At the beginning of the 20th century, two new theories completely changed our understanding of space, time, and reality. For over 75 years since then, we have continuously studied the issues related to these two theories in order to find a way to unify them into a single theory that describes everything in the universe.
These two theories are the general theory of relativity and quantum mechanics. The general theory of relativity examines space, time, and their curvature at a macroscopic scale caused by matter and energy in the universe. On the other hand, quantum mechanics investigates the microscopic scale. Quantum mechanics has the principle of uncertainty, which states that we cannot simultaneously determine the exact position and velocity of a particle: the more accurately we measure the position, the less accurately we can measure the velocity, and vice versa. There is always an element of uncertainty or randomness, which fundamentally affects the nature of matter at the microscopic scale. Almost single-handedly, Einstein was responsible for the general theory of relativity and played a crucial role in the development of quantum mechanics. He summarized his feelings about quantum mechanics with the phrase: “God does not play dice.” However, all experiments have shown that God is indeed the unchanging player, never missing an opportunity to throw the dice.
National laws apply only within a nation, but the laws of physics remain the same in England, the United States, and Japan. They are also the same on Mars and in the Andromeda Galaxy. Moreover, they are consistent regardless of our velocity of motion; whether we are on a high-speed train, a jet airplane, or standing still on the ground, the laws of physics remain the same. Certainly, a person standing still on the ground is moving at a speed of about 30 kilometers per second around the Sun; the Sun is moving around the Milky Way at hundreds of kilometers per second and continues to do so. Yet these motions do not alter the laws of physics; they are the same for all observers.
Galileo discovered the independence of motion from the speed of the system, establishing formulas that express the laws governing the motion of objects such as cannonballs or planets. However, a problem arose when applying this to light. Since the 18th century, it has been discovered that light does not travel instantaneously from the source to the observer: it travels at a determined speed of about 300,000 kilometers per second. But speed is relative to what? It was thought that there must be a medium filling the space through which light travels. This medium was called “ether.”
THE IDEA OF LIGHT WAVES TRAVELING at a speed of 300,000 kilometers per second relative to ether suggests that an observer stationary relative to ether would measure the speed of light as 300,000 km/s, while another observer moving relative to ether would measure the speed of light as either less or more than 300,000 km/s. Particularly, the speed of light would need to change as the Earth moves relative to ether in its orbit around the Sun. However, in 1887, through a very precise experiment, Michelson and Morley demonstrated that the speed of light is always constant. Observers in different states of motion always measure the same speed of light at 300,000 km/s. What is the truth behind this? Why do observers moving at different speeds measure the same speed of light? The answer lies in the fact that our conventional understanding of space and time is no longer adequate. In a famous work in 1905, Einstein pointed out that the previously mentioned observers can measure the same speed of light if they abandon the idea of a universal time. Each person has their own time measured by their own clock. The times measured by these different clocks will be almost the same if they move slowly relative to one another, but the difference will become significant if they move at very high speeds. This phenomenon has been practically observed by comparing two identical clocks: one stationary on the ground and one on a plane moving at very high speed. At normal speeds, the difference in time between the clocks is negligible.
Einstein first presented his theory of relativity in the famous work of 1905, which is now referred to as “Special Relativity.” This theory describes the motion of objects in space and time; it demonstrates that time is not a universal quantity, as time exists freely and independently of space. The future and the past; up and down, right and left, front and back, are all dimensions of what we call spacetime. One can only move forward in time at a specific angle to the time axis. Thus, time can flow at different rates.
Special relativity combined space and time, but they remained a fixed framework in which events occur. We can choose to follow different paths in spacetime, but everything we do cannot change the spacetime framework. Since 1915, when Einstein articulated the general theory of relativity, everything changed. He had a revolutionary idea that gravity is not merely a force acting within a fixed spacetime framework but that the presence of matter and energy distorts spacetime. Objects, cannonballs, or planets would ideally move in straight trajectories through spacetime, but those trajectories are curved because spacetime is no longer flat. The Earth attempts to move in a straight path through spacetime, but the mass of the Sun has curved spacetime and compelled the Earth to orbit around the Sun.
Similarly, light attempts to travel in a straight line, but the curvature of spacetime near the Sun causes light emitted from distant stars to be deflected when it approaches the Sun. Normally, it is impossible to distinguish distant stars aligned in the same direction as the Sun. However, during a solar eclipse, most of the sunlight reaching Earth is blocked by the Moon, allowing us to observe the light from those stars. Einstein established the theory of general relativity during World War I when scientific observations could not be conducted. Shortly after the war, a British expedition observing the solar eclipse in 1919 confirmed the predictions of general relativity: spacetime is not flat but curved in the presence of matter and energy. This was a significant victory for Einstein. His invention completely transformed our previous understanding of spacetime. Spacetime is not a passive framework in which events occur. We can no longer think of space and time as eternal and unchanging by the events occurring within the universe; instead, they become dynamic quantities that interact with events occurring within them.
One important property of mass and energy is that they are always positive. This is why the gravitational force between objects is always attractive. For example, the gravitational force of Earth pulls us towards it no matter where we are on its surface. The gravitational force of the Sun keeps its satellites in orbit around it and prevents Earth from drifting into the dark void between stars. If mass were negative, spacetime would curve in the opposite direction, as if the surface of a saddle. The positive curvature of spacetime, which manifests the gravitational force as an attractive force, posed a significant problem for Einstein. Generally, people thought of the universe as static, but if space, and particularly time, were folded, how could the universe continue to exist eternally in a state almost identical to its current state?
Initially, the equations of general relativity predicted a universe that was either expanding or contracting. Einstein added a new term—known as the cosmological constant—to the equation relating the mass and energy present in the universe with the curvature of spacetime. This cosmological constant caused a repulsive gravitational effect. Thus, the attractive force of matter and energy could balance with the repulsive force of the cosmological constant. In other words, the negative curvature of spacetime caused by the cosmological constant could counterbalance the positive curvature induced by mass and energy in the universe. This would create a model of the universe that could exist indefinitely in the same state. If Einstein had kept his original equations, he would have predicted that the universe was either expanding or contracting. Before 1929, when Edwin Hubble discovered that galaxies were moving away from us, no one thought that the universe changed over time. The universe is expanding. Later, Einstein declared the cosmological constant to be “the biggest blunder of my life.”
However, whether or not the cosmological constant exists, the issue of matter curving spacetime remains generally poorly understood; particularly, matter can curve a region into a point separated from the rest of the universe, which becomes a “black hole.” Objects can fall into a black hole but cannot escape from it; to escape a black hole, matter must move faster than the speed of light, which is not permitted by the theory of relativity. Thus, matter would be trapped in the black hole and self-compressed due to gravity into an unknown state with an extremely high density. Einstein was greatly perplexed by this gravitational collapse; he did not believe in the possibility of such a collapse. However, in 1939, Robert Oppenheimer demonstrated that an old star with a mass greater than twice that of the Sun, once it has exhausted its nuclear fuel, will inevitably undergo gravitational collapse.
Then World War II broke out, and Oppenheimer became engrossed in the nuclear bomb program, forgetting about gravitational collapse. Researchers focused more on physics that could be studied on Earth. They were skeptical about predictions concerning the boundaries of the universe because they seemed impossible to test through observation. However, in the 1960s, remarkable improvements in the range and quality of astronomical observations led to renewed interest in gravitational collapse and the early universe. The accurate predictions of general relativity in this context remained unclear until Roger Penrose and I proved several theorems. These theorems demonstrated that spacetime must be curved and lead to singularities, which are regions where spacetime has a beginning or an end. The beginning point is the Big Bang, which occurred approximately 15 billion years ago, and the endpoint would be at a star that is the victim of a collapse or with all matter falling into a black hole.
The prediction of the existence of singularities according to Einstein’s theory of general relativity has been confirmed but has led to a crisis in physics. At a singularity, the equations of general relativity, which relate the curvature of spacetime to the distribution of mass and energy, become undefined. This means that general relativity cannot predict what happens at a singularity. In particular, this theory is unable to predict how the universe must be born at the Big Bang. Thus, general relativity is not a complete theory. It requires an additional part to determine how the universe must be born and what happens when matter collapses under its own gravity. It appears that quantum mechanics is the necessary supplement. In 1905, the year Einstein wrote the theory of special relativity, he also published a paper on the “photoelectric effect.” He observed that when light shines on certain metals, it causes the emission of charged particles. What is puzzling here is that if the intensity of the light is reduced, the number of emitted charged particles decreases, but the speed of those particles does not decrease. Einstein proved that this can only be explained by considering the incoming light not as small continuously varying portions, as people then believed, but as discrete packets moving in a defined direction.
THE IDEA THAT LIGHT EXISTS ONLY IN QUANTA was proposed by German physicist Max Planck a few years earlier. Planck used the concept of quanta to explain why a piece of metal glowing red does not emit an infinite amount of heat, but he regarded quanta merely as a theoretical trick without any physical correspondence. In his work, Einstein demonstrated that quanta could be directly observed. Each particle emitted corresponds to a quantum of light striking the metal. This was recognized as a significant contribution to quantum theory, earning Einstein the Nobel Prize in 1922 (he should have received the award for his theory of general relativity, but at that time, the idea of curved space and time was considered too speculative, so he was awarded for his work on the photoelectric effect, which was also deserving of recognition).
It wasn’t until 1924 that people fully understood the photoelectric effect when Werner Heisenberg pointed out that this effect implies it is impossible to measure the exact position of a particle. To identify a particle, light must be shone on it. However, Einstein proved that one cannot use an arbitrary small amount of light; a minimum of a beam or a quantum is necessary. This beam of light striking the particle disturbs it and imparts some velocity. The more precisely we want to measure the position of the particle, the greater the energy of the incoming beam of light must be, which in turn causes a stronger disturbance to the particle. Regardless of how the particle is measured, the uncertainty in position multiplied by the uncertainty in velocity is always greater than a small constant. Heisenberg’s uncertainty principle demonstrates that one cannot precisely measure the state of a system, nor predict its future properties accurately. The best we can do is predict the probabilities of different possible outcomes. This random factor greatly troubled Einstein. He did not believe that physical laws could fail to provide a deterministic prediction; ambiguity was unacceptable. Yet, all tests have shown that quantum phenomena and the uncertainty principle are unavoidable, manifesting in every branch of physics.
Thus, Einstein’s theory of general relativity is a classical theory that has not been linked with the uncertainty principle. A new theory is needed that combines general relativity with the uncertainty principle. In most cases, the differences between the new theory and classical general relativity are extremely small since the uncertainty principle predicts quantum effects only play a significant role on small scales, while general relativity examines the structure of spacetime on very large scales. The theorems about singularities by Roger Penrose and myself have shown that spacetime can only bend significantly at very small scales. The consequences of the uncertainty principle will become very important, and the predicted results will be substantial.
One difficulty Einstein faced with quantum mechanics and the uncertainty principle stemmed from the common notion, consistent with intuition, that a system has a definite history. A particle must be either here or there; it cannot be half here and half there. Similarly, the event of an astronaut landing on the Moon either happens or it does not; it cannot occur halfway. One cannot be partially dead or slightly in the womb. Either one exists or does not exist. But if a system has a unique and definite history, then the uncertainty principle leads to all kinds of paradoxes, such as a particle being simultaneously present everywhere or an astronaut being half on the Moon.
American physicist Richard Feynman proposed a subtle way to avoid the paradoxes that troubled Einstein. Feynman became famous in 1948 for his work on quantum electrodynamics. He was awarded the Nobel Prize in 1965 alongside American Julian Schwinger and Japanese Shinchiro Tomonaga. Einstein disliked formality and trivial debates; he resigned from the National Academy of Sciences because he felt too much time was wasted on discussions about admitting new members. Feynman was renowned for his contributions to theoretical physics. He passed away in 1988. Among his contributions, the diagrams named after him—the Feynman diagrams—are fundamental for most calculations in particle physics. More importantly, his concept of summing over histories is key.
THE IDEA IS THAT A SYSTEM DOES NOT HAVE A SINGLE unique history in spacetime, as is conventionally accepted in non-quantum theory, but rather possesses all possible histories. For instance, at a point A at a certain verification time, there is a particle. Normally, we consider this particle moving along a straight line from A. However, according to the sum over histories, the particle can take every possible path starting from A. This occurs similarly to when we drop a drop of ink onto blotting paper. The ink particles will spread across the blotting paper in every conceivable direction. With each path or history of the particle, a certain number of path-dependent amplitudes are associated. We obtain the probability of the particle moving from A to B by summing all the amplitudes related to the paths from A to B. For most paths, the amplitude associated with a path will cancel with the amplitudes of nearby paths, contributing very little to the probability of the particle moving from A to B. However, for a straight path, its amplitude will enhance the amplitudes of nearby straight paths. Thus, the dominant contribution comes from the straight and nearly straight paths. This is why a primary particle, when moving in a bubble chamber, leaves a nearly straight trail. But if a barrier with a narrow slit is placed in the particle’s path, the trajectories of the particle will spread out behind the slit. Regardless of the straight path through the slit, elsewhere the chance of finding the particle will increase.
In 1973, I began researching the effects of the uncertainty principle on particles in curved spacetime near a black hole. I discovered the surprising fact that black holes are not completely black. The uncertainty principle allows particles and radiation to regularly escape from black holes. This result surprised me and many others and was received with skepticism. But looking back, it seems almost obvious. A black hole is a region of space from which nothing can escape unless it moves faster than the speed of light. However, Feynman’s sum over all histories states that particles can take every conceivable path in spacetime. Thus, a particle has the potential to move faster than light. The probability of a particle moving along trajectories faster than light is very small, but sufficient for the particle to escape the black hole’s grasp. Thus, the uncertainty principle allows particles to escape from a black hole—the most secure prison of all. The probability for a particle to escape from a black hole with a mass equivalent to the Sun’s mass will be very small because the particle must travel faster than light over many kilometers.
However, there may be many smaller black holes; they were created in the early universe. These primordial black holes are smaller than atomic nuclei but can weigh as much as a billion tons, like Mount Fuji. Such black holes could emit energy equivalent to that of a massive power plant. If we could find one of these black holes, we would have control over energy! Unfortunately, it seems that such black holes are scarce in the universe. The prediction of black holes emitting radiation is the first result of combining general relativity with quantum mechanics. It proves that gravitational collapse is not an absolute dead end as once thought. The particles of an incomplete black hole have not fully completed their history at a singularity. They can escape the black hole to continue their own histories.
Quantum mechanics may also imply that histories do not have a beginning in time, a point of creation at the Big Bang. This is an even more challenging question to address because it requires applying quantum mechanics to the structure of time and space, not just to the trajectories of particles in a specific region of spacetime. What we need to find is a way to sum over histories not only for particles but also for the entire backdrop of space and time. We still do not know how to compute this correctly, but there are some indications of what needs to be done. First, the summation will be easiest if we investigate histories in imaginary time rather than in conventional real time. Imaginary time is a difficult concept to grasp and certainly raises many questions for readers of the book: “A Brief History of Time*).
MY IDEA OF VIRTUAL TIME HAS received strong criticism from some philosophers. Why does virtual time have a connection to the real universe? I think these philosophers have not learned from history. In the past, people viewed the Earth as flat, and the Sun orbiting the Earth was taken for granted. Since the time of Copernicus and Galileo, we have had to accept the idea that the Earth is round and it orbits the Sun. Similarly, time was once thought to flow uniformly for all observers; however, since Einstein’s theory of relativity, we are compelled to accept that time flows differently for different observers. Likewise, the universe was seen to have a single history, but with quantum mechanics, we need to explore the universe with all its possible histories. For me, it seems that virtual time is something that also needs to be accepted. This represents a leap in knowledge comparable to when the Earth was recognized as round. I believe that virtual time today appears as naturally as the roundness of the Earth did in the past. No advanced individual now endorses the idea of a flat Earth. We can represent conventional real time as a straight line moving from left to right, but we can also explore another direction of time, which is from low to high. This is virtual time, which is perpendicular to real time.
What benefits does introducing the concept of virtual time bring? Why not stop at the conventional real time as we know it? The reason, as mentioned above, is that matter and energy warp spacetime. The dimension of real time inevitably leads to singularities, to regions where spacetime ceases to exist. At singularities, physical equations no longer hold; we cannot predict what will happen. However, virtual time is perpendicular to real time. This means that virtual time behaves similarly to the three dimensions of motion in space. Thus, the curvature of spacetime caused by matter in the universe can lead to three dimensions of space and one dimension of virtual time, forming a loop. They also create a closed surface like the surface of the Earth. The dimensions of space and virtual time will create a closed spacetime that has no boundaries, no edges, and no point that can be called a beginning or an end, just like on the surface of the Earth, which has no starting or ending point.
In 1983, Jim Hartle and I suggested that summing over the histories of the universe cannot be performed in real time but can be carried out in virtual time; the histories are closed upon themselves like the surface of the Earth. These histories have no singularities and no beginnings or ends; what will happen is entirely determined by the laws of physics. Thus, what will happen in virtual time can be computed; and if we know the history of the universe in virtual time, we can calculate its evolution in real time.
Therefore, we can hope to arrive at a completely unified theory, a theory that allows predictions of everything in the universe. Einstein sought such a theory in his later years but did not find it because he was skeptical of quantum mechanics. He was not willing to accept that the universe might have many interwoven histories, as in the summation over histories. We do not always know how to perform the summation over the histories of the universe, but it is almost certain that this summation will traverse virtual time and a closed spacetime. I believe that these concepts will enter future generations as naturally as the idea of the Earth being round. Virtual time has been a common ground for science fiction. But it is no longer science fiction or a mathematical trick; virtual time constitutes the universe in which we live.
(Stephen Hawking excerpt from “Black Holes and Baby Universes,” Odile – Jacob Publishing, 1994)
Translator: Nguyễn Đình Điện
