Quantum entanglement and its applications: Part 1

Reinhold Bertlmann

Credit: AB1927, Public domain, via Wikimedia Commons

Unlike classical physics and general relativity, which deal with the deterministic evolution of physical variables such as position and momentum, quantum mechanics deals with an abstract entity called the state vector. In general, the state vector resides in an infinite-dimensional complex Hilbert space. However, in the world of quantum information and quantum computing one deals mostly with state vectors that are finite-dimensional. For example, the spin of a particle or the direction of a superconducting current or energy state of a trapped ion has a state vector that is simply a vector residing in a 2-dimensional complex Hilbert (inner product) space.  By the Born rule (or Born postulate), the probability of an outcome during measurement of the physical variable (for example spin) is given by the square of the norm of the state vector. The evolution of the state vector can simply be represented by a 2 X 2 complex unitary matrix. In the absence of measurement, subjecting the system to conservative force fields simply results in the transformation of the state vector by a suitable unitary 2 X 2 matrix. The transformation of a state vector by these unitary matrics is called a quantum logic gate and can be represented graphically. One of the most unique and intriguing aspects of quantum mechanics is the phenomenon of entanglement. It deals with non-local correlations between measurements of complementary observables (such as position and momentum or spin directions) performed on parts of a system that are physically separated by a "large" distance. In the language of state vectors, it simply represents an indecomposable vector in the tensor product of two complex Hilbert spaces. The phenomenon of entanglement was first discussed by Einstein-Podolsky-Rosen in the famous EPR paper with a clearly stated goal of demonstrating the incompleteness of quantum mechanics as a theory of physical reality. EPR demonstrated that quantum mechanics had non-local effects, an anathema for Einstein as it seemed to violate special relativity. Actually, EPR only showed that quantum mechanics implies non-local correlations between measurements, but such a correlation is so counter-intuitive that it seemed to imply that there was more to quantum mechanics than the Copenhagen interpretation of quantum mechanics.  Little did Einstein know that John Bell would later show that non-locality was an essential component of quantum mechanics.  Bell showed that any local hidden variable theory would have to satisfy an inequality (known as Bell's inequality), which quantum mechanics did not satisfy. Bell wrote a wonderful paper called "Bertellman's Socks and the Nature of Reality", explaining the crux of the EPR paradox and Bell's inequality. 

Bertlmann and his socks

Credit: AB1927, Public domain, via Wikimedia Commons

Since then many people have recast the EPR paradox and Bell's discovery into different formats to convey the non-classical, counter-intuitive (Mermin's "Local Reality Machine"), and computationally powerful (CHSH Nonlocal game) nature of entanglement. Today, entanglement forms the foundation of modern quantum information theory and has applications to cybersecurity via schemes like quantum key distribution and quantum cryptography. The following presentation tries to give a flavor of the history and applications of quantum entanglement. I gave this presentation in an evening class on quantum computing that I took at the UW Physics department. 

In future posts, I hope to explain each of the slides on quantum entanglement (including Bell's paper on "Bertellman's socks") in simple terms. Stay tuned.

Entanglement and its applications from Prashanth Adhikari

Penrose's work on singularities: Part 2

 

In Part 1 of this article, we saw that by 1965 there was theoretical evidence that a sufficiently massive star that is spherically symmetrical would collapse to a black hole after it has spent all its nuclear fuel. Since General Relativity postulates that spacetime is curved by the presence of matter and energy, it is expected that there would be a severe distortion of spacetime when the matter is compressed to a high density. The Oppenheimer-Snyder model of gravitational collapse showed that for a spherically symmetric, homogenous, and static distribution of dust (with no internal pressure) of a sufficiently large mass, there is no known mechanism to prevent the compression of the matter to infinite density. This would result in the formation of a black hole with an event horizon at a radius of 2GM/c^2 (Schwarzschild radius) and a singularity behind the event horizon. It is customary to rescale the dimensions so that G/c^2=1, so the Schwarzschild radius is simply 2M. The singularity represents a point of infinite curvature and also a point where all future-directed paths and light rays come to an end. In 1965 Roger Penrose proved that even if we did not make any assumptions of spherical symmetry, the geometrical constraints imposed by a very strong gravitational field will inevitably result in spacetime singularities,

As discussed in Part 1, Roger Penrose formulated his groundbreaking theorem in terms of five assumptions, which he showed were collectively inconsistent with each other. Four of those assumptions are based on assumptions of reasonable "niceness" of spacetime, namely a) The "Past" and "Future" condition, which states that there exists a consistent definition of past and future everywhere, b) The "Null Completeness" condition, which states that all paths built out of light rays can be extended indefinitely at all points and in all directions, c) The "Cauchy Hypersurface" condition, which assumes the existence of a non-compact connected Cauchy hypersurface (to be defined below), d) The "Null Energy" condition, which states that the local energy at any point is always non-negative. The fifth condition is the only one that pertains to the conditions expected near a black hole where gravity is so strong that all light rays are bent towards each other. Penrose's fifth assumption is e) The "Trapped Surface" condition, which states that there exists a 2-dimensional compact surface (like a sphere) for which all light rays emanating from the surface are bent towards each other.   

Understanding the statement of the theorem and its proof entails absorbing a fair amount of geometry, topology, and terminology associated with the causal theory of spacetime and general relativity. In this article, I will attempt to explain the concepts in the simplest possible way without appealing to all the jargon that one would typically encounter in a rigorous exposition of the topic. We start with the causal theory of spacetime, which is fundamental to the entire subject.

Local Causal Structure of Spacetime

The origin of the causal theory of spacetime lies in Minkowski's reformulation of Einstein's special theory of relativity in terms of a four-dimensional spacetime. In special relativity, the speed of light occupies a very special place. Nothing can travel faster than the speed of light,  a universal constant independent of any inertial observer's frame of reference. Points in the Minkowski model are events whose separation is measured by the Lorentz metric, a quantity that is invariant under Lorentz transformations. The causal theory of what events can influence other events and the domain of influence is a feature unique to the special theory of relativity and is not present in classical Newtonian mechanics.

A fundamental geometrical object when studying causal theory is the "light cone". Imagine lighting a candle at a point on Earth. If you ignore the effects of gravity, light from the candle will spread out radially in all directions in straight lines. These straight lines will sweep out a sphere in 3-dimensional space. Now if we suppress one of the spatial dimensions (say the Z-axis), then we can visualize this as an expanding circle. If we choose the vertical axis to represent time and suppress one of the space dimensions (because we cannot really visualize a 4-dimensional object!), then we will see that as time progresses vertically, the wavefront of light will spread farther and farther, on the surface of a cone. A similar cone can be envisioned going back in time. 

The points in the interior and on the surface of the upper cone represent the causal future of a point (event) at the origin (at time t=0). Every event in the interior of the cone can be influenced by an object or signal traveling at a speed strictly less than the speed of light. Therefore, the interior of the cone is called the chronological future of the point at the origin. Points on the surface of the cone represent the boundary of the casual future. They are events that can only be influenced by signals traveling at the speed of light. The curves traced by such signals are called null geodesics the surface of the cone which is swept out by the null geodesics (light signals) is called a null hypersurface. 

Minkowski spacetime is a "flat" manifold, meaning it has no curvature. It represents an idealized condition where one is not subject to any force or influence of any kind and all entities are moving at a uniform speed relative to each other (inertial frames of reference).  For example, the spacetime very far away from any star could be approximated by Minkowski spacetime. In General Relativity, spacetime is curved by the presence of matter and energy. To quote John Wheeler, "matter tells spacetime how to curve, and spacetime tells matter how to move". The fundamental postulate of General Relativity known as the "Principle of Equivalence" states that near a point in spacetime the gravitational forces can be canceled ("transformed away by a coordinate change") by moving to a freely falling frame of reference. For example, if I am in a space station above the Earth I will experience a gravitational pull from the Earth. But if I step into a spacecraft and start freely falling in Earth's gravitational field, I will experience weightlessness. In geometric terms, this amounts to carrying out a coordinate transformation to a coordinate system wherein a small neighborhood spacetime looks and feels like Minkowski spacetime. Thus the Principle of Equivalence implies that as a manifold, spacetime is locally Minkowski at every point. So at every point, the local causal structure can be represented by a light cone. Depending on how spacetime curves due to the presence of matter, the light cones at different points might be pointing in different directions. 

Timelike Curves

As a particle moves in spacetime, its trajectory in spacetime is represented by a worldline (also known as a timelike curve). At each point of that curve is a lightcone which represents the boundary of all the different spacetime directions in which a signal can travel. If the particle is a photon (quantum of light), then the curve is built out of light rays, and in that case, the lightcone is tangential to the curve at every point in spacetime. Such a curve is called a lightlike or null curve. 

Lightlike (null) curves

So the concepts of timelike geodesics, null geodesics, chronological future, causal future, and the null hypersurface built out of the boundary carry over verbatim to curved spacetimes (also known as Lorentz manifolds).  An essential ingredient for this causal analysis is assumption a) of Penrose's theorem namely the "Past" and "Future" assumption. It is essential that there be a consistent way to define past and future across the spacetime manifold to avoid pathologies. 

For Penrose's theorem, it is important to consider not just the causal future of a point in spacetime, but the causal future of an entire "spacelike" surface in spacetime. Spacelike simply represents a slice of spacetime at a particular choice of time chosen uniformly across all points (the fact that you can do it is an assumption known as time orientability). In other words, a spacelike surface is just a region of space at a particular time. The spatial slice could not be curved. 

The chronological, causal, and null future of a set satisfies some easily provable topological properties. The chronological future is an open set, meaning that for every event in the chronological future, you can find a "ball" of neighboring events in spacetime that reside in the causal future. Similarly, for any point that is on the boundary, every ball of neighboring events will have an event that is in the chronological future (interior of the causal future). Moreover, the boundary if nonempty is a closed 3-dimensional achronal C^0 submanifold of the 4-dimensional spacetime. Achronal means that no two points of the boundary can be joined by a timelike curve (worldline of an object traveling at a speed smaller than lightspeed). C^0 submanifold means that for each point on the boundary of the causal future, there is a 3-dimensional neighborhood of a point on the boundary that is topologically equivalent to an open ball in R^3 (Euclidean 3-space). In general, the boundary of the causal future will not be a smooth manifold as is evident from the lightcone and the boundary of a disconnected set.

 

The C^0 (topological) manifold structure by taking the so-called Riemann Normal coordinates of 4-dimensional spacetime around any point p on the boundary of a causal future. For a sufficiently small neighborhood, one can choose one of the coordinates to be timelike (since the neighborhood can be chosen to be Minkowski). The integral curves of the tangent vector of this coordinate will intersect the boundary in exactly one point because the boundary is achronal (no two points are joined by timelike curves). So the remaining 3 coordinates can be used to define a homeomorphism to R^3. 

Such considerations will become important when we discuss Cauchy surfaces, trapped surfaces, and the proof of the Penrose theorem.

Global Causal Structure of Spacetime

Penrose theorem relies heavily on certain global assumptions about spacetime. Locally it is fairly clear what is happening topologically in spacetime given that it is a Lorentz manifold by the General Theory of Relativity. But when you stitch together these locally Minkowski spacetime neighborhoods, the resulting spacetime could have all sorts of pathological conditions. But when we look around with our telescopes we don't see any pathologies in spacetime. It is important to make an assumption that the spacetime starts out being nice and smooth and then determine what happens when gravity becomes too strong.  For example, an "asymptotically flat" spacetime is consists of a 3-dimensional space that extends out to infinity where the gravitational field (aka the curvature of spacetime) becomes negligible far away from the source of the field (typically a massive object such as a star). Near the source spacetime is curved but far away from the source spacetime is almost "flat" (hence the name "asymptotically flat").

Asymptotically flat spacetime

The Cauchy Hypersurface condition satisfies a global niceness condition. It states, that there is an initial connected (not broken up) smooth 3-dimensional space that is spread out infinitely from which all of the spacetime can be developed in a well-defined fashion. In fact, the assumption is that the entire spacetime can be "built" out of slices of spacetime at each instance of time. Asymptotically flat spacetime is a perfect example, of a spacetime satisfying the Cauchy Hypersurface condition. 

Cauchy Surfaces

A spacetime that satisfies the Cauchy surface condition has some nice properties. In fact, the technical definition of a Cauchy surface is a surface having the property that every timelike curve (a curve that is pointing in the chronological future of an event) will intersect it. It turns out (from the work of Choquet-Bruhat and Geroch) that spacetimes can be built smoothly from Cauchy surfaces. In addition, such spaces are also known to be "Globally Hyperbolic". Without getting too technical, it simply means that in such a spacetime you cannot go back in time (no closed timelike geodesics) and that there are no "holes" or gaps in the spacetime (the intersection of the causal future of an event p and the causal past of another event q that lies in the causal future of p is compact). It turns out that all Cauchy surfaces are topologically equivalent (homeomorphic to each other).

An intuitively obvious but crucial consequence of the Cauchy surface condition is that every point on the trajectory of a light curve in spacetime can be traced back to a point on the Cauchy surface using a timelike curve. 

In this picture, you have two Cauchy hypersurfaces and a light signal that goes from event P1 in one Cauchy surface to event P2 in the other. But the point P2 is also the evolution of a point that is the intersection of the perpendicular timelike curve with the first Cauchy surface. The timelike curves that are perpendicular to each Cauchy surface define a homeomorphism (1-1 topological equivalence) between the two Cauchy surfaces. We saw earlier, that the boundary of the causal future of a spacelike surface is an achronal C^0 manifold generates by null geodesics (light rays as above). The timelike curves coming down from a point P2 on the boundary to the Cauchy surface Sigma_1 will map open sets to open sets, so it is a homeomorphism onto its image. This fact will become important in the proof of Penrose's theorem.

Raychaudhuri's focusing equation

Until now we have not really discussed the effect of gravity on light rays and the curvature of spacetime. The first and most famous verification of the General Theory of relativity was the observation of the bending of light during a solar eclipse by Arthur Eddington in May 1919. The phenomenon of gravitational lensing is well known today. Light rays emanating from stars behind a massive object (like a black hole) will be bent when they pass near the massive object. 

Gravitational lensing (Credit: ESA/Hubble & NASA)

 

Raychaudhary was the first to study the implications of Einstein's equations for the collective behavior of families of geodesics in spacetime (such as families of light rays or families of trajectories of particles).  He showed that since gravity is an attractive force, neighboring geodesics are bent towards each other and will eventually intersect. The intersection of infinitesimally close neighboring geodesics has a very important consequence. They are known as focal points or conjugate points. They have been studied extensively in the context of Riemannian differential geometry and they usually have important consequences for the global differential geometry of surfaces. Penrose and Hawking were the first to study them in the context of relativity and spacetime.

A familiar situation in ordinary Riemannian geometry where geodesics intersect is the example of a sphere. 

Credit: Hawking-Penrose

If a geodesic in Riemannian geometry has a conjugate point, then it cannot be a length minimizing geodesic as seen on the sphere. If a great circle from a point p to q encounters a conjugate point r before reaching q, then it will not be the shortest path between p and q, because there will be another shorter great circle that directly joins p and q. 

There are analogous implications of the existence of conjugate points for timelike and null geodesics in spacetime. The Raychaudhuri equation helps determine the conditions under which geodesics will encounter conjugate points. This brings us to assumption d) of Penrose's theorem - the "Null Energy" condition. The null energy condition (also known as the Weak Energy Condition) states that the local energy at any point is always non-negative. From Einstein's field equations it turns out that the Null energy condition is equivalent to the last term in the right-hand side of the Raychaudhuri equation being positive. This means that the entire right-hand side of the equation is bounded below by the square of the convergence.  By solving the Raychaudhuri inequality (ignoring the shear terms and the energy term both of which are positive), one can show that the convergence factor is bounded below by a function of the affine parameter that depends on the initial convergence factor and the initial parameter value.  

Now by itself, the Raychaudhari equation does not imply that all null geodesics will have conjugate points. After all light rays tend to spread out, so the initial convergence factor is usually negative. If you light a candle or if a star explodes in a large flash of light, the light rays will expand out spherically. If the initial convergence factor is negative, then even if gravity tries to focus the light rays back, it may not be enough to make them meet. That is unless gravity is so strong that the convergence factor starts out being positive. It seems counter-intuitive to imagine, but that is exactly what happens with a trapped surface. If the initial convergence factor is positive, then in a finite period of time (measured by the affine parameter), the convergence factor will blow up to infinity, meaning one will encounter a focal point.

  Closed Trapped Surfaces

The final assumption of Penrose's theorem, which is based on the fact that you are in a situation of very strong gravity is e) The "Trapped Surface" condition. The following two diagrams will illustrate the contrast between a "normal" 2-dimensional surface in spacetime and a "trapped" 2-dimensional surface.

Credit: Hawking-Penrose

With a close trapped surface, you have a compact 2-dimensional surface (with no boundary) such as a sphere from where the light cones at the surface will always be tipped inwards, which is the same thing as saying that the initial convergence factor is positive (or equivalently the null expansion factor is negative). Since the Raychaudhuri equation shows that the convergence factor for light rays emanating from the trapped surface is bounded below by the initial convergence factor, the light rays will always have a positive convergence factor. If light rays (or null geodesics) can be extended indefinitely, which is assumption b) of Penrose's theorem, then eventually geodesics will have to encounter a focal point after a finite period of time. The terms caustics, conjugate points, and focal points are all used in the literature to describe the same concept. 

There has been a tremendous amount of research on the conditions under which trapped surfaces will form. The most obvious example is in the case of a spherically symmetric black hole when the event horizon is formed. Compact spacelike surfaces behind the event horizon are trapped surfaces. Even if you deform the spacetime so that it is not spherically symmetric, the trapped surfaces will continue to form.  

However, there are theorems that show that even in the absence of spherical symmetry, trapped surfaces can form during gravitational collapse or conditions of strong gravity.

Putting it all together - the punchline

To summarize, we know from the local causal theory of spacetime that the boundary of the chronological future of a spacelike surface is generated by null geodesics (think of the light cone in Minkowski space whose surface consists of light rays). Think of the surface of a star that is undergoing a supernova explosion. The particles emanating from the surface are moving into the chronological future of the star and the light rays from the boundary of the chronological future in spacetime. In particular, the boundary of the chronological future of a trapped surface is generated by null geodesics.  What is different about a trapped surface as opposed to any other surface (such as the surface of a star) is that the light rays will all start focusing on each other. If you let the light rays travel indefinitely, they will have to intersect their infinitesimal neighbors at some point, so there will be a focal point on each geodesic. Penrose proves that this contradicts the Cauchy surface condition. The easiest way to visualize the proof is to examine the following diagrams. They are due to Penrose and are taken from his 2020 Nobel lecture.

Penrose uses normal null geodesics that are emanating from the trapped surfaces and shows them converging. A key fact is that any null geodesic that encounters a focal point stops being null and becomes timelike after it crosses the focal point. Hence it must enter the interior of the chronological future (see the image below). 

The proof of this fact is a bit subtle (see Hawking-Ellis Proposition 4.5.12 or Witten Section 5.2).  The heuristic argument is that if you take a geodesic \gamma from p to r containing a focal point at q will allow an infinitesimally nearby geodesic that will also join p and q. Then the neighboring geodesic plus the segment qr will have a "kink" at q, which means that this new curve is not a geodesic. But this curve has the same length as \gamma. By smoothening out the kink we can create a path that reaches its destination to the past of q. This implies that the original curve \gamma is timelike.  

The fact that the null generators of the boundary leave the boundary and enter the interior after a finite amount of time implies that the boundary itself has to have a finite extent. In other words, the null boundary of the chronological future of the trapped surface is compact as can be seen in Penrose's diagrams above.  Now Penrose claims that this contradicts the Cauchy hypersurface (global hyperbolicity) condition.  Compare Penrose's conical diagram above with the Cauchy hypersurface diagram below.

Credit: Wald GR

  

The light ray from P1 to P2 is one of the generators of the cones shown above. It turns out that the null boundary of the causal future of any surface can be mapped down to the Cauchy surface using timelike geodesics that are orthogonal to the Cauchy surface. If you can extend the null geodesics indefinitely, then they would form a hypersurface that is topologically equivalent to the Cauchy surface. So the null boundary of the chronological future must be a Cauchy hypersurface. But that is not possible if the null boundary "closes up" onto itself. The technical way to state this is that the focusing of null geodesics results in the null boundary of the causal future of the trapped surface being compact. But the original assumption was that the Cauchy surface was non-compact (meaning extending out to infinity). 

Mapping of future null boundary to Cauchy surface 

(Credit: Sayan Kar, IIT KGP)

So the boundary curling up into itself to form a compact hypersurface cannot happen if the initial Cauchy hypersurface is non-compact. This means that the null geodesics generating the boundary of the causal future of a trapped surface must not be extendible beyond a certain point. This is incompleteness. The light rays are moving in direction of a singularity, but will never reach it. For example, the squiggly line below represents r=0 which cannot be reached by the light rays. The incompleteness (presence of singularity) means you can imagine slicing the surface along the singularity and spreading the null boundary out to map to the initial Cauchy hypersurface.  

Credit: Wald GR

The technical proof arrives at a contradiction by showing that the timelike mapping to the initial Cauchy hypersurface is compact and open. Being open and closed the image of the mapping is the entire Cauchy hypersurface since the latter is connected. But that is a contradiction because the Cauchy hypersurface is non-compact. Again the perfect example of a non-compact Cauchy hypersurface is the asymptotically flat spacetime surrounding an object such as a star.

Closing Remarks

Penrose's result was a turning point in the study of collapsed objects and the subsequent work by him and Hawking started a revival of interest in General Relativity. The Hawking-Penrose singularity theorems represent a landmark in the history of General Relativity. The developments spurred by their work would have possibly shocked Albert Einstein, the discoverer of relativity who always believed that singularities were a mathematical anomaly. But the exciting aspect of singularities is that they provide a hint of new physics that is yet to be developed. The study of black holes and singularities is an active area of research in theoretical physics, astrophysics, and mathematics.

Any discussion of black holes and singularities would be incomplete without a discussion of Penrose's cosmic censorship conjecture. Since Penrose's (and Hawking's) results show that singularities are inevitable when gravity is very strong, why is it that we don't encounter or observe singularities in the universe? Remember all the evidence for black holes is about dark supermassive objects that exercise enormous gravitational influence on their neighborhood. All known models of black holes have an event horizon. The images taken of the region around black holes show either gravitational lensing of light from stars behind the black hole or light spinning around the black hole near the event horizon (photosphere) or the accretion disk, which is a region near the event horizon where the matter is sucked into the black hole. 

  

It would be rather disconcerting if there were singularities just lying about in spacetime, but their invisibility led Penrose to make this conjecture.

Weak Cosmic Censorship Hypothesis: Nature abhors a naked singularity.

In other words, even though singularities are inevitable in general relativity, they are always hidden behind event horizons. No observer from outside (at "Null Infinity") can see a singularity. It turns out that this allows one to develop a nice theory of black holes since a lot of physics can be done without worrying about the singularities. Proving or disproving the cosmic censorship hypothesis is one of the central problems of mathematical general relativity.

References:

General Relativity by Robert M. Wald 

The large scale structure of space-time by S. W. Hawking and G. F. R. Ellis

Light Rays, Singularities, and All That by Ed Witten

Penrose's Work on Singularities: Part 1

Image of a black hole (2019)

Credits: Event Horizon Telescope collaboration et al.

The most beautiful thing we can experience is the mysterious. It is the source of all true art and science.  - Albert Einstein      

The year of black holes

For most people, 2020 will be remembered as the year that a pandemic raged across the globe killing hundreds of thousands and disrupting many lives. It was the year of social and political upheaval culminating in a contentious US election. But it was also the year that Black Hole research received the recognition that it deserved. The 2020 Nobel Prizes in Physics were awarded to Sir Roger Penrose, Andrea Ghez, and Reinhard Genzel for their pioneering work on Black Holes. In announcing the prizes the Nobel committee stated that half of the Nobel Prize was awarded to Roger Penrose "for the discovery that black hole formation is a robust prediction of the theory of relativity".  This post will try to unpack this statement and provide an intuitive feeling for the uninitiated reader of Penrose's remarkable work.  The discovery in question refers to a short 2-1/2 page paper entitled "Gravitational Collapse and Singularities" that Penrose published in January of 1965 in the journal "Physical Review Letters". In this paper, Penrose provided rigorous mathematical proof that under certain conditions the formation of a singularity in space-time is unavoidable. What is a singularity and what does it have to do with black holes? A singularity is a place and time where something really "bad" happens. It could be things like the curvature of space-time "blowing up" to infinity or a "tear in the very fabric of space-time". For example, a sufficiently massive object could collapse under its own gravitational force, and if there is nothing to resist the collapse it could distort space-time so badly that its curvature could end up becoming infinite. But such descriptions are somewhat misleading. What it really means is that there is a breakdown in the physical theory, and that a broader theory is needed to explain what is going on. Singularities don't have to be associated with points of infinite curvature. For example, in the case of the Big Bang, things seem to come out of nowhere, meaning particles don't have a history beyond a certain point in time in the past. Similarly, in the interior of a black hole particles or light rays could reach a point beyond which spacetime simply ceases to exist. 

Penrose Lecturing on the Big Bang in Berlin, 2015

Incidentally, any mention of Penrose's work and singularities would be incomplete without mention of Stephen Hawking. As portrayed in the movie "The Theory of Everything" both were young graduate students at Cambridge when Penrose made his discovery on singularities. Hawking immediately understood the significance of Penrose's work and applied it to Cosmology and the Big Bang. By essentially reversing the time direction of Penrose's argument, Hawking was able to prove that there had to be a singularity at the time of the Big Bang (the birth of space and time!). Both Penrose and Hawking were awarded the prestigious Adams Prize in 1966 for their research. They then went on to collaborate and publish a series of singularity theorems that are now collectively known as the Hawking-Penrose theorems. Hawking became an iconic figure in Science who overcame a debilitating disease (ALS) to make groundbreaking discoveries in physics. It is unfortunate that Hawking died in 2018, else he would have surely shared the Nobel prize with Penrose.   

In the next couple of sections, we will discuss the history of black holes and singularity research prior to Penrose's publication of his 1965 paper.

The Schwarzschild Singularities

In 1915 Albert Einstein made history when he presented his General Theory of Relativity to the Prussian Academy of Sciences. Newspapers around the world hailed the discovery as the most important since Newton and Einstein became a household name. It heralded the dawn of a new era in Science with a new and transformed understanding of the universe. Just one month after the publication of his results on space, time, and gravitation, Einstein was stunned to receive a postcard from a lieutenant in the German army containing the first-ever exact solutions to his field equations of gravitation. Einstein's equations are highly non-linear differential equations and notoriously difficult to solve. Einstein had only been able to supply an approximate solution in the context of the planetary motion of Mercury.  But here was a postcard from someone posted at the Russian front that said "As you see, the war treated me kindly enough, in spite of the heavy gunfire, to allow me to get away from it all and take this walk in the land of your ideas."  Karl Schwarzschild was the Director of the Astrophysical Observatory in Potsdam, but as a patriot had decided to join the army to fight in the war. During breaks from the fighting on the Russian front, he had managed to find the time to not only read Einstein's latest papers but also solve Einstein's equations for the space-time surrounding a spherically symmetric, non-rotating, non-charged body. Einstein was impressed and replied "I have read your paper with the utmost interest. I had not expected that one could formulate the exact solution of the problem in such a simple way. I liked very much your mathematical treatment of the subject. Next Thursday I shall present the work to the Academy with a few words of explanation". 

Schwarzschild's postcard to Einstein and his metric

Elegant and beautiful as Schwarzschild's solution was, it had a problem. There were two singularities in it, one at the center of the body at radius r=0 and one at the radius r=2GM/c^2, known today as the gravitational radius (or Schwarzschild radius). Here G is Newton's gravitational constant, M is the mass of the body and c is the speed of light. At these two points the solutions "blew up", meaning that they shot up to infinity. Einstein did not consider these singularities as physically meaningful. In fact in the spirit of classical electrostatic and Newtonian gravitational potentials, he made the assumption that the Schwarzschild solution applied only outside the spherical region of radius r=2GM/c^2. It is unclear what he expected to happen inside that sphere, but he considered them mathematical pathologies that had no physical meaning. In fact, Einstein was wrong to dismiss the singularity at the Schwarzschild radius. Today, the sphere at this radius is known as the "event horizon" of a black hole. Inside this spherical region, nothing can escape the gravitational attraction towards the center. The escape velocity exceeds the speed of light, so even light cannot escape, hence the term "black hole". As for the singularity, it was discovered by Arthur Eddington and David Finkelstein that it was indeed just an artifice of the choice of coordinates used by Schwarzschild. Eddington and Finkelstein showed that the singularity could be "transformed away" by simply choosing a different set of coordinates (now known as the Eddington-Finkelstein coordinates). However, r=0 was still a bonafide singularity and could not be transformed away. That did not cause any worry for Eddington or anyone else since the mass was centered at r=0, so the belief was that the field equations did not apply there. 

Collapsing Stars

The first sign of trouble came in 1929 when a 19-year old Indian astrophysicist named S. Chandrashekhar performed some calculations on the final fate of stars on his sea voyage from India to England. The prevailing wisdom at that time was that a star that had spent all its nuclear fuel, would start collapsing causing outgoing shockwaves that would eject its outer shell in a "supernova explosion". 

SN2018gv observed by the Hubble telescope

The inner core of such a collapsing star would settle into a stable object known as a white dwarf. It was believed that the white dwarf was prevented from further collapsing by something known as the "electron degeneracy pressure". In essence, the electron degeneracy pressure is a consequence of the Pauli exclusion principle in quantum mechanics which states that no two electrons can be in the same quantum state at the same time. So if you squeeze a collection of cold electrons in a small space, their repulsion due to Pauli exclusion principle and the electrostatic repulsion would result in an outward-facing pressure. The famous astrophysicist Ralph Fowler had shown that the electron degeneracy pressure was sufficient to resist the gravitational force and prevent the white dwarf from collapsing into itself.  In doing so he ignored the relativistic motion of particles. Chandrashekhar produced a "relativistic degeneracy formula" that showed that if the star's mass was greater than 1.4 times the solar mass, then the electron degeneracy pressure was insufficient to prevent the star from collapsing beyond a white dwarf. This implied that the star would keep shrinking and collapsing ad infinitum. This was a startling and highly disconcerting discovery. While most experts including Fowler believed that Chandra's results were correct, Arthur Eddington who was highly influential at that time reacted with derision. At a conference in 1935, Eddington told his audience that Chandrasekhar's work “was almost a reductio ad absurdum of the relativistic degeneracy formula. Various accidents may intervene to save a star, but I want more protection than that. I think there should be a law of Nature to prevent a star from behaving in this absurd way!”  Roger Penrose gave a nice talk on the topic, where he made the point that even though Chandrashekhar was correct in his calculations, Eddington was also right in believing that something in nature should prevent a star from collapsing indefinitely. Penrose does point out that Chandrashekhar (who was of a conservative bent of mind) was careful not to speculate about the eventual state of such an endlessly collapsing star. And it is known today that Eddington was wrong about the relativistic degeneracy formula. By dismissing Chandra's work, Eddington may have delayed much-needed research in the area of stellar collapse. 

Penrose talking about Chandra

Chandrashekhar was eventually vindicated in his work and today the size of 1.4 solar mass is called the Chandrashekhar limit (Chandra also won the Nobel prize in Physics in 1983 for his work on the structure and evolution of stars).  Meanwhile, Walter Bade and Fritz Zwicky proposed the existence of a neutron star just two years after the discovery of the neutron by James Chadwick in 1931. They predicted that a bigger star could squeeze the electrons and protons together to form neutrons which would generate a "neutron degeneracy pressure" that would resist runaway gravitational collapse in a manner similar to the electron degeneracy pressure. In 1939, Oppenheimer and Volkoff calculated an upper bound to the mass of cold, nonrotating neutron stars, analogous to the Chandrashekhar limit for white dwarf stars. This is known today as the Tolman-Oppenheimer-Volkoff limit and is estimated to be between 1.5 and 3 solar masses.  

But what if the mass was greater than 3 solar masses. Was there a "law of Nature" as Eddington expected that would prevent a star from collapsing indefinitely? In a 1939 paper entitled "On Continued Gravitation Contraction", Oppenheimer and Snyder showed, that a spherically symmetric ball of gas of sufficient mass would necessarily collapse beyond the stage of a neutron star. In their words "When all thermonuclear sources of energy are exhausted a sufficiently heavy star will collapse. Unless fission due to rotation, the radiation of mass, or the blowing off of mass by radiation, reduce the star's mass to the order of that of the sun, this contraction will continue indefinitely." They further showed that "the radius of the star approaches asymptotically its gravitational radius; light from the surface of the star is progressively reddened and can escape over a progressively narrower range of angles".  In essence, they showed that for a spherically symmetric ball of gas, the gravitational collapse will result in infinite density and the creation of an event horizon in finite time. 

The 1965 Paper

The reception to the Oppenheimer-Snyder paper was lukewarm due to the assumptions made about spherical symmetry. Ironically, at about the same time Einstein published a paper claiming that singularities could never form in General Relativity. His paper contained a mistake. Until Penrose's 1965 paper, there was a question as to whether objects like black holes and singularities were mathematical pathologies that could never exist in nature. The Russians Kalatnikov and Lifshitz claimed to have proved that singularities could not occur in cosmology. Their paper contained an error that was later corrected by Belinski. But it did not constitute a categorical proof that singularities could not occur and Penrose himself was skeptical of their methods. The objection towards assumptions made in prior work is best articulated by Penrose himself in his 1965 paper: 

"The question has been raised as to whether this singularity is, in fact, simply a property of the high symmetry assumed. The matter collapses radially inwards to the single point at the center, so that a resulting space-time catastrophe there is perhaps not surprising. Could not the presence of perturbations which destroy the spherical symmetry alter the situation drastically? The recent rotating solution of Kerr [also possesses a physical singularity, but since a high degree of symmetry is still present (and the solution is algebraically special), it might again be argued that this is not representative of the general situation. Collapse without assumptions of symmetry will be discussed here."  

There are a few notable facts about Penrose's result. 

• Penrose makes a very generic argument without making any assumptions of symmetry.  Unlike prior results on black holes and singularities, which rely on explicit solutions of Einstein's equations, Penrose's work makes use of differential topology and global methods in geometry. 
• Second, Penrose gives a very precise definition of singularity which is broader than the usual definition based on infinite curvature. Specifically, Penrose uses a concept in differential geometry called "geodesic incompleteness" as a proxy for the presence of singularities. Incompleteness means you cannot go past a certain point in space-time, which is an indication of a breakdown in the predictability of space-time. 
• Third, Penrose's result is a negative statement in the sense that it simply says that under certain reasonable assumptions about space-time and gravitational collapse, space-time has to become incomplete. It says nothing about where exactly the completeness breaks down and it does not even make any claim about the nature of the singularities that would lead to such incompleteness. However, what Penrose's paper does do is provide rigorous mathematical proof that incompleteness is inevitable if certain reasonable conditions are met.

With regard to the implications of his result, Penrose makes this intriguing remark. "If, as seems justifiable, actual physical singularities in space-time are not to be permitted to occur, the conclusion would appear inescapable that inside such a collapsing object at least one of the following holds: (a) Negative local energy occurs. (b) Einstein’s equations are violated. (c) The space-time manifold is incomplete. (d) The concept of space-time loses its meaning at very high curvature – possible because of quantum phenomena. In fact (a), (b), (c), (d) are somewhat interrelated, the distinction being partly one of attitude of mind."

Geodesic Incompleteness

Geodesics refer to the "straightest possible" curves on a surface. For example, a straight line is a geodesic on a flat plane. The longitudinal lines on a sphere (great circles) are geodesics. They are curves of extremal (maximum or minimum) length between two points.  Typically one thinks of the shortest path between two points. However, in space-time, it is more appropriate to consider the time it takes for an object or a signal to travel between two points (events). In General Relativity, freely falling bodies in a gravitational field follow geodesics. In addition light rays also follow geodesics. They are the analogs of "straight" lines on the plane.   

Penrose's diagram showing a singularity

A  surface or a manifold is said to be geodesically complete if starting at any point p you can follow a smooth path indefinitely in any direction. A plane and a sphere are both geodesically complete. But if you remove a point from the plane ("punctured plane") you get a geodesically incomplete space. If you follow a straight line going towards the missing point at a certain speed, then after a finite amount of time you will hit the puncture ad you cannot go any further. You can think of the point that is removed as the singularity. Of course, a punctured plane is an artificial example, because it resides in an ambient smooth space namely the plane and one can remedy the incompleteness by simply adding back the point. But in general, that may not be possible. In differential geometry, you can define manifolds without immersing them in a larger space. If you follow a geodesic path and the path cannot be continued after a certain point, then the manifold is said to be geodesically incomplete. For example, a light ray or a spaceship moving towards the center of a Schwarzschild black hole will not have a future after a finite amount of time because it will encounter the singularity at r=0. Time and space literally come to an end at that point. Penrose has a rather amusing footnote in relation to his reference to space-time incompleteness: "The “I’m all right, Jack” philosophy with regard to singularities would be included under this heading!"   "I'm all right, Jack" is a well known English expression indicating smug and complacent selfishness. It was also the title of a well known British movie starring Peter Sellers. Clearly, singularities are not very accomodating when it comes to letting things and signals from getting through. Technically speaking Penrose's singularity theorem should truly be called "Penrose's Incompleteness Theorem".  

The Singularity Theorem

Penrose's paper is a mathematician's dream to dig into. But sadly for the uninitiated, it would be daunting to comprehend as it uses sophisticated mathematical concepts. The physicist Ed Witten said jokingly in a lecture that there are a small set of ideas in the paper that if understood would make even the uninitiated an expert. However, Witten's comment is addressed to his colleagues at the Institute for Advanced Study who could be called anything but uninitiated. 

Witten's lecture at IAS on singularities

Regardless of my misgivings about Witten's assessment, I will attempt to convey the key ideas Penrose's theorem and its proof. Penrose lists five assumptions, which by themselves are reasonable but together lead to a mathematical inconsistency. The five assumptions are as follows:

1. "Past and Future": Space-time is a smooth manifold with a clear definition of past and future everywhere.
2. "Null Completeness":  Every path built out of light rays can be extended indefinitely into the future. 
3. "Cauchy hypersurface condition": Initial condition of space based on the distribution of matter allows one to determine its evolution over time (in relativity space-time is a dynamical entity that evolves with time).  A Cauchy surface is a "nice" initial surface in space-time that can be used to predict the future dynamic evolution of spacetime. Penrose makes a crucial assumption that there exists a "non-compact" Cauchy hypersurface. Non-compact Cauchy hypersurface means a 3-dimensional surface that extends out to spatial infinity. For example, if there is a "non-compressed" spatial distribution of matter and energy, then locally space will be curved but it will be surrounded by "flat" space that extends out to infinity.
4. "Non-negativeness of local energy": The local geometry of space-time is affected by the local distribution of energy. The non-negativeness of energy implies the non-negativeness of curvature. This is essentially stating that gravity is an "attractive force". 
5. "Trapped surface": This is the one and only assumption that is unique to a black hole situation. Trapped surfaces are special types of spherical surfaces where all light rays that emerge from them start bending towards each other. In general, outgoing light rays will spread out from the surface of a sphere. But behind an event horizon gravity is so strong that space itself is shrinking with time. If the shrinking of space is faster than the spreading of light, then over time the light rays will start focusing towards each other. The great advantage that trapped surfaces have over previous approaches to singularities is that they are robust to small perturbations away from spherical symmetry. So even if the space-time is distorted so that it is not spherically symmetric, trapped surfaces will continue to form behind the event horizon. 

As a consequence, if we assume that assumptions 1, 3, and 4 are true and if we assume the presence of a trapped surface (assumption 5), then we have to conclude that assumption 2 is false, which implies that incompleteness of space-time is inevitable. So stated more simply Penrose's theorem states that for any reasonable space-time where the matter is distributed in a concentrated region of space and space-time becomes close to "flat" as we move away from that region, the presence of trapped surfaces will inevitably result in the incompleteness of space-time (aka singularities).  We will discuss these assumptions in more detail and the proof of Penrose's theorem in Part 2 of the article. 

    

Talking about relativity to high school students.

Sometime in March of 2019, my colleague Howard Dierking asked me if I was interested in talking about my mathematical journey to a group of high school students. His wife Jennifer taught at the Bear Creek School in Redmond. She was looking to expose her kids to some outside perspectives on academia and research. She had a good reason for it - the students in her class were precocious. Rather than talk purely about my journey into Math, I thought I would mix relativity and geometry and throw in some fun facts about topology, gravitational waves, and GPS satellites. These were all topics that I was fascinated by. I was intrigued by the fact that an esoteric subject like General Relativity has a very practical application to location tracking via GPS satellites. Anyone who has used turn-by-turn directions on their smartphone knows the importance of accurately tracking one's location. Moreover, if you are in a plane that is landing using autopilot you know your life depends on the accuracy of those GPS satellites. But did you know that there are corrections that need to be made to the formulas to account for Special and General Relativistic effects? Time slows down when things travel fast such as the GPS satellites for instance. Time also goes faster the farther you are from the earth's gravitational field. For accurate triangulation of the location of a device, the devices on the GPS satellites will apply these relativistic corrections. Else your location will be subject to drift and you may end up very from where you need to be! To learn about some of these topics take a look at this slide presentation.

    

The Road to Relativity - High School Presentation from Prashanth Adhikari

Einstein's road to General Relativity

One of the most fascinating aspects of the Einstein Conference in Berlin was an exhibition at the basement of the Harnack Haus where Einstein's original manuscripts on GR were put on display. These manuscripts were loaned by the Hebrew University of Jerusalem to the Max Planck Society for the occasion. It was a joy to browse through the manuscripts and see the development of ideas that led to Einstein's discovery.  It contains Einstein's explorations, false starts, mistakes, corrections, and his eventual triumphant discovery of the field equations of gravitation. It is a remarkable record of the creative process of a genius whose discovery made front-page news around the world and shook up the world of physics. The Exhibition shows Einstein having the "happiest thought of his life" when he realized that the principle of equivalence was really the reflection of a geometric property of space-time. Einstein realized that he needed new mathematics to turn his ideas into equations. Einstein had dismissed his teacher Hermann Minkowski's geometric reframing of special relativity in terms of 4-dimensional space-time and light cones as "superfluous erudition". Maybe it was because Minkowski had called him a lazy dog with a lackadaisical attitude towards Mathematics. But the problem of Gravitation made Einstein appreciate the value of the geometrical approach. In fact, it became his main approach to his attempts to unify gravitation and electromagnetism which lasted all the way until his death. The manuscripts show Einstein learning the basics of Riemannian geometry with the help of his childhood friend Marcel Grossman. His early attempts at constructing the equations of gravitation using the scalar curvature failed. He started again and as he was getting close, he realized to his alarm that the great David Hilbert was close to finding a solution to the problem. Hilbert had started working on uncovering the field equations for gravitation after listening to some lectures by Einstein in Goettingen. Hilbert's approach was based on variational principles. Einstein's realized to his dismay that his decade long mission to find new equations of gravitation would be completed by Hilbert. A lot was at stake, so he doubled down. In the end, they were both very close to each other in publishing their results. But Hilbert was gracious enough to give Einstein the full credit for the discovery. Einstein also later published a variational derivation of the field equations of gravitation. In fact, the name Einstein-Hilbert action is given in honor of both of these giants of science.   

I created this slide presentation for a college reunion in 2016 in honor of a beloved Mathematics professor who was retiring that year. Several of Dr. Shobha Madan's former students who were accomplished mathematicians had gathered at IIT Kanpur to present talks in their areas of expertise. I was the only "non-academic" person and the only one to attend from outside India. And I chose to speak on an academic topic. So go figure. But it was a wonderful experience. Anyway, here is the slide deck.

The Road To Relativity from Prashanth Adhikari

What is Perspicacia (Part 2)?

Some of the greatest insights in physics have come out of paradoxes and incompatibilities between well-established theories of nature and newly observed phenomena. In Part 1 of this blog, we saw the development of classical physics up to the 19th century. At the dawn of the 20th century, it was widely believed that the end of physics was near. Lord Kelvin, a pre-eminent physicist of the day is said to have famously pronounced (in a possibly apocryphal event) "There is nothing new to be discovered in physics now, all that remains is more and more precise measurement". Newton's laws were spectacularly successful in describing the motion of bodies, both earthly and celestial. The macroscopic and thermodynamic properties of matter could be explained in terms of the statistical behavior of atoms that obeyed Newtonian mechanics. Maxwell's equations gave a complete mathematical description of electromagnetic phenomena. While Faraday's original formulation of electromagnetic theory did not postulate any medium for the transmission of electromagnetic fields, it was widely believed that an all-pervading material called the aether was the medium for electromagnetic phenomena. Even Maxwell is said to have believed that electromagnetic waves traveled as oscillations of the aether.  Also, classical physics at the turn of the century rested on the twin foundations of causality and determinism. These principles demanded that given initial conditions, the dynamic evolution of any system is governed by physical laws that can be expressed by differential equations, whose solutions allow precise calculation of positions and velocities of particles involved in the dynamics of the system.

However, there were some ominous clouds gathering over the prevailing sense of completeness and finality in the subject. Attempts to fit the Faraday-Maxwell theory of electromagnetism into the mechanistic framework of the aether were ad hoc at best. No experiment could detect the all-pervading aether. Precise experiments such as those of Michelson and Morley failed to detect any changes in the speed of light due to the earth's motion through the aether (unlike for example the change in the speed of sound due to the motion of the sound source through the air). There were other vexing astronomical anomalies such as the perihelion motion of Mercury's orbit. On the earthly front, the existing wave theory of light could not explain the photoelectric effect, where light shining on a metal resulted in the production of an electric current whose energy depended only on the frequency of the light as opposed to the intensity of the light. There were still many questions about the existence of atoms, their structure, and their relationship to radiation phenomena. But what triggered the quantum revolution was the problem of "black body" radiation.

The Quantum Revolution

A black body is one that absorbs all the light falling on it. When a black body is heated it starts to radiate and a major problem in theoretical physics was to derive the observed spectrum of the radiation using known laws of physics. A theoretical model created by Rayleigh and Jeans based on the classical theory of electromagnetism produced the paradoxical result that the energy emitted by the body would grow indefinitely as a function of frequency. This divergence was called the 'ultraviolet catastrophe' by the physicist Paul Ehrenfest. Max Planck overcame this paradox in 1900 by postulating that energy could only be exchanged in discrete units called "quanta". A quantum of energy is related to the frequency of the emitted radiation via the famous Planck's constant h. Using his equation Planck was able to produce the observed distribution of radiation energy as a function of radiation frequency and temperature. Planck was of a conservative bent of mind and by his own admission hardly a revolutionary. But he had the courage to go where the problem took him and make a counter-intuitive postulate in order to resolve a paradox. But Albert Einstein in a fashion typical of him took what was an ad hoc postulate and generalized it to a universal property of light itself (and all electromagnetic radiation). He postulated that light consists of packets of energy called photons, which carry momenta and energy that can knock electrons off certain metals resulting in the photoelectric effect. This was a dramatic development since Thomas Young had disproved in 1800 Newton's assertion that light consisted of corpuscles by demonstrating the interference of light using the double-slit experiment. The wave theory of light had been an established fact for more than half a century and had gotten an additional boost from Maxwell's theory of electromagnetism. But here was Einstein postulating that light consisted of particles. Einstein of course knew that this implied the paradoxical statement that light was both a wave and a particle. He hoped that a deeper theory of nature would resolve this paradox. Little did he know that the theory that would eventually emerge would go against his deepest beliefs about nature.  Quantum mechanics got another early boost from Niels Bohr's work on atomic theory. Early models of the atom such as Rutherford's planetary model suffered from the uncomfortable fact that they violated known laws of electromagnetism. Unlike planets revolving around the sun, an electron revolving around a nucleus should radiate energy and fall into the nucleus. This led Bohr to conclude that the stability and spectrum of atoms could best be explained by postulating that electrons in an atom can assume only certain discrete energy states. The observed line spectra of gases such as Hydrogen could then also be explained as the result of transitions between these discrete energy states (quantum jumps). The hallmark of early quantum theory was the realization that when it comes to atomic and radiation phenomena, energy is not a continuous entity (as in classical physics) but comes in discrete "lumps" (terminology used by Richard Feynman - see Feynman Lectures in Physics Vol. 3). 

Numerous attempts to fit this realization into known classical models of electromagnetic theory and Newtonian mechanics met with failure. In the early 1920s, de Broglie and Schrödinger developed a wave theory of matter that carried a tremendous intuitive and visual appeal to classical physicists. Schrödinger's famous equation was also a major milestone that allowed one to describe the time evolution of a system using a "state vector" known also as the wave function. Schrödinger's interpretation of the wave function was in terms of physical "matter waves". But it became clear over the next 2 decades largely due to the work of Heisenberg, Pauli, Max Born, and most importantly Niels Bohr, that all visual and classical ideas of nature had to be abandoned when it came to atomic phenomena. The consensus that emerged (known as the Copenhagen Interpretation) was that wave and particle were complementary aspects of nature that manifested themselves depending upon the kind of experiment that was being performed. All measurements carry an inherent uncertainty where dual observables such as position and momenta cannot be simultaneously measured with unlimited precision. The wave function itself is an "infinite-dimensional" mathematical entity, whose only physical interpretation is that its "modulus square" represents the probability of finding a particle at a location during a measurement. Interestingly the author of this probabilistic interpretation of quantum mechanics (Max Born) attributed the idea originally to Einstein, who had suggested in an unpublished manuscript that the wave nature of light actually represented a "wave of probabilities" of photons. However, Einstein believed that these probabilities applied to an ensemble of photons and not to an individual photon. When it came to individual particles, Einstein firmly believed that deterministic laws would apply. On the other hand, Max Born's interpretation of the wave function and the later Copenhagen interpretation of quantum mechanics asserted that determinism did not apply to atomic phenomena. When it came to atomic measurements one could only speak in terms of probabilities. Einstein never accepted this "probabilistic" interpretation of quantum mechanics and maintained till the end of his life that "God does not play dice with the universe". The debates that raged between Bohr, Heisenberg, Schrödinger, and Einstein could be counted as one of the greatest philosophical debates of all time. The Copenhagen interpretation of quantum mechanics is widely accepted by physicists today with some notable exceptions. 

As Feynman stated in his numerous lectures, it is safe to say that nobody really understands quantum mechanics. Niels Bohr, one of the founding fathers of quantum theory stated that anyone who is not shocked by quantum mechanics has not really understood it. But it is indeed one of the most successful theories of nature and is the foundation of much of modern physics and technology. Quantum mechanics plays a foundational role in solid-state physics, condensed matter physics, the theory of superconductivity, nuclear physics, stellar evolution, particle physics, modern chemistry, and even modern biology. Today's technology including semiconductors, lasers, MRI machines, electronics, etc. would not be possible without quantum mechanics. 

Even when he was wrong, Albert Einstein was remarkably prescient and insightful. His ingenious thought experiments aimed at showing that Quantum Mechanics was incomplete, culminated in the famous EPR (Einstein-Podolski-Rosen) paper that highlighted the "spooky non-local" nature of quantum entanglement of particles separated by large distances. However, entanglement is a fundamental and experimentally verified aspect of quantum mechanics. It is an active area of research in both physics and quantum information theory today. Quantum mechanics is also penetrating other areas of knowledge such as information science and computing. Research labs around the world are racing to build the first quantum computer. Such a development would transform computing and communication as we know it. 

Relativity

At the turn of the 20th century, it was realized that Maxwell's equations had the strange property that the speed of an electromagnetic wave (such as light for example) seemed to be the same in all frames of reference. This was clearly incompatible with the intuitively obvious concept of relative motion. For instance, two trains moving at the same speed in the same direction appear to be at rest relative to each other. Not so with light, which seemed to travel at a fixed constant speed of 186,000 miles/sec irrespective of the speed of the observer. The speed of light (c) shows up in Maxwell's equations as a constant number that can be computed from the electrical permittivity and magnetic permeability of empty space. Maxwell himself assumed that this speed must be relative to an absolute frame of reference namely the aether. The Michelson-Morley experiment of 1887 failed to measure any movement of the earth relative to the aether. Their experiment showed that light traveled at exactly the same speed both parallel and perpendicular to the motion of the earth (which contradicted the hypothesis of a drag in the aether caused by the earth's motion). Nobody appreciated the constancy of the speed of light more deeply than Albert Einstein. After all, as a teenager, he had performed a Gedankenexperiment (thought experiment) of racing behind a ray of light and wondering what he would see. Common sense indicated that he should see a standing light wave, but that seemed to violate Maxwell's equations. Einstein was also aware of Galileo's discovery of "inertial frames of reference" and the "principle of relativity". Galileo had shown that there was no fundamental difference between a frame of reference at rest and one at uniform motion relative to it. In other words, no experiment conducted within a uniformly moving frame of reference can tell whether it is at rest or at motion. So all uniform motion seems to be relative and all laws of physics should be the same in all inertial frames of reference. Einstein in 1905, resolved the paradox of the constancy of the speed of light in all frames of reference by developing a new theory of space and time called the "Special Theory of Relativity". By elevating the "Principle of Relativity" and "the Constancy of Speed of Light" to the status of fundamental postulates, Einstein derived a host of consequences that transformed our understanding of simultaneity of events, lengths of objects, and time intervals. In Einstein's theory, space and time lost their absolute and independent nature and blended into a 4-dimensional continuum called spacetime whose geometry was such that no signal could travel faster than the speed of light. The mathematician Hermann Minkowski announced rather dramatically in a 1908 speech, "henceforth space by itself and time by itself are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality".  Some of the remarkable consequences of this theory were length contraction of moving objects, time dilation (slowing down of moving clocks), and the famous equivalence of mass and energy given by the equation E = mc^2. 

Revolutionary as the special theory of relativity was, there were some unsatisfactory aspects to it, at least as far as Einstein was concerned. The circular nature of the definition of an inertial frame had not escaped his attention. Clearly, the notion of an inertial frame was a kind of idealization or approximation. If you were far away from all other objects and if there were no measurable forces then you could treat your frame of reference as an inertial one. The special status given to inertial frames irked him and made him ask the question as to why the laws of physics would not be the same in ALL frames of reference (and not just inertial frames of reference). In this, he was influenced by the philosophical ideas of Ernst Mach. Further, he puzzled over the nature of gravity. Newton's law of gravity suggests an "action at a distance" that is transmitted instantaneously across distances between two masses. This clearly violates the special theory of relativity where nothing can travel faster than the speed of light. There was also the mystery of the equivalence of inertial and gravitational mass as discovered by Galileo (which is equivalent to the statement that all objects of any mass accelerate at the same rate near the earth). All of this set the stage for arguably one of the greatest leaps of human intuition since the time of Newton, namely the discovery of the General Theory of Relativity.  Based not just on empirically observed facts, but on phenomenological and philosophical considerations involving frames of reference, Einstein developed a dynamic theory of space, time, and gravitation that completely revolutionized our understanding of the universe. In the General Theory of Relativity, Einstein declared that the laws of physics were the same in all frames of reference (Principle of General Covariance). Moreover, spacetime in general relativity is a dynamic entity that is curved by the presence of matter. The curvature of spacetime manifests itself as the force of gravity. Bodies in motion follow geodesics (paths of least "distance") in curved spacetime. The geometry of spacetime in the vicinity of matter can be best understood using the mathematics of non-Euclidean geometry developed by Bernhard Riemann in 1854. The language and formalism of tensor analysis were used by Einstein to derive his famous field equations of gravitation. These equations provide a precise relationship between the geometry of spacetime and the distribution of matter and energy in a region. Einstein was able to show that Newton's Law of Gravitation appeared as a limiting case when gravitational fields were weak.  Einstein was able to precisely account for the observed perihelion motion of Mercury thus solving a longstanding mystery in astronomy. More remarkably he was also able to predict effects such as the bending of light due to gravity (gravitational lensing) and gravitational waves (recently detected by the LIGO observatories). 

General relativity has withstood the test of time and continues to be tested by precise experiments conducted on earth and space. It is one of the two pillars of modern physics today, the other being quantum mechanics. Einstein's theory has paved the way for much of the dramatic developments in modern cosmology and astrophysics including the theory of black holes, the Big Bang theory, and the theory of the expanding universe. 

However, lest one should think of Einstein's theory as an esoteric theory with no practical applications, it should be pointed out that both special relativity and general relativity play a critical role in modern GPS technology. The time dilation of atomic clocks due to the motion of satellites (a special relativistic effect) and the speedup of the clocks due to the location of the satellites above the earth (a general relativistic effect) impact the precision of calculation of the location of objects on earth. Modern GPS software accounts for the relativistic effects of geolocation. Anyone who questions the practical value of pursuing theoretical physics must stop to think about how the smartphone is able to provide precise turn-by-turn directions!

Remarkable applications of relativity and quantum mechanics can also be found in the health sciences - especially in the science behind MRI (Magnetic Resonance Imaging) and PET (Positron Emission Tomography). Amazing things happen when special relativity and quantum mechanics meet each other. The first one to discover this was Paul Dirac. Special relativity demands that equations of physics be written in such a way that they maintain their form in all inertial frames of reference (assuming that the gravitational effects are weak). The Schrödinger equation did not meet this requirement.  Many attempts to address this problem failed until Dirac produced his relativistic quantum equation in 1928. Two remarkable consequences of the Dirac equation were the discovery of spin angular momentum of a particle and the discovery of the positron. The former is the source of magnetic phenomena such as those used in MRI. The latter is a remarkable example of a theory predicting the existence of a completely unknown and new type of matter known as an anti-particle. A positron is an "anti-electron", which would annihilate an electron if it came into contact with it resulting in radiation energy. Positrons are used in the health sciences in PET scans (Positron Emission Tomography) using radioactive materials called tracers.  These are again examples of fundamental science having a profound impact on modern society. 

Physics Today

The last century has seen remarkable progress in the development of the physical sciences. The discovery of an expanding universe by Hubble and detection of the cosmic microwave background radiation (CMB) have allowed us to create an effective model of the universe and its evolution from the time of the Big Bang. Precision cosmology has also provided strong evidence for the inflationary expansion of the universe right after the Big Bang when the universe supposedly underwent a very rapid expansion and slowed down to allow for the creation of galaxies. X-Ray, Gamma Ray, and Infrared astronomy have provided strong evidence for the existence of extraordinarily compact astrophysical objects such as Neutron Stars and Black Holes. The recent dramatic discovery of gravitational waves by the LIGO observatories in Washington and Louisiana have provided us a window into the strange and violent nature of the universe. However, in many ways, we are in the same situation as the Greeks. Our model of the universe is incomplete and many problems remain unsolved. Two great puzzles of cosmology are the apparent presence of dark matter and dark energy. Dark matter refers to invisible matter whose presence is needed to explain the stability of spinning galaxies. Dark energy refers to a mysterious force that seems to cause the universe to expand at an increasingly rapid pace (as observed by the Hubble space telescope). It is estimated that roughly 5% of the universe is visible matter, 27% is dark matter and the rest (68%) is dark energy. However, we don't know what dark matter is and nor do we have a clear framework to describe dark energy. There are theoretical proposals such as WIMPs (Weakly Interacting Massive Particles) designed to explain dark matter.  But they have not been detected in any experiment. Dark energy is supposed to be the intrinsic energy of vacuum (represented by Einstein's cosmological constant) that causes a negative pressure leading to the expansion of the universe. However, most quantum field theories predict that the energy of the vacuum has to be 100 orders of magnitude larger than what is required to explain the observed rate of expansion. Again many alternative theories have been proposed but none are conclusive.

Quantum mechanics and special relativity have been remarkably successful in explaining the properties of matter and its internal constituents. A central problem of theoretical physics is Einstein's dream of unification of the forces of nature. The Standard Model of Particle Physics has been successful in unifying the electromagnetic, strong, and weak nuclear forces into a common theoretical framework. Experiments at the LHC (Large Hadron Collider) have verified the Standard Model by detecting the Higgs Boson. However, the force of gravity has resisted attempts at unification with the rest of the forces. String theory is a potential approach to unification, but many mysteries remain and string theory is far from being testable (let alone tested). Any such unification would require a quantum theory of gravity.  When gravitation and quantum mechanics meet in the strong field regime of a black hole, very interesting puzzles emerge. The most interesting of these is the apparent loss of information due to the evaporation of a black hole due to Hawking radiation. All of these puzzles, the proliferation of proposed theories to resolve them and the experimental attempts to probe higher energies and deeper realms of the cosmos suggest that we are at the cusp of another paradigm shift in our understanding of the universe.

Perspikacia

From ancient times to the present day, deep contemplation, precise reasoning, mathematical analysis, careful observation, and sophisticated experimentation have shown us that the world is not as it seems to be nor is it as one may imagine or wish it to be. Time and again visionary individuals such as Galileo, Newton, Faraday, Maxwell, Einstein, Bohr, Dirac, Feynman, Hawking, and many others have used their perspicacity to give us dramatically new perspectives on the inner workings of nature. In Science as in great art, it is individual perspicacity that produces shifts in human consciousness that deepens our understanding of the universe. While in some cases these individuals are driven by just "the pleasure of finding things out" (Feynman), in other cases they are propelled by the desire to "know the mind of God" (Einstein). There is something in human consciousness that takes pleasure in understanding the order of things. 

"The scientists’ religious feeling takes the form of a rapturous amazement at the harmony of natural law, which reveals an intelligence of such superiority that, compared with it, all the systematic thinking and acting of human beings is an utterly insignificant reflection."  - Albert Einstein