeatedly confirmed Einstein’s theories, the abstract

M. Kersting (�)Department of Physics, University of Oslo, Oslo, Norway

ARC Centre of Excellence OzGrav, Swinburne University of Technology, Hawthorn, Australiae-mail: magdalena.kersting@fys.uio.no

© Springer Nature Switzerland AG 2020B. Sriraman (ed.), Handbook of the Mathematics of the Arts and Sciences,https://doi.org/10.1007/978-3-319-70658-0_120-1

1

2 M. Kersting

character of this physical knowledge contradicts the common sense of many.Based on the physics of relativity and the mathematics of differential geometry,scientists have developed visualizations and representations of spacetime tomake Einstein’s ideas more intelligible. This chapter explores the links betweenthe mathematics of space and time and our historic struggle to visualizethese concepts. Technology serves as the lens to unpack the fruitful interplaybetween mathematics, physics, and arts that has shaped our understanding ofspacetime. Linking mathematical concepts with physical intuition and artisticvision, imaginative thinkers developed representations that led from simplespacetime diagrams and analogies to powerful numerical simulations and vir-tual environments that allow exploring the extreme physics of black holesand gravitational waves. Visualizations of spacetime continue to be an activefield of research that is driven by interdisciplinary efforts to understand thecosmos.

Keywords

Spacetime · Special relativity · General relativity · Numerical relativity ·Technology · Visualization · Imagination · Interdisciplinary

Introduction

Before dealing further with the special theory of relativity, I want to try to convey to thereader what is involved in the new phrase “spacetime,” because that is, from a philosophicaland imaginative point of view, perhaps the most important of all the novelties that Einsteinintroduced. (Russell, 1925)

For many physicists, the theory of relativity is one of the most beautifulmathematical descriptions of our cosmos. In a stroke of genius, Albert Einsteinmerged space and time into spacetime, a four-dimensional entity that takes anactive role in shaping the laws of physics. Einstein’s field equations govern thedynamics of spacetime physics. The gravitational field influences the flow of time,and massive cosmic collisions let space vibrate and ripple. Yet, the abstract natureof four-dimensional spacetime has challenged the imaginative faculties of scientistsand laymen alike.

Even though the theory of relativity firmly established a new scientific worldviewand had vast philosophical repercussions, Einstein’s ideas of curved space andwarped time are still perceived as elusive, counterintuitive, and at times evenparadoxical (Kersting and Steier, 2018). In their quest to understand our universe,generations of scientists have faced the challenge of visualizing four-dimensionalspacetime. Based on the physics of relativity and the mathematics of differentialgeometry, scientists have developed representations that illustrate features of atheory whose geometry continues to confound.

This chapter explores the links between the mathematics of space and timeand our historic struggle to visualize these concepts. To unpack this struggle,

Visualizing Four Dimensions in Special and General Relativity 3

technology serves as the lens through which we view possibilities of visualizationsin the domain of relativity. According to Martin Heidegger (1977), technology isa way of seeing and understanding the world. In line with this view, the historicstruggle to obtain representations of spacetime translates to revealing unknownfeatures of spacetime in order to push the boundaries of our understanding. Thisquest has propelled the development of relativity from its historic origins intothe age of scientific visualizations and numerical relativity. Investigations into thephysical properties of spacetime and its mathematical structure continue to motivatescientists in their wish to understand the Universe.

Mathematics of Space and Time

The views of space and time which I want to present to you arose from the domain ofexperimental physics, and therein lies their strength. Their tendency is radical. From nowonwards space by itself and time by itself will recede completely to become mere shadowsand only a type of union of the two will still stand independently on its own. (Minkowski,1909)

In the following, we briefly present the mathematics of spacetime to introducespecial and general relativity as the stage where century-old ideas of space and timeinteract with new technologies. Although Einstein published his theory of relativitymore than a century ago, much progress has only been made in recent years. As asystem of coupled, nonlinear partial differential equations, Einstein’s equations arenotoriously difficult to solve. Consequently, scientists study relativistic phenomenathrough numerical methods. Numerical relativity provides sophisticated visualiza-tion techniques that allow visualizing dynamic spacetime phenomena such as binaryblack hole mergers and gravitational waves (Baumgarte and Shapiro, 2010).

In line with the historic development of spacetime, our presentation in this sec-tion follows a progression from the mathematical foundations of four-dimensionalspacetime and special relativity to the formulation of Einstein’s field equations andgeneral relativity before we turn to numerical relativity and present-day challengesof visualizing spacetime. The exposition of this chapter addresses graduate studentsor advanced undergraduate students in physics or mathematics without priorknowledge of relativity and differential geometry. We only expect readers to befamiliar with linear algebra, basic group theory, and classical mechanics. Followingthe conventional formalism, we provide a brief introduction to the central conceptsof special and general relativity. A more detailed introduction to these topics andmore examples can be found in numerous textbooks, for example (Baumgarte andShapiro, 2010; Carroll, 2003; Guidry, 2019; Hartle, 2003). Once the mathematicalstage of special and general relativity is set, our focus shifts to visualizations ofrelativistic phenomena. Specifically, we show how technology and interdisciplinaryhave played important roles in understanding and visualizing spacetime.

It is interesting to trace the historic origins of spacetime because the pioneers ofrelativity were very much led by visual-geometric thinking. Many of these pioneers,among them Hermann Minkowski and Henri Poincaré, were mathematicians who

4 M. Kersting

translated their mathematical insights into a physical theory of space and time(Galison, 1979). Minkowski went so far as to claim that the theory of relativitydiscovered by Einstein could have been formulated by mathematicians already inthe late nineteenth century when it became popular to study geometries via theircharacteristic groups of transformations. Others agreed with this judgment. In hispopular scientific account of relativity, Bertrand Russell (1925) stated that there waslittle in the theory of relativity that could be regarded as physical laws or physics inthe strict sense.

Yet, relativity is a branch of physics, not of pure mathematics. Of course,the conclusions and implications of the theory could not have been obtainedwithout the aid of abstract mathematical reasoning (Durell, 1926). Still, Minkowski(1909) repeatedly stressed the importance of experimental physics in deducingthat “three-dimensional geometry becomes a chapter in four-dimensional physics.”For Minkowski and others, geometry was not merely an abstraction from physicallaws but constituted the very nature of natural phenomena (Petkov, 2014; Walter,2014). Thus, experimental results and observations played a pivotal role in thedevelopment, interpretation, and visualization of spacetime physics: mathematicsand physics have always been tightly interwoven in the domain of relativity.

Four-Dimensional Spacetime and the Special Theory of Relativity

According to Minkowski (1915), “the world in space and time is, in a certainsense, a four-dimensional non-Euclidean manifold.” The goal of this introductionto the mathematics of space and time is to explain Minkowski’s statement and tosupplement it with a precise mathematical formalism. We introduce some necessarynotation first. Spacetime is a four-dimensional set in which we combine the time t

and the position vector �x ∈ R3 into a four-vector

xμ = (t, �x) = (x0, x1, x2, x3)

that labels a point (sometimes called “event”) in spacetime. We use small Greekletters to denote spacetime indices and small Roman letters to denote spatial indicesof such four vectors, i.e., μ, ν = 0, 1, 2, 3 and i, j = 1, 2, 3. Since most of ourequations will be equations between 4 × 4 matrices, it is convenient to adopt suchan index notation where superscripts and subscripts label elements of vectors andmatrices. We define the spacetime interval, often also called the line element by

ds2 = −dt2 + dx2 + dy2 + dz2. (1)

As is common in relativity, we use natural units in which the speed of light c is setequal to 1. Some authors prefer to define the line element with the c-factor included:

Visualizing Four Dimensions in Special and General Relativity 5

ds2 = −cdt2 + dx2 + dy2 + dz2

This notation makes explicit that c is a conversion factor between space and time.Thus, in our notation that uses natural units, time is measured in length units.

The importance of the spacetime interval in relativity is that it is invariant underchanges of inertial reference frames, i.e., reference frames that are either at rest orthat move at a constant velocity. One can write the spacetime interval in a morecompact form by introducing a 4 × 4 matrix, the Minkowski metric, via

ημν =

⎛⎜⎜⎝

−1 0 0 00 1 0 00 0 1 00 0 0 1

⎞⎟⎟⎠ .

Here, the two lower indices label the components of the metric. Introducingthe standard summation convention, in which indices appearing both assuperscripts and subscripts are summed over, we can now express the lineelement as

ds2 = ημνdxμdxν. (2)

Equations (1) and (2) are therefore equivalent; equation (2) just makes use ofthe conventional summation formalism that one can find in many textbooks.Minkowski space is the four-dimensional real vector space of all points xμ

equipped with the Minkowski metric. Minkowski space presents a special caseof a four-dimensional non-Euclidean manifold. In the next section, we properlyintroduce the notion of a differentiable manifold. In this section, we take acloser look at the non-Euclidean nature of Minkowski space. The geometry ofMinkowski space is determined by the Poincaré group, the group of all lineartransformations that leave the spacetime interval invariant. The Poincaré groupis a ten-parameter non-abelian group that consists of translations, rotations, andboosts. Boosts illustrate the non-Euclidean nature of Minkowski space quite wellbecause they can be thought of as rotations between space and time. Thus,in contrast to a four-dimensional Euclidean space, Minkowski space does notfeature an absolute notion of simultaneous events. It depends on the observerand the choice of coordinates whether or not two events occur at the sametime.

The set of spatial rotations and boosts form the so-called Lorentz groupunder matrix multiplication. If �μ

ν is a Lorentz transformation (i.e., a 4 × 4matrix labeled by two indices μ, ν) that corresponds to the coordinate transforma-tion

6 M. Kersting

xμ �→ xμ′ = �μ′νx

ν,

then we can express the condition that �μν be a Lorentz transformation as

follows

ηρσ = �μ′ρ�ν′

σ ημ′ν′ . (3)

One simple example of a boost in the x-direction is given by

�μ′ν =

⎛⎜⎜⎝

cosh(φ) − sinh(φ) 0 0− sinh(φ) cosh(φ) 0 0

0 0 1 00 0 0 1

⎞⎟⎟⎠ (4)

where φ is the boost parameter. It is a simple exercise for the reader to check thatthis matrix does, indeed, satisfy equation (3). We get back to this specific boosttransformation in the next section when we visualize Lorentz transformations viaspacetime diagrams.

There is a close analogy between the Lorentz group and the rotation group inthree-dimensional Euclidean space. Geometrically, Lorentz transformations can beconstructed as a rotation about the origin of coordinates in a four-dimensional vectorspace with three real axes and one imaginary axis. Of course, it is challenging tobuild intuitions of such a four-dimensional geometry. Yet, from a physics perspec-tive, it is relatively straightforward to interpret Lorentz transformations. Lorentztransformations are those transformations that leave the form of the equation of apropagating light wave invariant. The structure of Minkowski spacetime is thereforea natural consequence of special relativity that postulates that there exists a fixedvelocity c = 1 at which electromagnetic waves propagate in vacuum.

Publishing his special theory of relativity, Einstein (1905) embraced the impli-cations of the light postulate: while absolute motion, absolute space, and absolutetime do not have any physical significance, it is the relation between these items inform of the spacetime interval that has physical significance. More than anythingelse, special relativity is therefore a theory of the structure of spacetime.

Gravity, Geometry, and the General Theory of Relativity

Special relativity replaced the Euclidean geometry of three-dimensional space withthe geometry of a four-dimensional non-Euclidean differentiable manifold. In thelast section, we encountered already first examples of non-Euclidean features ofspacetime, for example, the observer-dependence of simultaneity. In this section,we turn to the definition of differentiable manifolds and show how gravity can beinterpreted as the curved geometry of four-dimensional spacetime. Our introductionfollows Carroll (2003) and Wald (1984), and the interested reader can find moredetails in those two textbooks. The knowledge of the structure of spacetime served

Visualizing Four Dimensions in Special and General Relativity 7

as the point of departure for Einstein’s discovery of general relativity (Petkov andAshtekar, 2014). In general relativity, four-dimensional spacetime is no longer flatbut identified with a curved manifold. Despite the need to introduce a certainamount of mathematical formalism to discuss four-dimensional spacetime curvaturein a quantitative way, the basic notion that the curvature of spacetime leads togravitational phenomena is quite simple and elegant (Carroll, 2003).

A differentiable manifold (or just a manifold) generalizes the notion of Euclideanspaces. Manifolds are sets that look locally like flat (Euclidean) space Rn but thatcan have a different (non-Euclidean) global geometry. To define manifolds we firstformalize the notion of a coordinate system (sometimes also called chart) and anatlas. For a given set M , a coordinate system consists of a subset U ⊂ M and aone-to-one map φ : U −→ Rn such that φ(U) ⊂ Rn is open. A (smooth) atlas is anindexed collection of charts (Uα, φα) such that

1. the union of all Uα covers M ,2. if Uα ∩ Uβ = ∅, then the map (φα ◦ φ−1

β ) is smooth and takes points in φβ(Uα ∩Uβ) ⊂ Rn onto an open set φα(Uα ∩ Uβ) ⊂ Rn.

We can think of an atlas as a collection of coordinate systems that describeindividual regions of M and that are smoothly related on their overlap. Since a set M

can be covered in many different ways using different combinations of coordinatesystems, an atlas is not unique. Two atlases of M are equivalent if their union isalso an atlas of M . We define a maximal atlas as the equivalence class containingthe given atlas under this equivalence relation, i.e., the atlas containing all possiblecoordinate systems consistent with this atlas. The maximal atlas of a set M is unique.The notion of a maximal atlas finally allows us to define a manifold. A differentiablemanifold is a set M together with a maximal atlas of M . With this definition we havenow formalized Minkowski’s statement that the world in space and time is a four-dimensional non-Euclidean manifold.

What makes manifolds and four-dimensional spacetime interesting is of coursenot their locally flat character but their global geometry that often deviates starklyfrom flat geometry. We therefore turn to the metric tensor (often just called themetric) as our next object of study. As a generalization of the flat Minkowski metricημν , the metric tensor gμν is one of the most important objects in general relativitybecause it completely determines the geometry of spacetime. Among other things,the metric tensor allows defining geodesic curves, i.e., the spacetime analogue ofshortest distance paths, and it provides the basic building blocks to construct thecurvature tensor. To define the metric, we first introduce tangent spaces which arereal vector spaces associated with each point of a given manifold M .

Intuitively, we can think of tangent spaces as the spaces of all possible directionsin which one can tangentially pass through a point x on M . Tangent vectors,elements of this vector space, are thought of as the velocity of a curve passingthrough x. However, this intuitive picture makes use of the fact that we often thinkof a manifold as embedded into an ambient vector space Rm (Fig. 1). It is desirable,though, to define the notion of a tangent space based solely on intrinsic propertiesof the manifold. To that end, we define a tangent vector as an equivalence class

8 M. Kersting

Fig. 1 A pictorial representation of the tangent space of a single point on a sphere. Intuitively, thetangent space contains all possible directions in which one can tangentially pass through a pointon the sphere. (Credits: Alexwright at English Wikipedia [Public domain])

Fig. 2 The tangent space TxM of a manifold M at the point x ∈ M is defined via tangent vectorsv ∈ TxM along a curve γ (t) that passes through x. (Credits: derivative work by McSush; theoriginal uploader was TN at German Wikipedia [Public domain])

of curves passing through a point x ∈ M in the following way. Pick a coordinatechart φ : U −→ Rn where U is an open subset of M containing x. For two givencurves γ1, γ2 : (−1, 1) −→ M that satisfy γ1(0) = x = γ2(0) and for whichφ ◦ γ1, φ ◦ γ2 : (−1, 1) −→ Rn are smooth functions, we say that γ1 and γ2 areequivalent at 0 if and only if the derivatives of φ ◦ γ1 and φ ◦ γ2 coincide at 0. Thisdefines an equivalence relation on the set of all curves γ : (−1, 1) −→ M thatsatisfy γ (0) = x and φ ◦ γ : (−1, 1) −→ Rn is smooth. We define tangent vectorsof M at x to be such equivalence classes of curves and we denote the tangent vectorby γ ′(0).

In a next step, we define the tangent space of M at x, denoted by TxM , to be theset of all tangent vectors at x (Fig. 2). The vector space operation on TxM is givenby a map dφx : TxM −→ Rn that we define via

dφx(γ′(0)) := d

dt

[(φ ◦ γ )(t)

]∣∣∣t=0

Visualizing Four Dimensions in Special and General Relativity 9

One can check that these definitions do not depend on the choice of coordinatechart φ : U −→ Rn. It is natural to think of tangent vectors as directionalderivatives, and indeed, this identification allows us to specify a basis of the tangentspace for a given coordinate system. In classical multivariable calculus, for a vectorv ∈ Rn, one defines the directional derivative at a point x ∈ Rn by

(Dvf )(x) := d

dt[f (x + tv)]

∣∣∣t=0

=n∑

i=1

vi ∂f

∂xi(x) ∀f ∈ C∞(Rn).

To extend this notion to more general manifolds, we think of a tangent vector v ∈ M

as the initial velocity of a curve γ , i.e., v = γ ′(0). Then we define the covariantderivative of v ∈ TxM at x ∈ M by

Dv(f ) := (f ◦ γ )′(0) ∀f ∈ C∞(M).

Using this identification, we can now construct a basis of the tangent space in thefollowing way. For a given coordinate system φ = (x1, . . . , xn) : U −→ Rn with

p ∈ U , we define a basis((

∂∂xi

)p

)n

i=1of TpM by

∀i ∈ {1, . . . , n},∀f ∈ C∞(M) :( ∂d

∂xi

)p

:= (∂i(f ◦ φ−1))(φ(p)).

Using the coordinate system φ = (x1, . . . , xn) : U −→ Rn, we can write everytangent vector v ∈ TpM as a linear combination of the basis tangent vectors(

∂∂xi

)p

∈ TpM in the following way:

v =n∑

i=1

v(xi)( ∂

∂xi

)p

Since tangent spaces are attached to specific points x ∈ M , it is useful to look at setsof tangent vectors with exactly one tangent vector at each point on the manifold.Such vector fields are generalizations of the velocity field of a particle moving inspace. A vector field X attaches to every point x ∈ M a tangent vector X(x) ∈TxM in a smooth manner. More precisely, using the above decomposition of tangentvectors as linear combinations of basis vectors, we say that a vector field X on M

is smooth if for any coordinate system φ = (x1, . . . , xn) : U −→ Rn, we have, forany point a ∈ U ,

X(a) =n∑

i=1

f i(a)( ∂

∂xi

)a

10 M. Kersting

for some smooth functions f i : U −→ R. We are now finally in the positionto define the metric tensor g as a family of bilinear, symmetric, nondegeneratefunctions

gx : TxM × TxM −→ R, x ∈ M

such that for every pair of vector fields X, Y on M, the map

x �→ gx(X(x), Y (x))

defines a smooth function M −→ R. We write g(X, Y )(p) = gp(X, Y ) for p ∈M . For a given coordinate system (x1, . . . , xn) on an open set U ⊂ M and the

corresponding basis of vector fields(X1 = ∂

∂x1 , . . . , Xn = ∂∂xn

), the metric g has

components gij = g(

∂∂xi ,

∂∂xj

). These n2 functions form the entries of an n x n

symmetric matrix. In the context of relativity where our manifold corresponds tofour-dimensional spacetime, the metric is usually denoted gμν following standardindex notation. For the rest of this chapter, we will focus on the specific case offour-dimensional spacetime. Since the metric tensor is nondegenerate, there existsan inverse metric tensor gμν such that gμνgνσ = δ

μσ . Here, δμ

σ denotes the Kroneckerdelta

δμ′μ =

{0 if μ = μ′,1 if μ = μ′.

We think of the metric as an inner product on the tangent space at each point of themanifold that varies smoothly from point to point. This construction allows us, inparticular, to define local notions of angles or lengths of curves on our manifold.We will soon see how the metric tensor replaces the classical gravitational fieldin Einstein’s theory of gravity, thus linking differential geometry to the physicsof general relativity. However, before we turn to gravity, we first formalize theidea of curvature that is intimately linked to the metric tensor. The basic idea ofthe construction is to express the intrinsic curvature of a manifold by assigninga curvature tensor to each point of the manifold. This tensor measures the extentto which the metric is not locally isometric to the metric of Euclidean space. Thecurvature tensor relies on a geometric object that connects nearby tangent spacesby relating tangent vectors in these different spaces. This object is the Christoffelconnection λ

μν defined by

λμν = 1

2gλσ (∂μgνσ + ∂νgσμ − ∂σ gμν).

Using the Christoffel connection, we define the Riemann curvature tensor Rρσμν

by

Visualizing Four Dimensions in Special and General Relativity 11

Rρσμν = ∂μ ρ

νσ − ∂ν ρμσ +

ρμλ

λνσ −

ρνλ

λμσ .

The Riemann curvature tensor completely encodes the curvature of spacetime.One can show that if a coordinate system exists in which the components of themetric are constant, the Riemann tensor will vanish. Conversely, if the Riemanntensor vanishes, one can always construct a coordinate system in which the metriccomponents are constant (Carroll, 2003). This relation between the metric and thecurvature tensor shows again why the Minkowski metric ημν does indeed describethe flat spacetime of special relativity. Let us now turn to the curved spacetime ofgeneral relativity and to Einstein’s elegant idea that the curvature of spacetime leadsto gravitational phenomena.

Generally, there are two parts to describe the physics of gravitational interactions:first, how the gravitational field influences the movement of matter and, second, howmatter determines the gravitational field. Classically, there are two equations thatcorrespond to each part. One equation describes the acceleration a of an object in agravitational potential φ

a = −∇φ,

and the second equation expresses the gravitational potential in terms of the matterdensity ρ and Newton’s gravitational constant G

∇2φ = 4πGρ.

In what physicist Max Born (1968) described as “the greatest feat of humanthinking about nature, the most amazing combination of philosophical penetrationphysical intuition, and mathematical skill,” Einstein (1915) identified the curvatureof spacetime with the gravitational potential. It is the curvature of spacetime thatacts on matter; energy and matter, in turn, influence the geometry of spacetime.Einstein’s field equations describe this dynamic interplay and make up the heart ofgeneral relativity:

Rμν − 1

2Rgμν = 8πGTμν

Here, the left-hand side of the equation encapsulates the curvature of spacetimewhere Rμν = Rλ

μλν is the contraction of the Riemann curvature tensor and R =gμνRμν is the trace of this contraction. The right-hand side of the equation describesthe energy and matter content of the spacetime region. Tμν is the stress-energy tensorthat describes the density and flux of energy and momentum in spacetime. Just as themass density is the source of the gravitational field in classical physics, the stress-energy tensor is the source of the gravitational field in general relativity. G is againthe gravitational constant that also appears in the classical equation of gravity.

12 M. Kersting

Black Holes and Numerical Relativity

One of the most exotic and spectacular consequences of general relativity is theexistence of black holes (Guidry, 2019). Black holes are regions in spacetime thatbecome so curved that they trap light. The mathematical description of a black holeis surprisingly simple and given by the Schwarzschild metric, the unique sphericallysymmetric vacuum solution of Einstein’s field equations. In spherical coordinates{t, r, θ, φ}, the Schwarzschild solution reads

ds2 = −(

1 − 2GM

rdt2

)dt2 +

(1 − 2GM

rdt2

)−1dr2 + r2d�2 (5)

where d�2 is the metric on a unit two-sphere

d�2 = dσ 2 + sin2(θ)dφ2.

The Schwarzschild metric describes empty spacetime around a gravitating objectsuch as the Sun or the Earth. In this context, the constant M is interpreted as the massof this object. The most interesting property of the Schwarzschild solution is thatthe line element in equation (5) contains two singularities at r = 0 and r = 2GM ,that is, two points at which the metric is not defined. The quantity RS = 2GM iscalled the Schwarzschild radius, and it plays a central role in the description of thespacetime geometry around a black hole (Guidry, 2019). The Schwarzschild radiusis often called the event horizon because it defines the radius at which spacetimecurvature is so strong that the escape velocity exceeds the velocity of light. Sincelight is trapped inside this radius, the region interior to RS is called a black hole.While RS is a coordinate singularity that can be removed by choosing differentcoordinates, r = 0 is a physical singularity whose meaning is hard to interpret bothmathematically and physically.

By their very nature, black holes are difficult to visualize and pose severe chal-lenges to our ability to find representations of these extreme spacetimes. Moreover,black hole phenomena that are astrophysically realistic such as the spiraling andmerging of two black holes are only accessible through numerical simulations. Inthese scenarios, the exact form of spacetime is no longer known, and all analyticalmethods break down (Varma, 2019). Studying black holes is one of the main goals ofnumerical relativity, which is the art and science of developing computer algorithmsto solve Einstein’s equations for astrophysically realistic systems (Baumgarte andShapiro, 2010). In fact, much insight into the physics of space and time has onlybeen gained in the age of numerical computing. For most of the twentieth century,computer technology was not advanced enough to support numerical solutions toEinstein’s equations. By and large, progress in numerical relativity was impededby lack of computers with sufficient memory and computational power to performwell-resolved calculations of realistic spacetime scenarios (Anninos et al., 1995).Not only are the equations difficult to solve because they are multidimensional,nonlinear, coupled partial differential equations in space and time, but they present

Visualizing Four Dimensions in Special and General Relativity 13

additional complications due to the existence of singularities. Even today numerical-relativity simulations of merging black holes might take months of computationaltime on powerful supercomputers (Varma et al., 2019).

Historically, first steps toward numerical approaches were taken by Arnowittet al. (1962) who proposed a decomposition of spacetime back into separated spaceand time parts. Such a decomposition in form of sequences of three-dimensionalspace-like hypersurfaces allows reformulating Einstein’s field equations as an initialvalue problem suitable for numerical solutions. However, computer technologyat that time was still in its infancy, and it was only in the 1980s that Starkand Piran (1985) attempted the first realistic calculations of a rotating blackhole. In the following three decades, both computers became more power, andresearchers developed new computational techniques to avoid the interior spacetimesingularities in the course of their simulations (Baumgarte and Shapiro, 2010). Forexample, the so-called excision technique does not evolve portions of spacetimeinside the event horizon surrounding the singularity of a black hole; rather, one onlysolves the equations outside of the event horizon (Alcubierre and Brugmann, 2001).Alternatively, the puncture method factors the solution of Einstein’s equationsinto an analytical part that contains the black hole singularity and a numericallyconstructed part which is free of singularities (Brandt and Bruegmann, 1997). Thebreakthrough in numerical relativity came in 2005 when Pretorius (2005) achievedthe first successful numerical relativity simulation of the merging of two black holes.Based on an improved version of the puncture method that allowed punctures tomove through the coordinate system, accurate long-term evolutions of two blackholes orbiting each other became possible for the first time.

Today, numerical relativity and visualizing the results of such relativistic sim-ulations are fields which still offer many challenging and exciting problems forresearchers (Ruder et al., 2008; Varma et al., 2019). Often, sophisticated visual-izations of binary back holes serve as a reality check of numerical simulations.For example, field lines of spacetime curvature elucidate the nonlinear dynamics ofcurved spacetime in merging black-hole binaries (Owen et al., 2011). Nowadays,the use of computers and methods of computer graphics has greatly increasedthe potential and scope of visualization of spacetime, not only for the purposeof disseminating science but also as a tool for researchers to develop an intuitiveunderstanding of their results (Ruder et al., 2008; Varma et al., 2019).

Revealing Spacetime Through Technology

No man can visualize four dimensions, except mathematically. We cannot even visualizethree dimensions. I think in four dimensions, but only abstractly. The human mind canpicture these dimensions no more than it can envisage electricity. Nevertheless, they are noless real than electromagnetism, the force which controls our universe, within, and by whichwe have our being. (Einstein 1929)

Technology has always provided crucial perspectives on how we navigate,experience, and understand the world around us. Thus, it seems natural to take

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technology as the lens through which we view possibilities of visualizations inthe domain of relativity. The point of entrée into our explorations of relativisticvisualizations is the observation that technology, at its heart, is a way of seeingand understanding the world. Indeed, according to Heidegger (1977), technologymust be understood as “a way of revealing.” In line with Heidegger’s famousanalysis of technology, we understand technology as a particular approach oforienting ourselves to the world so that reality is brought forth through the act ofrevealing. This perspective on technology aligns well with the task of visualizingfour-dimensional spacetime to reveal the basic structure of our cosmos.

In the regimes of everyday life, the non-Euclidean character of our worlddoes not become apparent. Visualizations can help us reeducate our intuitionswith respect to a world that is best described by a four-dimensional curvedmanifold (Chandler, 1994). More generally, visualizations can change the wayscientists think about the world. In contrast to the common belief that visualizingand analyzing scientific phenomena are separate tasks, these two go hand inhand when scientists explore abstract concepts and phenomena (Goodman, 2012).Often, emerging technologies such as skills, techniques, and knowledge createnew access to known concepts, allow for deeper insights, and facilitate physicalintuition.

By adopting a broad definition of technology as a way of revealing, experiencing,and understanding reality, we explore how century-old ideas of space and timehave interacted with evolving tools and technical expertise in relativity. For theremainder of this chapter, we explore some of these attempts with a particular focuson how different visualizations revealed new insights into the structure of space andtime.

Imagination and Artistry

Imagination lies at the heart of science and art because it involves interactingwith situations that are different from the present reality. Before the age ofcomputers, scientists had to rely on imagination, artistry, and analogies to visu-alize space and time. Doing so, they took on a technological approach towardunderstanding four dimensions because they facilitated their ways of seeing byentering into different relations with the abstract concept of spacetime. Orient-ing themselves to particular spacetime features, scientists, mathematicians, andartists were able to reveal spacetime in new ways by relying on their imagina-tion.

It takes imaginative efforts to explore the physical implications of a mathematicalstructure that asks us to let go of absolute space and universal time, concepts thatseem integral to our understanding of the world (Woodhouse, 2014). Imagininginvolves efforts to bring forth the unexperienced, immaterial, or non-present bybringing together disparate aspects of the object of imagination into a perceivablewhole (Steier and Kersting, 2019). Literature served as an excellent medium tobring together our everyday experience of motion and the existence of a universal

Visualizing Four Dimensions in Special and General Relativity 15

speed limit. Relativistic effects become noticeable when one approaches velocitiesclose to the speed of light. An early literary example that explores such relativisticmovement to imagine the strange nature of Minkowski space is George Gamow’spopular scientific book “Mr Tompkins in Wonderland” (Gamow, 1940). In thisbook, the title character Mr Tompkins dreams of an alternative world wherethe speed of light is set to a lower numerical value. In this dream world, therelativistic concepts of space, time, and motion are accessible to our ordinarysenses:

A single cyclist was coming slowly down the street and, as he approached, Mr Tompkins’seyes opened wide with astonishment. For the bicycle and the young man on it wereunbelievably shortened in the direction of the motion, as if seen through a cylindrical lens.The clock on the tower struck five, and the cyclist, evidently in a hurry, stepped harder onthe pedals. Mr Tompkins did not notice that he gained much in speed, but, as the result ofhis effort, he shortened still more and went down the street looking exactly like a picturecut out of cardboard.

Gamow chose a literary approach to explore how an observer perceives relativis-tic length contraction, the phenomenon that a moving object’s length is measuredto be shorter than its length as measured in the object’s own rest frame. Eventhough Gamow failed to recognize the difference between measuring and visuallyperceiving relativistic phenomena (Kraus, 2008), he was an early popularizer of theidea that the division of spacetime into space and time is a choice we make for ourown purposes, not something intrinsic to the world.

Visualizations of black holes serve as another example of the fruitful interplaybetween artistic, imaginative, and mathematical explorations of spacetime. As oneof the flashier fruits of Einstein’s century-old insights, black holes are deeplyembedded in the popular imagination and show up regularly in movies, books,and TV shows (Cole, 2019). Presumably no other scientific concept has fueledthe imagination of scientists and artists in the same way because black holes pushthe known laws of physics to their limits (Fig. 3). According to astrophysicistChandrasekhar (1992) ,“the black holes of nature are the most perfect macroscopicobjects there are in the universe: the only elements in their construction are ourconcepts of space and time.”

Naturally, artists imagined possibilities of black hole physics long before super-computers were able to simulate spacetime around black holes. Figures 4 and 5show artistic illustrations of black holes that reveal several features of our conceptsof space and time around these exotic objects. Figure 4 shows an artistic illustrationof two spiraling black holes that orbit one another in a plane. These two blackholes have different orientations relative to the overall orbital motion of the system.In the artist’s conception, swirling gas around the black holes helps visualize thespiraling motion of the merging process. Also Fig. 5 uses swirling gas in form ofan accretion disk to visualize a supermassive black hole at the core of a young,star-rich galaxy. This illustration also shows how supermassive black holes candistort space around them in a phenomenon called gravitational lensing. We explorevisualizations of binary systems and gravitational lensing in more detail in the nextsection.

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Fig. 3 According to physicist John Wheeler (1998), “[the black hole] teaches us that space can becrumpled like a piece of paper into an infinitesimal dot, that time can be extinguished like a blown-out flame, and that the laws of physics that we regard as ‘sacred,’ as immutable, are anything but.”In this visualization, a digital artist has imagined a scene that pushes the known laws of physics totheir limit: a black hole is about to swallow a neutron star creating a trail of swirling gas during theprocess. Credit: Carl Knox/OzGrav, https://www.ozgrav.org

Analogies and Metaphors

Closely related to imagination and artistry is the generation of analogies andmetaphors that play a central role in scientific practice, thought, and creativity(Kapon and DiSessa, 2012). By constructing similarities between two objects,analogies and metaphors enable new ways of revealing aspects of the world. Notonly does science produce these figures of thought for subsequent developmentin the arts. The technological importance of analogies and metaphors lies in theirability to serve as frames through which we perceive and make meaning of theworld (Hesse, 1953).

The rubber sheet analogy is one of the most prevailing popular visualizations ofspacetime and presumably also one of the oldest representations in the context ofgeneral relativity. According to Einstein’s field equations, spacetime tells matterhow to move, and matter tells spacetime how to curve (Wheeler, 1998). Therubber sheet analogy captures this basic mechanism of gravitational attraction bycomparing the four-dimensional fabric of spacetime to a two-dimensional stretchedrubber sheet. The dynamic interplay between the movement of massive objects andthe curvature of spacetime is illustrated by heavy objects placed on the rubber sheetthat warp the sheet and that influence the movement of other objects on the sheet(Fig. 6).

Visualizing Four Dimensions in Special and General Relativity 17

Fig. 4 An artist’s conception shows two merging black holes which will ultimately spiral togetherinto one larger black hole. In this illustration, the black holes are spinning in a nonaligned fashion,which means they have different orientations relative to the overall orbital motion of the pair.(Credit: Courtesy LIGO/Caltech/MIT/Sonoma State (Aurore Simonnet))

Even though Einstein did not explicitly use the rubber sheet analogy to explainhis field equations, he did use the analogy of a soft cloth to illustrate boundaryproblems in his gravitational theory. In a letter to his colleague Willem de Sitter, hewrote in 1917: “Our problem can be illustrated with a nice analogy. I compare thespace to a cloth floating (at rest) in the air, a certain part of which we can observe.This part is slightly curved similarly to a small section of a sphere’s surface.”(Hentschel, 1998)

Although the rubber sheet analogy is simple, it exhibits great explanatory powerby revealing the dynamic interplay between gravity, space, and time. Illustrating thegeometric and universal nature of gravity, the analogy can visualize orbital motions,curved space, and photon trajectories in intuitive ways (Kersting and Steier, 2018).At the same time, the explanatory scope of the analogy is limited because itsimplifies four-dimensional spacetime to a two-dimensional spatial fabric. Theanalogy suggests that space curves into an unseen dimension while neglectingthe role of warped time all together. Since the human mind cannot visualizefour dimensions and much less curvature of a four-dimensional entity, analogieswill always have limitations in their ability to reveal the nature of spacetime(Fig. 7).

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Fig. 5 This artist’s conception illustrates a supermassive black hole (central black dot) at thecenter of a young galaxy. As the drawing shows, gas swirls around a black hole in what is calledan accretion disk. Space and light around the black hole are distorted which is illustrated by thewarped stars behind the black hole. (Credit: Courtesy NASA/JPL-Caltech)

Spacetime Diagrams

In the early days of relativity, imagination and analogies served as one routetoward finding useful visualizations of four-dimensional spacetime. Another routeemployed two-dimensional representations in form of spacetime diagrams. As oneof the earliest scientific visualizations of Minkowski space, spacetime diagramsrevealed the causal structure of our four-dimensional world. To this day, spacetimediagrams are a routine way of introducing students to the physics of space and timebecause they are useful tools to reason with relativistic kinematics.

Historically, Minkowski had to persuade physicists of the value of his spacetimeapproach since his treatment of spacetime was couched into the abstract language offour-dimensional vector calculus (Walter, 2014). To provide visual aids, Minkowskiemployed spacetime diagrams as geometrical interpretations that offered diagram-matic readings of the Lorentz transformations and illustrated the light-cone structureof spacetime.

The underlying idea is to suppress two spatial dimensions and only consider thetime axis t and one spatial axis x at right angles (Fig. 8). The starting point for thedevelopment of special relativity was the observation that the speed of light is finite.

Visualizing Four Dimensions in Special and General Relativity 19

Fig. 6 The rubber sheet analogy is a two-dimensional representation that visualizes how our Sunand Earth warp spacetime. The green grid illustrates that spacetime is a malleable fabric on whichmassive objects move along paths that are determined by the geometry of spacetime. (Credit:LIGO/T. Pyle)

Fig. 7 The rubber sheet analogy is a ubiquitous visualization of spacetime but it hasa limited explanatory scope. (Credit: https://xkcd.com/895/, Creative Commons Attribution-NonCommercial 2.5 License)

This very same observation gives guidance when constructing spacetime diagrams.The paths that correspond to travel at the speed of light c = 1 are given by x = t . Ifwe imagine adding one more spatial coordinate, these two diagonal lines will forma cone (Fig. 9). This light cone describes the set of points that are connected to asingle spacetime event by straight lines at 45◦ angles.

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Fig. 8 A spacetime diagram is a two-dimensional representation of four-dimensional spacetimethat offers illuminating visualizations of Lorentz transformations that relate the (t, x)-coordinatesto the (t ′, x′)-coordinates. In accordance with the light postulate of special relativity, light conesremain unchanged under these transformations

Fig. 9 Light cones inspacetime diagrams visualizethe causal structure ofMinkowski space by dividingthe set of spacetime eventsinto timelike, lightlike, andspacelike separated points

Light cones naturally visualize causality and the physics of simultaneity becausethey are divided into future and past. The set of all points inside the future and pastlight cones of a given spacetime event are timelike separated from this event, thoseoutside the light cones are spacelike separated, and those on the cones are lightlikeseparated. Using the definition of the line element in equation (1), we see that the

Visualizing Four Dimensions in Special and General Relativity 21

interval between timelike separated points is negative, between spacelike separatedpoints positive, and between null separated points zero.

Spacetime diagrams offer illuminating visualizations of Lorentz transformations.If we return to the previously introduced Lorentz transformation in equation (4) thatdescribes a boost in the x-t-plane, we see that the transformed coordinates t ′ and x′are given by

t ′ = t cosh φ − x sinh φ

x′ = −t sinh φ + x cosh φ.

Therefore, the point defined by x′ = 0 moves with the velocity

v = x

t= tanh φ.

If we replace φ = tanh−1 v and set γ = 1√1−v2

, we recover the familiar form of

Lorentz transformations that are conventionally used to derive length contractionand time dilation:

t ′ = γ (t − vx)

x′ = γ (x − vt).

To see how this Lorentz transformation rotates the space and time axes into eachother, we just have to look at the new axis x′ which for t ′ = 0 is given by

t = x tanh(φ).

Similarly, the t ′ axis for x′ = 0 is given by

t = x

tanh(φ).

Thus, contrary to our Euclidean intuition, the transformed space and time axesscissor together; it is impossible to say whether a point that is spacelike separatedfrom a spacetime event is in the future, past, or simultaneous to that event (Fig. 8).Although different observers can clearly distinguish between space and time, thedistinction drawn by one observer is not the same as that drawn by another. Whatone observer measures as “time,” another one might measure partly in space andpartly in time (Durell, 1926).

Attempting to visualize four dimensions in special relativity amounts to findingrepresentations of a geometry that is governed by the Lorentz transformations.The basic postulate of special relativity that the speed of light is constant finds aprominent expression in the fact that Lorentz transformations leave the paths x = t

invariant; the paths defined by x′ = t ′ are precisely the same as those defined by

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x = t . Thus, spacetime diagrams are an early example of the usefulness of visual-geometric thinking in special relativity and present a powerful tool to reveal thestructure of four-dimensional spacetime.

Relativistic Ray Tracing and First-Person Visualizations

There is a subtle but important difference between attempts to visualize spacetimefrom an exocentric and an egocentric perspective. Spacetime diagrams are exocen-tric visualizations because they place the observer outside of the concept that is tobe visualized. A more intuitive approach to visualizing four dimensions adopts anegocentric perspective that corresponds to a first-person visualization (Kraus, 2008;Weiskopf et al., 2006). The key idea of egocentric visualizations is to translate thegeometry of spacetime into an experienceable relativistic scenario similar to thedream worlds of Mr Tompkins. First-person visualizations of relativistic movementoften reveal counterintuitive features of our four-dimensional world that can beperplexing even to experts in the field (Kraus, 2008).

Placing the observer directly into the scene, a key technique to visualizerelativistic phenomena is relativistic ray tracing. Based on light propagation throughspacetime, the basic idea is to follow light rays that travel backward in time froman observer to the spacetime scene in question. These light rays generate a map thatproduces an image of objects seen in the scene (James et al., 2015). In the languageof general relativity, each light ray follows a geodesic curve through spacetime.Geodesic curves are spacetime generalizations of straight paths. A parameterizedcurve xμ(λ) is a geodesic if

d2xμ

dλ2 + μρσ

dxρ

dλ

dxσ

dλ= 0.

where λμν is the Christoffel connection that we defined in the previous section.

One of the most famous examples of relativistic ray tracing is the visualizationof a wormhole in the Hollywood blockbuster Interstellar that built science into itsvery fabric (James et al., 2015). Interstellar was the first movie to correctly depicta wormhole as it would be seen by a nearby observer. To visualize the appearanceof objects under the influence of a strong gravitational field, the simulation hadto take into account the bending of light through curved spacetime around thewormhole. The team of physicists behind these visualizations wrote an instructionalpaper in which they explained the main steps of the ray tracing process includingits numerical implementation (James et al., 2015). Interstellar provides us with yetanother example of how artistic approaches toward exploring spacetime phenomena– in this case film-making – inspired scientists to push the boundaries of our physicalknowledge.

Numerical first-person visualizations also allow us to revisit the strange dis-tortions that Mr Tompkins observed when a fast-moving object approached himin his dream world (Galison, 1979). Visual observations arise from the photons

Visualizing Four Dimensions in Special and General Relativity 23

that simultaneously reach the eye of an observer. In everyday life, we assumethat such photons have left the observed object simultaneously as well. However,this assumption does no longer hold at relativistic speeds. If the relative velocitybetween observer and object is comparable to the speed of light, one has to take intoaccount the finite light travel time that leads to distortions in the direction of light(Woodhouse, 2014).

Kraus (2008) explored the visual distortions of moving objects in great detail. Asan illustrative example, she chose a row of cubes all facing toward an observer withthe number “3.” A second row of such cubes moves toward the observer at 90% ofthe speed of light as illustrated in Fig. 10. Quite contrary to our everyday experience,the simulation reveals that the apparent shape of the cubes has changed and that wecan see the rear sides of the cubes as well. In this case, it is not the relativistic lengthcontraction but the finite travel time of light signals that explains the surprisingvisibility of the rear side of the cubes. Photons from the rear are able to reach theobserver because the cubes move fast enough “out of the way” of these photons.The team around Kraus (2008) has created many visualizations of relativisticphenomena that are accessible at https://www.spacetimetravel.org. A short movieof the moving cubes is available at the website: https://www.spacetimetravel.org/tompkins/tompkins.html.

Computer-generated first-person visualizations provide insights into the structureof spacetime that are complimentary to exocentric visualizations such as Minkowski

Fig. 10 Cubes are set up in a row (bottom). A second row of cubes moves along the first row at90% of the speed of light (top, motion is from left to right). All cubes, whether moving or at rest,have the same orientation: the face with the “3” is in front, the “4” is on the rear side. The factthat we can see the rear sides of the moving cubes is a consequence of the finite light travel time.(Credit: Ute Kraus, Institute of Physics, Universität Hildesheim, Space Time Travel (https://www.spacetimetravel.org/), Attribution-ShareAlike 2.0 Germany (CC BY-SA 2.0 DE))

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diagrams. Even though both types of visualizations share the common assumptionthat the speed of light is finite, they reveal quite different consequences: Minkowskidiagrams reveal the causal structure of spacetime and illustrate Lorentz transforma-tions; numerical first-person visualizations reveal the distortions that a relativisticobserver would perceive.

Gravitational Lensing and Astrophysical Observations

While relativistic ray tracing provides a numerical procedure to visualize spacetimebased on the propagation of light, the bending of light in strong gravitationalfields offers real-world visualizations of curved spacetime as well. Telescopes andastrophysical observations reveal intriguing manifestations of general relativity. Infact, the light deflection of distant stars around the Sun that Arthur Eddingtonmeasured during a total eclipse in 1919 was the first experimental evidence forgeneral relativity (Fig. 11).

Gravitational lensing, the bending of light due to spacetime curvature aroundmassive objects such as our Sun, black holes, or distant galaxies, provides visu-alizations of the curved geometry of our universe. Light that comes from a

Fig. 11 This picture is anegative of the 1919 solareclipse taken from the reportof Sir Arthur Eddington onthe expedition to verifyEinstein’s prediction of thebending of light around theSun. (Credit: F. W. Dyson, A.S. Eddington, and C.Davidson [Public domain])

Visualizing Four Dimensions in Special and General Relativity 25

Fig. 12 This illustration of gravitational lensing shows how light from a distant source bendsaround a massive object. The light rays (the gray arrows) from the distant galaxy are bent whenpassing a large gathering of mass – such as the galaxy cluster symbolized by the ball with blueglow in the center. When the light finally arrives at the Earth, telescopes observe it as coming froma slightly different direction (the red arrow). One might say that the cluster has acted like a giantmagnifying glass, or gravitational lens, in space – focusing, magnifying, and distorting the imagesof the galaxy. (Credit: NASA/the Space Telescope Science Institute (STScI))

distant light source gets deflected around a massive object before getting refocusedagain (Fig. 12). Today, astronomers use gravitational lensing to obtain informationabout the gravitational lens or to reconstruct properties of the background objects.Figure 13 shows an Einstein ring that was taken with the Hubble Space Telescope’sWide Field Camera 3 based on data from the Sloan Digital Sky Survey. Einsteinrings are lensing phenomena that occur when the source, the lens, and the observerare aligned in such a way to make the distorted light take the shape of a ring.

In 2019, the Event Horizon Telescope Collaboration revealed the first-ever takenpicture of the shadow of a supermassive black hole (Fig. 14). When surrounded bya transparent emission region, black holes are expected to reveal a dark shadowcaused by gravitational light bending and photon capture at the event horizon(Akiyama et al., 2019). Up to this point, only artists and numerical simulationshad produced visualizations of gravity in its most extreme limit. Interestinglyenough, astronomers compared their images to an extensive library of ray-tracedgeneral-relativistic simulations of black holes. Finding that the observed image was

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Fig. 13 Pictured above is theimage of a galaxy that isbeing magnified by thegravity of a massive cluster ofgalaxies situated in front of it.This phenomenon is calledgravitational lensing. Here,the lens alignment is soprecise that the backgroundgalaxy is distorted into ahorseshoe – a nearly completeEinstein ring. (Credit: NASA,ESA, J. Richard (Center forAstronomicalResearch/Observatory ofLyon, France), and J.-P.Kneib (AstrophysicalLaboratory of Marseille,France))

Fig. 14 The Event Horizon Telescope, a planet-scale array of eight ground-based radio telescopes,captured the first direct visual evidence of the supermassive black hole and its shadow. In this imagetaken on 11 April 2017, the shadow of a black hole is the closest we can come to an image of theblack hole itself, a completely dark object from which light cannot escape. (Credit: Event HorizonTelescope [CC BY 4.0 (https://creativecommons.org/licenses/by/4.0)])

Visualizing Four Dimensions in Special and General Relativity 27

consistent with expectations for the shadow of a rotating black hole as predictedby general relativity, present-day astronomy shows how closely visualizations,numerical simulations, and the mathematics of four-dimensional spacetime interactto produce new knowledge of our cosmos.

Numerical Simulations of Gravitational Waves

General relativity predicts fluctuations in the metric of spacetime that can propagateat the speed of light as gravitational waves (Guidry, 2019). Gravitational waves are“ripples” in space time that are produced by some of the most violent events inthe cosmos, such as the collision of binary black holes and neutron stars (Figs. 15and 16). With the first direct observation of gravitational waves and the subsequentbirth of gravitational wave astronomy (Abbott et al., 2016), numerical relativity andvisualizations of numerical simulations have gained much impetus in contemporaryresearch. These explorations promise unique insights into the nature of spacetimeand allow testing general relativity in the highly dynamic and nonlinear strong fieldregime that has been largely unexplored until very recently (Varma et al., 2019).Testing spacetime physics in the strong-gravity limit is crucial because there isalways the possibility that general relativity might be valid for weak gravity butbreaks down in not yet explored realms of strong gravity (Guidry, 2019).

Modeling gravitational waves is important to interpret experimental observationsfrom the LIGO and Virgo gravitational wave detector networks. Visualizations of

Fig. 15 Neutron stars are the remnants of dead stars that were not heavy enough to collapse intoblack holes. This visualization depicts a binary system of a black hole and a neutron star and thegravitational waves that ripple outward as the two objects spiral toward each other. (Credit: MarkMyers/Ozgrav, https://www.ozgrav.org)

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Fig. 16 Even though the spiraling and merging of cosmic objects are among the most violentevents in our Universe, they produce only the tiniest ripples in spacetime. In this illustration, adigital artist has attempted to visualize the elusive nature of gravitational waves. (Credit: CarlKnox/OzGrav, https://www.ozgrav.org)

these models, in turn, help scientists develop intuitions of the complex dynamicsof binary black hole systems. Moreover, visualizations of gravitational wavesare instrumental in disseminating gravitational wave discoveries for outreach andeducational purposes (Key and Hendry, 2016; Varma et al., 2019). Yet, the complexdynamics of binary black hole systems make them hard to model. Often, obtaining asingle prediction for the wave form of a merging system can take several months ofcomputational time on powerful supercomputers (Varma et al., 2019). In response tothese numerical challenges, scientists have started to develop surrogate models thatuse interpolation models to accurately reproduce the result of gravitational wavesimulations in fractions of a second (Varma, 2019).

One example of such a surrogate model is the binary black hole explorer thatoffers on-the-fly visualizations of precessing binary black hole systems (Varmaet al., 2019). These binary systems are characterized by a misaligned orbitalangular momentum similar to the artistic visualization in Fig. 4. The binary blackhole explorer offers an interactive scientific tool to visualize the merging ofbinary black holes, the emitted gravitational wave forms, and the black holeremnant properties. Figure 17 shows such a visualization that can be generated

Visualizing Four Dimensions in Special and General Relativity 29

Fig. 17 This visualization of a numerical simulation shows the complex dynamics of a precessingbinary black hole merger. The black holes are shown as oblate spheres, with arrows indicatingtheir spins. The orbital angular momentum is indicated by the pink arrow at the origin. Similar toelectromagnetic waves, gravitational waves have different polarizations. The colors in the bottom-plane encode the values of the plus polarization of the gravitational wave as seen by an observerat that location. In the subplot at the bottom, one can see the plus and cross polarizations asseen from the camera viewing angle. (Credit: Varma et al. (2019), https://vijayvarma392.github.io/binaryBHexp/)

on a laptop using an easy-to-install-and-use Python package that is available atvijayvarma392.github.io/binaryBHexp.

Before the first observation of two merging black holes in 2015, gravitationalwaves had been the last prediction of general relativity that had not been testeddirectly. Combining state-of-the-art detector technology, numerical relativity, andscientific visualization techniques, gravitational wave astronomy promises com-pletely new ways of revealing the nature of spacetime and our cosmos.

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Virtual, Augmented, and Mixed Reality

Ever since the historic origins of spacetime, technology has facilitated attempts tovisualize four dimensions. The latest advancements in scientific visualizations buildon virtual, augmented, and mixed reality applications that merge real and virtualworlds. By reaching new contexts that are far from our possibilities in the real world,these technologies have the potential to deeply transform the way scientists andthe public engage with spacetime phenomena. Instead of being a mere observer orinterpreter, scientists become able to directly experience and manipulate spacetime.

While the scientific use of virtual and augmented reality is still in its infancy,science education and outreach teams around the world have started to capitalize onthe unique opportunities to blend physical and digital objects in real time. Often,these teams are highly interdisciplinary and combine the expertise of scientists,programmers, digital artists, and game designers to increase the public’s scientificliteracy through immersive technology. For example, the Gravitational Wave Groupat the University of Birmingham founded the community interest company LaserLabs that develops educational apps such as Pocket Black Hole and Stretch andSqueeze (Carbone et al., 2012). These apps invite users to add black holes orgravitational waves to their real-world environments (Fig. 18). Using the app, userscan explore distortions of space and time and manipulate effects of gravitationallensing in interactive and immediate ways.

In a similar vein, the educational app Science in VR (SciVR) invites users toexperience spacetime scenes and to explore the virtual universe either using virtual

Fig. 18 The app Pocket Black Hole allows users to explore distortions of space and time byplacing black holes into their immediate environment. Here, a black hole was placed on top ofa walking trail near the Swedish town of Karlstad. (Credit: Pocket Black Hole/Laser Labs, https://www.laserlabs.org)

Visualizing Four Dimensions in Special and General Relativity 31

Fig. 19 Science in VR contains a universe of virtual reality content, from explorations of ourMilky Way and the solar system to black holes colliding. The app offers visualizations of spacetimephenomena that users can enjoy at home with accompanying audio explanations. (Credit: OzGrav,https://www.scivr.com.au)

reality cardboards or devices in 3D mode with accompanying audio explanations(Fig. 19). The developers behind SciVR, the Education and Public Outreach Teamat the Australian Research Council Centre of Excellence for Gravitational WaveDiscovery (OzGrav), combine classic scientific modeling with immersive virtualreality to make gravitational wave physics more accessible to school studentsand the general public. Using the freedom of virtual reality, the OzGrav teamreimagines what it means to reveal spacetime phenomena from a first-personperspective (Fig. 20). The OzGrav virtual reality environments offer different andcomplementary approaches to the scientific visualizations of gravitational waves inthe previous section. Virtual reality headsets detect the movement of the wearerand allow for realistic embodied experiences of gravitational waves. Figures 21, 22,and 23 show screenshots of virtual scenes in which users can freely move aroundand lean into the spacetime scene to explore gravitational waves.

According to the mission of OzGrav, the center wishes “to capitalize on thehistoric first detections of gravitational waves to understand the extreme physics ofblack holes and warped spacetime and to inspire the next generation of Australianscientists and engineers through this new window on the Universe.” Just asgravitational waves have opened a new window into the Universe that revealsthe physics of space and time, so have emerging new technologies opened newpossibilities to alter one’s perception of four-dimensional spacetime. Advancementsin technology and visualizations of four-dimensional spacetime drive change ineach other. Virtual, augmented, and mixed reality applications mediate a completelydifferent quality of spacetime experiences. The promise to provide people with an

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Fig. 20 Mission Gravity is a virtual reality environment that allows school students to explorespacetime phenomena in interactive and immersive ways. Wearing virtual reality headsets, studentscan see and interact with a world that would otherwise not be accessible to them. In this scene,students sit inside a virtual spaceship to study the properties of a giant blue star in the background.(Credit: Mark Myers/OzGrav, https://www.ozgrav.org)

Fig. 21 In this virtual reality scene, two spiraling black holes have been placed next to the Earthso that users can observe and experience the squeezing and stretching of our own planet due to theripples in space. (Credit: Mark Myers/OzGrav, https://www.ozgrav.org)

Visualizing Four Dimensions in Special and General Relativity 33

Fig. 22 Similar to electromagnetic waves, gravitational waves can have different polarizations.In this virtual reality visualization, users can observe these different polarizations from differentangles while the gravitational waves move toward or away from the observer. (Credit: MarkMyers/OzGrav, https://www.ozgrav.org)

Fig. 23 This screenshot of a dynamic virtual reality visualization shows a binary system of ablack hole and a neutron star that spiral around each other. In the virtual reality environment, theobservers can view the scene from different perspectives to explore gravitational waves. (Credit:Mark Myers/OzGrav, https://www.ozgrav.org)

34 M. Kersting

intuitive feel for the extreme physics of warped spacetime and our four-dimensionalreality has truly propelled spacetime explorations into the twenty-first century.

Conclusion

According to the valid observation of Pushkin, imagination is as necessary in geometry asit is in poetry. Everything that requires artistic transformation of reality, everything that isconnected with interpretation and construction of something new, requires the indispensableparticipation of imagination. (Vygotsky, 1998)

Modern physics unfolds on the stage of four-dimensional spacetime. To thenon-mathematical mind, the abstract character of such a description may seemunsatisfactory (Russell, 1925). Yet, it is this very abstract character of our physicalknowledge that has inspired our scientific, imaginative, and artistic attempts topush the boundaries of what we know. Nowhere does the fruitful interplay betweenmathematics, sciences, and arts feature as prominently as in the historic struggle tovisualize four dimensions as part of the quest to understand our cosmos.

Special and general relativity present unique challenges to the human mindbecause these theories contradict the commonsense understanding of many. To alignthe perceptual qualities of spacetime physics with our experiential understandingof the world, we must draw on a repertoire of visualizations, representations, andmodels (Kersting and Steier, 2018). In the course of this chapter, we have exploredsome of these visualizations through the lens of technology. Since the stage ofspecial relativity is flat Minkowski spacetime, visualizations of special relativisticphenomena are usually restricted to visual perceptions of the distortions of objectsdue to movement at high speeds. In general relativity, in contrast, the dynamicinterplay between spacetime and massive objects allows for a greater variety ofspacetimes that can be studied and visualized. Exotic phenomena such as blackholes and gravitational waves push the laws of physics to their limits and challengeour imaginative faculties.

The historical origins of spacetime and the century-long efforts to visualizerelativistic phenomena testament to the power of using technology as a mediatorof our understanding. Along with technological progress throughout the twentiethcentury, computer simulations started to replace artistic illustrations of four-dimensional spacetime with more accurate visualizations based on the mathematicsof differential geometry. In this process, interdisciplinary has been a drivingforce and a recurrent theme. The historic origins of spacetime sprung from afruitful bringing together of mathematical prowess, visual-geometric thinking,and physical insight. Today, interdisciplinary teams with expertise in relativisticphysics, computer simulations, user interfaces, and digital artistry come together todevelop visualizations that allow probing extreme phenomena of four-dimensionalspacetime (James et al., 2015; Weiskopf et al., 2006).

It is only through interdisciplinary efforts that imaginative leaps become possible.Linking mathematical concepts with physical intuition and artistic vision, theseimaginative leaps have led from simple spacetime diagrams and rubber sheet

Visualizing Four Dimensions in Special and General Relativity 35

analogies to sophisticated first-person visualizations and powerful virtual environ-ments that allow exploring the extreme physics of black holes and gravitationalwaves. In the history of spacetime, interdisciplinary has been the driving force andmathematics and technology the means through which imagination has explored ourworld for new knowledge.

Cross-References

�Artistic Manifestations of Topics in String Theory�On the Reciprocal Influence of Mathematical Cosmos Representations and Artis-

tic Creation�Visual Experiment and the Case for Unconventional Tools in Mathematics

Research

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