What will come after Einstein?

We have come a long way since Galileo Galilei was persecuted by the Catholic Church for his findings that the Earth revolves around the motionless Sun [1]. Yet people of all cultures and religions are still oended by actions and facts that contradict their belief system. Scientists on the other hand are happy to admit that what they know is not — and might never be — the final truth. They accrete knowledge through empirically observable results of reproducible experiments, which is known as the scientific method. Hence one of the fundamental differences between religious (or pseudoscientific) beliefs and scientific theories is the disposition of scientists to willingly enhance, replace, or discard an idea when new evidence contradicts the old findings. This was beautifully summarized by Albert Einstein.

No amount of experimentation can ever prove me right;
a single experiment can prove me wrong.
— Albert Einstein [2]

Experimentation can only strengthen the likelihood that an idea is correct, but nothing can ever truly prove it. The strongest support comes if one can predict a result. This is how various hypotheses on relativity, space-time, and the relationship between mass and energy became established. Yet however elegant the idea, if nature is shown not to conform then the idea is wrong.

In fact, Galileo was wrong with his statement that the Sun is motionless. It rather orbits the center of our galaxy at a zippy 225 kilometers per second. Just recently NASA’s Interstellar Boundary Explorer (IBEX) was able to map the solar system’s tail of cosmic dust following behind [3]. Even our entire Milky Way galaxy is heading towards a collision with the Andromeda galaxy at roughly 110 kilometers per second [4], and our sun is dragged along. And then there is the ‘Great Attractor’, a gravity anomaly that attracts our entire Local Group of galaxies at roughly 600 kilometers per second [5]. All of these forces on our Solar System add up to a velocity relative to the cosmic microwave background — which is as close as we can get to a rest frame of the universe — of 371 km/s. So you see that Galileo’s main finding about celestial mechanics is still valid: the Earth is by no means the center of anything.

What is Gravity?

Gravity, the everyday force that keeps us to the ground, is the dominating force in the universe. It can bind systems as small as a city district and is responsible for the dynamics in our solar system, but also reaches out to the largest structures known to us. Have a look at this impressive overview of systems bound by gravity. Our own Milky Way galaxy is part of the Laniake Supercluster. Traces in the picture represent the movement of galaxies in the direction of the Great Attractor. The nature of this gigantic unseen mass some 250 million light years from our Solar System remains one of the great mysteries of astronomy.

In 1687 English physicist Sir Isaac Newton tried to explain the motions of moons and planets and found that two masses attract each other by a force proportional to the product of the masses and inversely proportional to the square of the distance between them. The proportionality factor in Newton’s inverse-square law of gravity [6] is an empirical physical constant measured to be 6.67384e-11 N(m/kg)² with a relative standard uncertainty of 0.12%. All superclusters of galaxies, stellar associations, and planetary systems are bound by this invisible gravitational force that dominates all structures in the universe.

As you would expect from a good scientist, it was Newton himself who questioned his own theory. For him, the assumption that gravity acts instantaneously, regardless of distance and even through a vacuum, was “so great an absurdity that, I believe, no man who has in philosophic matters a competent faculty of thinking could ever fall into it.” [7] Finally, a discrepancy in Mercury’s orbit pointed out that Newton’s theory must be wrong [8]. However, it provides a very accurate approximation and is still used today for most physical situations including calculations as critical as spacecraft trajectories.

Until the 19th century, many physicists tried to come up with a mechanical explanation of gravity without the troubling ‘action at a distance’. Many of them included some kind of aether, a space-filling substance orfield [9]. But all of these theories were overthrown by observations.

Einstein’s General Relativity

In 1907 Albert Einstein started working on a hypothesis as to the cause of the gravitational force. Nine years later he published his geometric description of gravity: the general theory of relativity [10]. His conclusion: gravity does not propagate through space, and it is not a force of a field, or substance penetrating empty space. Instead gravity is mediated by the deformation of spacetime, a mathematical model that combines the three dimensions of space and the one dimension of time into a single four-dimensional continuum. While mass, energy, momentum, pressure, or tension (all combined in the stress-energy tensor Tµν which measures the matter content) curve spacetime, matter simply follows the geodesics of spacetime.

This main message of general relativity is illustrated in the figure below. The geometry of spacetime is described in the Einstein tensor Gµν, which measures its curvature. Each of the two tensors has 10 independent components, the relationship between both was formulated in the Einstein field equations. G is the same gravitational constant as in Newton’s law and c is the speed of light. This equation makes gravity a fictitious force where free falling reference frames are equivalent to an inertial reference frame. The Λ is the cosmological constant, an energy density in otherwise empty space influencing the metric tensor gµν (or, simplified, the gravitational field).

In a static and never changing universe, as assumed by Einstein, a well chosen cosmological constant could counteract gravity and prevent the universe from falling in on itself. After astronomers like Georges Lemaître [13] or more famously¹ Edwin Hubble [14] discovered that the universe is expanding and must have been created in a big bang from a ‘primeval atom’, it was not needed to artificially stabilize the universe any more, but: as it turns out, the same constant Λ can be used to describe the accelerated expansion observed in our universe [15]. As you see, the Einstein field equations are capable of much more than just explaining gravity; they have given us a tool set to understand the workings of the universe.

¹: Lemaître derived Hubble’s law and provided the first observational estimation of the Hubble constant in his original 1927 paper, but these parts were lost in translation for an English publication in 1931 [12].

The Nature of the Universe

When you assume a homogeneous and isotropic universe, you can derive a set of equations, called the Friedmann equations, that govern the expansion of space [16]. These equations tell you that a universe within the context of gravitational relativity could either be flat (i.e. Euclidean space), a closed 3-sphere of constant positive curvature, or an open 3-hyperboloid with a constant negative curvature [17]. As derived from one of the Friedmann equations, in a flat universe without a cosmological constant the mass density ρ would equate the critical density ρc =3H²/8πG. Hence with no more than a given Hubble constant H (speed of expansion of the universe) and a gravitational constant G, you are able to conclude the shape, curvature, and fate of the universe out of its mass density. For ρ < ρc the universe would be an open 3-hyperboloid and expand forever. For ρ > ρc it would be a closed 3-sphere and eventually stop expanding, then collapse under its own gravity. It would also be ofinfinite size and, due to its curvature, in the end traveling far enough in one direction will lead back to one’s starting point. The special case of ρ = ρc results in a flat (or Euclidean) and static universe as described above. These three cases are commonly expressed by the density parameter Ω ≡ ρ/ρc with Ω < 1, Ω > 1, and Ω = 1 respectively, as illustrated in the figure below.

It is important to note that there is a huge discrepancy between the baryonic matter density, ρb with Ωb ≡ ρb/ρc, and the mass density calculated through general relativistic means. Baryonic matter accounts for all ‘ordinary’ matter and is usually referred to as visible or luminous matter. The observed gravity within large-scale structures in the universe is much stronger than what could be accounted for by visible matter. Gravitational lensing — background radiation curved by gravitational fields — points to a total mass six times larger than what can be observed directly. Since free photons and cosmic neutrinos — that once accounted for big parts of the mass and energy distribution in the early universe — are entirely negligible nowadays, cosmologists hypothesized that this excess gravity is caused by an yet unknown form of ‘dark’ matter that does not interact electromagnetically. Current models assume that it consists of slowly moving particles which interact very weakly with electromagnetic radiation [18]. Thus these “cold dark matter” (CDM) particles are almost invisible and can currently only be observed through gravitational interaction. The CDM density, ρcdm with Ωcdm ≡ ρcdm/ρc, would also explain other mysteries like the “flat” rotation curves of galaxies [19] or the evolution of large-scale structure of the universe [20].
Considering a positive cosmological constant (or vacuum energy density) the situation gets even more complex. Here you could have a closed, spherical universe where the vacuum energy density, ρΛ with ΩΛ ≡ ρΛ/ρc, is part of the total mass density. Usually gravity dominates in the long run and causes a spherical universe to contract eventually. In the case of Λ > 0 however, the negative pressure of the vacuum energy prevents that from happening and drives an accelerated expansion of the universe. Hence the expansion of such a universe will continue forever, up to a point where the observable part of the universe would be quite empty.

The concept of a field with negative pressure that accelerates the expansion of the universe is generally referred to as “dark energy”, where the positive cosmological constant is just its simplest form. Cosmological models in which the universe contains such kind of a field are collectively subsumed under the heading Lambda-CDM (or ΛCDM) model. It is the current “standard model” of cosmology because of its precise agreement with observations. Of course, this model allows many different flavors of the universe to exist within the borders of Einstein’s field equations. Naturally, scientists are eager to measure the parameters of our universe to determine its shape.
Today we know that our universe has a baryonic matter density of Ωb = 0.0456 ± 0.0016 that is missing a CDM density of Ωcdm = 0.227 ± 0.014 to account for all observed gravitational effect. Furthermore we see that the universe we live in expands with a Hubble constant of H = 70.4+1.3−1.4 km/s per megaparsec. High-precision measurements show that this expansion rate changes over time and reveal that the rate of expansion is accelerating from 7.5 billion years after the big bang onwards. We can conclude a vacuum energy density of ΩΛ = 0.728+0.015−0.016 and determine the age of the universe to be 13.75 ± 0.11 billion years [21].

It is most fascinating to see that — at least in our local observable universe — the total mass density equates the critical density as exactly as Ω = Ωb + Ωcdm + Ωλ = 1.0006+0.0306−0.0316. This would mean that we live in a totally flat universe. The positive cosmological constant produces an ever accelerating expansion and points to a very unpleasant fate of the universe. Some time in the distant future the Hubble constant might become so large that even stars in galaxies are torn apart and the observation of distant stars would become physically impossible. The metric expansion of space might not even stop at atoms and subatomic particles, breaking all bonds in matter and creating a huge, dark, cold and empty universe [22]. But: The ultimate fate of the universe not only depends on the shape of the universe and the vacuum energy density, but also on the role vacuum energy will play as the universe ages. It is still unclear whether the total energy is conserved in general relativity as the universe expands.

Reality is More Complicated

All cosmology described before is solely based on the Einstein field equations. It turns out, our universe not only seems to be flat, but could actually have zero total energy [23]. This leads to the speculation that it may have been created in a coincidental quantum fluctuation. Such a closed system would not require any higher structure to provide a trigger mechanism for the big bang. Yet predictions of quantum mechanics seem to contradict predictions from general relativity, and suddenly the picture becomes much more complicated.

Black holes for example are a solution of the Einstein field equations and describe a massive singularity surrounded by a gravitational field so strong that — within a well-defined surface known as the event horizon — even light cannot escape. Nevertheless, the quantum field theory predicts that black holes evaporate over time due to quantum vacuum fluctuations (creation of particle-antiparticle pairs of virtual particles) at the black hole’s event horizon. The escaping particle is known as Hawking radiation [25] and causes the black hole to lose mass and energy. It can be shown with currently accepted theories that this particle must be entangled with its infalling antiparticle that is swallowed by the black hole, as well as with all the Hawking radiation previously emitted by the black hole [26]. Since quantum mechanics forbids any particle to be fully entangled with two independent systems, the combination of general relativity, quantum mechanical unitarity, and quantum field theory creates a paradox [27]. The very same Hawking radiation might even prevent black holes from forming in the first place [24], yet we know from observations that black holes do exist.

There is another — much simpler — gedanken experiment that also tells us that the universe cannot be described by Einsteins field equations alone: the horizon problem. Naturally, we can only retrieve information from within a certain volume that is defined by the cosmological horizon which represents the boundary of the observable universe. Due to the nature of an expanding universe, this horizon has a radius of 46.2 billion light years [28] although light from that distance only traveled for 13.75 billion years (which represents the age of the universe). Light from outside this horizon had no chance to reach us yet. Thus it is obvious that we are embedded in a much larger unobservable structure that could have a different shape and where our local geometry only seems to be flat. Other parts of this larger structure, beyond our cosmological horizon, might host additional local universes of widely differing curvatures. The figure below shows Region A and Region B which both lie within our observable universe, but the local universes for both regions cover different parts of the overall universe and do not fully include each other. (Please note: These many local universes are very different to the parallel universes as a result of the many-worlds interpretation of quantum mechanics [29] where new universes pop into existence for every possible outcome whenever an observation is made.)

This cosmological horizon imposes another paradox. We know that — due to its small size — the early universe was so dense and therefore so hot that photons scattered at free electrons. It took until 380,000 years after the big bang for protons and electrons to finally combine and form neutral hydrogen atoms. It was at this time that the universe became electromagnetically transparent. Today the light from this last scattering can be observed — now red shifted due to the cosmic expansion — in the cosmic microwave background (CMB) radiation. It tells us details about the conditions 380,000 years after the big bang. (This time represents the earliest direct observation currently possible as we have no way to observe the universe prior to that time.)
Due to the random nature of the initial conditions the temperature of this radiation should be very different for different directions in the sky. Similar to the figure below, distant regions in space back in the time of the last scattering had no causal contact. Light sent out from opposite patches of the origin of the CMB just reached Earth, which is positioned half way between them. Thus those patches can not know anything about each other. The distance between Earth and the origin of the CMB expanded to 46 billion light years. Thus even with a constant expansion of space at the current Hubble constant, light that just reached Earth could never reach the other side of the CMB: space in between Earth and the origin of the CMB expands so fast that both points seem to recede from each other faster than the speed of light. There is no change that an equilibrium was formed or could ever form and the initial temperature fluctuations should still be observable. Yet the CMB radiation has a surprisingly uniform temperature, isotropic to roughly one part in 100,000 over the entire sky, with a very fine fluctuation pattern that may have seeded the growth of structure in the universe. This cannot be explained by the standard ΛCDM model.

What Now?

We know, that Einstein’s field equations of general relativity cannot fully describe the universe we observe today. The resolution to the paradox described in the previous parts has to be found somewhere before the time of last scattering, in the first 380,000 years after the big bang.

We can simulate the hot young universe from a second after the big bang and follow its evolution over time while it expands and cools down. The predictions of such simulations are at overwhelming agreement with detailed observations all the way to the present time. To explain the uniform CMB temperature physicists hence aim at the first fractions of a second after the big bang. This first second is poorly understood. The unexplained imbalance in baryonic matter and antibaryonic matter for example originates from the same time. The closer you get to the big bang, quantum effects become more and more important, but the theories we use to extrapolate back in time were not developed to include quantum physics. Until today, there is no “theory of everything” that describes gravity, space, time, and the shape and evolution of the universe, as well as physical phenomena at nanoscopic scales with all known quantum effects. It is impossible to tell if the properties of our universe gave birth to the laws of natural, or if there is an underlying fundamental set of rules that determines both, general relativity and quantum mechanics. Maybe the shape and nature of our universe is even somehow independent from the interactions of matter and energy within.

Currently two favored theories exist that extend the standard ΛCDM model to explain the observed uniform CMB temperature. The more conservative theory is the cosmic inflation model [30]. It postulates an inflationary epoch that lasted from 10􀀀^-36 seconds to roughly 10􀀀^-32 seconds after the Big Bang where space expanded exponentially. In this model, tiny quantum fluctuations during inflation became magnified to cosmic size, and all other inhomogeneities were smoothed out. It explains the uniform CMB temperatures including its pattern, and predicts that the total mass density equates the critical density, as can be observed today. In this model, in the beginning there was nothing. The universe started out of a singularity that expanded exponentially for a very short time span and follows the rules of general relativity ever since, including an ever accelerating expansion dominated by dark energy as shown in the figure. Inflation though was not the same for the whole universe — other parts would undergo different inflationary epochs or might even still inate today, producing many local universes with very different properties.

In M-theory — a unifying string theory and promising candidate for a unified theory of general relativity and quantum mechanics — a very different kind of universe is possible. This theory even raises the question if the big bang was the real beginning of our universe as it can also describe a cyclic (or ‘ekpyrotic’) universe that did not start with a singularity but with two branes moving apart [31]. Branes are higher-dimensional objects described by M-theory that exist in space-time and follow the rules of quantum mechanics. Space in between both branes cannot be accessed by any object situated in either brane — like us — but both branes can move along this extra dimension. This opens the possibility for the big bang to be a collision of both branes, caused by a spring-like force between them [32]. In this collision hot matter and radiation was created. Both branes move apart while the two universes begin to expand. M-theory predicts that the accelerated expansion cannot last forever since within this context, dark energy is associated with the spring-like force between both branes. Eventually, it will bring both branes back together again. In the subsequent collision all kinetic energy is converted to new matter and radiation and the cycle starts all over again. This is illustrated in the figure.
Strictly speaking this is not a full cycle as the universe contracts only in the extra dimension and continuously expands in all other dimensions. Thus the overall Universe becomes bigger and bigger. Nevertheless, the local universes from an observer’s perspective remain the same for each cycle. During the process of contraction — which is the alternative to inflation and lasts for about 10 billion years — the universe smooths out until quantum fluctuations take over. Consequently, the branes are slightly wrinkled and do not collide everywhere at the exact same time. Some regions of space bounce off earlier (heat up sooner) than others. Simulations predict that this would cause exactly the same uniform CMB temperature with its distinct pattern. This model also provides an explanation for the nature of dark matter: it merely is the influence of matter from the distant brane felt in our local universe.

Both models result in the universe that we observe today. The nature of the universe though is fundamentally different for both theories. In one, the CMB pattern is caused by quantum fluctuations shortly after the big bang during the time of cosmic inflation, while in the other one the same pattern arises from quantum fluctuations prior to the big bang which merely was the most recent collision of two higher-dimensional branes. The amount we already know about our universe speaks volumes of our ingenuity and science, but if we want to find answers about the origin of our universe, we need a new kind of observational cosmology. The only way to distinguish between both models lies in one single differentiation: strong gravitational waves should have been created during rapid inflation, but almost zero gravitational waves would originate from a slow collision of two branes. We just need to push today’s technology a little bit further to detect these gravitational waves.

Gravitational Waves?

Besides black holes and gravitational lensing the maybe most exciting consequence of Einstein’s theory is the postulation of gravitational waves [33].

The Einstein field equations predict that accelerated matter (or energy in general) emits gravitational quadrupole radiation as illustrated in the figure. These waves stretch and compress spacetime perpendicular to the direction of travel and cause directly observable distance fluctuations between freely falling objects. Let’s assume we have a ring of cubes freely floating in the xy-plane and a gravitational wave propagates along the z-direction. As illustrated in figure, the distance between the masses oscillates with time. The direction of this oscillation depends on the polarization of the gravitational wave. The usual basic set of polarization states are plus (+) and cross (×) polarization, others can be formed by linear combinations of these two.

While the strength of the gravitational field falls off with the square of the distance, this effect, an amplitude, falls off linearly proportional to the distance [34] and even sources located at the other end of the observable universe can produce relative distance fluctuations on the order of 10􀀀^-20 or more, depending on the frequency of the signal.

Example: On a distance of 4 kilometers, relative distance fluctuations of 10^-􀀀20 correspond to 40 attometers. This is much smaller than an atomic nucleus. For a distance of 1 million kilometers, the same fluctuations correspond to 10 picometers, which is a factor of ten below the diameter of a hydrogen atom.

This effect, as tiny as it might seem, can tell us about electromagnetically invisible objects and has a huge discovery potential for new physics. Gravitational radiation travels unaffected throughout the entire universe; in contrast to electromagnetic radiation that interacts strongly with matter and hence can be distorted or blocked. Gravitational waves were able to propagate unimpeded even in the young hot universe prior to 380,000 years after the big bang. Thus gravitational waves are a superior messenger that holds complimentary or even otherwise completely unobtainable information about processes in the universe. Gravitational wave observatories are expected to bring the next big revelations in astronomy, cosmology, and fundamental physics alike.
General relativity passed all tests with flying colors. For over 25 years we had indirect yet irrefutable proof of the existence of gravitational waves [35]. Just recently, the Laser Interferometer Gravitational-wave Observatory (LIGO) directly detected gravitational waves [36]. More observatories are under construction and very soon we will have a sensitive network of many different gravitational wave antennas on ground and in space to detect gravitational waves over a wide frequency range. Research teams also look into indirect evidence for gravitational waves produced during cosmic inflation, now red-shifted to a static polarization pattern imprint in the cosmic microwave background radiation [37, 38]. The clear detection or non-detection of a primordial gravitational wave background from the inflationary epoch would rule out one of the two leading theories about the origin of our universe. This — for me — is the most fascinating prospect of gravitational wave astronomy: we might not only determine the shape and structure of our own universe, but also learn about the nature of the global universe it is embedded in.

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