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Here's a look at another interesting topic in physics besides quantum mechanics: relativity.

*This is the fourth in a series of articles that take us in the realms of quantum physics. **You can read Part 1 here, Part 2 here, and Part 3 here.*

Besides quantum mechanics, another interesting topic in physics is relativity.

Let’s take a look at it!

Of all the force fields, the gravitational field is the one man has known for the longest time. One of its fundamental and most important properties (which today goes by the name of the principle of equivalence) was recognized by Galileo at the beginning of the 17^{th} century and its fundamental law was established by Newton towards the end of the same century. We will analyze the principle of equivalence in a while, but it is good to remember that this is one of the pillars of the theory of general relativity: At the base of everything, this is the key element according to which the gravitational field imparts the same acceleration locally on all bodies, given the equivalence between inertial mass and gravitational mass.

The gravitational force is still studied along with the other fundamental forces. The reason is that the gravitational force is extremely weaker if compared with the others, in the order of 10^{38 }compared to the electromagnetic force, for example. Therefore, an experiment that aims to make gravitational measurements is disturbed by the presence of other forces, unless we consider very large mass bodies: In this case, the gravitational forces, which are only attractive and long range, reach a high value, while the other forces tend to become negligible because they have a short range of action, or because, being both attractive and repulsive, the net effect is of zero average.

The theory of general relativity predicts the existence of gravitational waves. The relative weakness of gravitational force has both experimental advantages and disadvantages. The disadvantage is the great difficulty of detecting it, as can easily be deduced from the considerations made above. The advantage is the fact that, once the sensitivity necessary to detect gravitational waves such as the current Virgo and LIGO experiments has been reached, a probe will be available to explore matter and, in particular, the Universe in its most hidden accesses. Gravitational waves are able to pass through large amounts of matter without almost losing their energy. The Virgo and LIGO experiment succeeded in detecting gravitational waves.

The gravitational wave information reflects the large-scale distribution of matter; unlike the electromagnetic ones which provide information on an atomic scale. And all of this makes it very difficult to model stellar-sized objects. The discovery of gravitational waves has characterized a new experimental phase in which their properties such as waveform, spectrum, intensity, polarization and direction of origin will have to be studied. In a more distant future, when physicists have total control of the experiment, gravitational waves could be produced in a laboratory and used for practical purposes such as electromagnetic waves. In this case, we will need some gravitational field transmitting antennas. Is it possible that extraterrestrials are able to communicate with gravitational waves instead of electromagnetic ones? The search for gravitational waves is of fundamental importance for the understanding of the structure of the universe and its origin.

In 1905, Albert Einstein published an article in which he expounded the special relativity principle (RR) as a modification of the previous Galilean principle of relativity with the addition of the constancy of the speed of light. According to Einstein, space and time should no longer be considered as absolute, separate entities. On the contrary, space and time are part of a single reality, 4-dimensional space-time. The extension of the RR goes under the name of the principle of general relativity (GR). The problem of defining the GR was complicated by the fact that the RR only spoke of the electromagnetic field while the other force field known at that time, the gravitational field, remained completely excluded. How can the gravitational field also be included in the theory?

Newton’s gravitational field formula was known. It expresses the force with which any two masses attract each other. The gravitational force is a universal force that acts between all bodies, be they small or large. It is a very weak force that becomes appreciable only between bodies of great mass. Einstein devoted his life to solve the problem with defining GR from 1905 to 1916 and solved it starting from a brilliant intuition on a very simple fact: All bodies “fall” with the same acceleration regardless of their mass (not considering the friction of the air and the fact that the earth rotates on itself). This phenomenon goes by the name of the principle of equivalence and expresses the fact that the inertial mass is equal to the gravitational mass, or that two bodies gravitationally attract each other with the same mass with which they react to any force, of any type. The principle of equivalence is the basis on which the GR is based. A gravitational field is then equivalent to a non-inertial frame of reference in which space-time is modified, influenced, curved by the masses that generate it. Thus, the non-inertial reference systems are incorporated in the theory as a generalization of the inertial ones in the presence of the gravitational field. The 4-dimensional space-time is thus perturbed and bent by the masses that generate the gravitational field. This means that the space-time of physical reality is not flat, i.e., it is not Euclidean. The rules of Euclidean geometry no longer apply. In a non-Euclidean space, for example, the sum of the angles of a triangle is other than 180 degrees. Real space-time is not Euclidean. This data may seem surprising because Euclidean geometry is well satisfied in common experience up to very significant distances. The curvature of space-time begins to be felt over very large distances. The concept of non-Euclidean space-time is very difficult to imagine because it is difficult for us to visualize spaces with more than 3 dimensions. The definition of the curvature of space-time occurs through Einstein’s equation, an extremely complex equation that is able to describe any type of gravitational field and even the universe as a whole. Einstein’s equation is:

where R_{ik} is the Ricci tensor, g_{ik} is the covariant metric tensor, R is the curvature (Ricci’s scalar), G the gravitational constant and T_{ik} is the energy-momentum tensor.

Ricci’s tensor and scalar provide an indication of the curvature of space-time. They are related to the metric tensor. The unknowns in Einstein’s equation are the g_{ik }and T_{ik} or the field and distribution of mass and energy. Due to its complexity, it is soluble exactly only in very few simple cases. When Einstein expounded the theory of general relativity, there was no experimental evidence that required a new theory of gravitational phenomena. The Einsteinian theory explained some small corrections which were amply confirmed: An example is the deflection of the trajectory of light rays by large mass bodies. Subsequently, Einstein introduced the cosmological constant in order to obtain a static model and bring the universe back into equilibrium:

In any case, there were studies that highlighted the structure of the homogeneous and isotropic universe but in continuous expansion. The cosmological constant has taken on various roles over time, attempting to explain this acceleration as a fluid identified as the density of matter in the universe. One accredited model is that of the vacuum energy predicted by quantum mechanics.

In Einstein’s equations, the energy-momentum tensor plays the role of source of the gravitational field and, in the case of a weak and stationary field, it represents the static mass distribution. We, therefore, think that when a mass distribution evolves over time, this variation must modify the field and the curvature of space-time. As in all interactions existing in nature, this variation cannot be detected instantaneously in all space, but it will be linked to the propagation with finite speed in space-time of a perturbation which will be called a gravitational wave. These are metric waves and when they propagate in spacetime, the geometry and thus the distance between the points in 4-dimensional spacetime changes over time. From the Einstein field equations, we have to derive the g_{ik}, i.e., to obtain solutions able to describe the gravitational field.

Gravitational-wave (GW) sources can be classified into 4 types based on the frequencies of the radiation. The first type of source emits waves in the frequency band from 2 to 10 Hz, and it is believed that the initial density for the turbulences associated with the inflation of the universe introduces a gravitational influence in this frequency band. The second type of sources are those with a “very low“ frequency (from 10-7 to 10-9 Hz). The predicted sources are super-massive black holes. The third type sources, from 10^{-6} to 1 Hz, include the gravitational collapse of super-massive stars and cosmic gravity. Sources of the fourth type, high frequency from 1 Hz to 10 kHz, are the fusion of binary objects with stellar mass and supernovae. In general, heavier sources correspond to lower frequencies. And the same physical objects as they evolve can generate GW radiation at different frequencies.

A few years ago, the LIGO / Virgo experiment, or rather the LIGO / Virgo interferometer, detected the gravitational signal related to the collisions of two black holes, which occurred exactly one billion years ago. LIGO as well as its twin Virgo in Italy is a Michelson interferometer. A laser beam is separated into two different beams. These two rays travel through two different optical paths and are made to interfere, producing interference fringes. Due to the passage of a gravitational wave that propagates in the direction perpendicular to one of the two paths, the length of one of the two arms varies, and thus a variation of the interference fringes is obtained and can be measured with a photodiode. One of the problems encountered is the statistical fluctuation in the number of photons emitted by the laser source. In order for a gravitational wave to be detected, it is necessary to build interferometers with arms of enormous length; to circumvent this problem, interferometers called multipasses with Fabry-Perot cavities are used, in which the effective length of the arms is increased by means of multiple reflections until the desired length is reached.

Different sources of noise can introduce spurious signals that can hide the gravitational wave. The main types of noise that perturb the measurement with the interferometer are: shot noise, due to the statistical fluctuation of the number of photons; Brownian noise of the mirrors, a residual pressure of the gas in the vacuum tube crossed by the laser beams which causes the refractive index to vary; seismic noise due to earthquakes. Some of them affect the position of the optical components (displacement noise), others are intrinsic noises of the reading system (phase noise). In any case, these noises affect the phase shift recorded by the photodiode at the output of the interferometer. The data recorded by a gravitational wave detector essentially consists of noise, from which it is necessary to distinguish the contribution of any and very weak signals of gravitational origin. This is the task of data analysis, which makes use of the theoretical foundations provided by Information Theory (detection of signals in the presence of noise) and the support of computers capable of performing sophisticated simulations.

An example of an implemented algorithm is the Kalman filter which allows to estimate the state of a system starting from the measured data. The algorithm consists of two processes: One makes a prediction of the state of the system, while the other optimizes the measurement based on the noise. Mathematically, it is something very simple, X_{t + 1} = Ft* Xt + Wt, where Xt is the state vector while the term Wt is the noise that defines the observability statistics. The optimality of the Kalman filter assumes that the errors are Gaussian. The story of the creator of this filter, Rudolph Kalman, is related to the mission to the Moon (Apollo mission) and addresses some problems about the impossibility of carrying out the mission safely, in particular the problem of recovering the capsule upon re-entry. The problem was this: Given a series of points such as radar tracks, and given the dynamic model of the capsule’s reentry, identify the most likely splashdown point and the area of scattering. Rudolph Kalman, a very young NASA researcher, presented a solution to this problem. But his solution was not initially regarded as the first. After other unsuccessful solutions were tested, the Kalman filter contributed effectively.

Let’s go back to relativity.

According to the general theory of relativity, the intense accelerations of masses emit gravitational waves: That is, distortions of space-time that propagate at the speed of light. One billion years ago, two black holes collapsed sending a gravitational “shiver” across the universe: ripples in the fabric of space-time. LIGO and Virgo managed to measure it.

In the scientific article cited at the end of the article, in the references section, it is indicated that the signal has an overall duration of the fraction of a second with a high value of statistical “certainty”, thus confirming Einstein’s theory but above all the existence of black holes, that is space-time passages. Then, measured energy was enormous, equal to about three solar masses (one solar mass is about 10^{30} Kg). The two black holes formed a binary system in rotation with each other with a mass that is 36 and 29 times that of the Sun (about 2×10^{30} Kg). They approached at a speed close to the speed of light and the collision produced a black hole releasing gravitational waves. The discovery made it possible to define the existence of black holes, whose existence would lead to many scientific reflections, starting from time travel. For science fiction lovers, that day could be a very special day, with the discovery of gravitational waves and perhaps with an in-depth knowledge about space-time tunnels. Just remember how Marty and Doc traveled between “black holes” at 88 miles per hour.

We have all dreamed of traveling in time in the DeLorean, perhaps getting the Grays Sports Almanac to become millionaires like Biff Tannen in “Back to the Future Part II”. The “Back to the Future” trilogy would not have been possible without the invention of time travel. The scientific equation has calculated 88 miles per hour. It was the speed needed to reach quantum spacetime and fly into another time era. Time travel is obviously science fiction, although Professor Ronald Mallett, a professor of physics at the University of Connecticut, is convinced that one day we will travel in time. The personal tragedy of his father’s death from a heart attack when he was very young marked the beginning of the “timeline” adventure with a specific mission: go back in time, warn him of what was happening and try to save him.

Einstein’s special theory of relativity tells us that time is affected by speed. According to this theory, the time for a moving clock slows down. The faster the clock moves, the slower time slows down. This has been proven by many experiments. In particular, on a microscopic scale, there are subatomic particles that only live for a very short period of time and then disintegrate. When particle accelerators such as the Large Hadron Collider (LHC) at CERN in Geneva, Switzerland, accelerate these subatomic particles to a speed close to the speed of light, it turns out that the internal particle clock slows down, and the particles can live by 20 to 30 (or more) times longer than normal. This effect is technically called time dilation. On a large scale, this slowing down of time with speed was also demonstrated in an experiment that was done at the United States Naval Observatory. This experiment involved two atomic clocks. One of the clocks was held at the Naval Observatory, the other was put on a passenger jet that flew around the world at the speed of sound. When the clock on the passenger jet was compared to the clock at rest, it was found that the clock on the passenger jet had slowed down as Einstein had predicted. It should be remembered that the human heart is a clock. This means that the heartbeat behaves differently if a person tends to move fast. The person would not notice it, but a person at rest would notice it by looking at the heart rate of the moving person. The heartbeat and metabolism of the person moving on a rocket traveling at a speed close to the speed of light would slow down. If he or she had only traveled for a few years, many decades could have passed on earth on his or her return. He or she would discover that he or she had traveled to the future of the Earth. It is for this reason that Einstein’s theory of relativity predicts the possibility of time travel to the future.

The possibility of time travel is based on Einstein’s general theory of relativity, developed in 1915. This is due to the prediction of the general theory of relativity that time is affected by gravity. According to this theory, the stronger the gravity, the more time slows down. This has been demonstrated experimentally and forms the basis of the GPS system, which takes into account the fact that clocks on the surface of the earth, where gravity is strong, are slower than clocks aboard satellites that are far above the earth, where gravity is weak. Theoretically, it has been shown that a rotating gravitational field, such as that of a rotating black hole, could allow for quantum perturbations that could lead to time travel even to the past. Time travel to the past could lead to paradoxes such as the grandfather paradox. If one travels back in time to prevent the grandparents from meeting, then the parents were not born and if the parents were not born, then one was not born. So, if you weren’t born, how can you go back in time to stop your grandparents from meeting? It has been speculated that quantum mechanics can solve this paradox through the interpretation of many worlds or parallel worlds. According to the interpretation of parallel worlds, if you go to the past, you reach the past version of a parallel universe. In that parallel universe, you can prevent grandparents from meeting, but that doesn’t affect the original universe you came from.

Light to matter could create a gravitational vortex that would alter time. Mallett idealized a tunnel-like laser machine to twist time. The professor was able to mathematically demonstrate that, by using a beam of laser light in circulation, it is possible to have a twist of space and time. I won’t show the equation discovered by Mallett, but I recommend watching his videos on YouTube. Mallett’s time machine can send information in the form of neutrons into the past / future. If the theory was valid, Ronald could only receive data from the future (and send it into the past), i.e., “activate” a timeline the moment it is turned on. Unlike the movie “Back to the Future”, Mallett does not intend to be catapulted from one side to the other but to convey information on a timeline (similarly to the plot in the movie “Timeline”). Time travel fascinates us, and almost everyone would like to have the chance to go back and change something, defying one of the biggest unknowns in physics: time. Different time travel theories have taken different positions on the possibility of changing the past. There are three main theories: the theory of the multiverse, the theory of the dynamic timeline, and the theory of the fixed timeline.

We have all dreamed of traveling through time, if only for a minute, to have the chance to meet some of our ancestors. Maybe talk to him/her another time, listen to his/her teachings again. Suppose for a moment that you have a time machine with the possibility of taking a trip, where would you like to go? Personally, I would like to go back to greet my grandparents with whom I spent the best moments of my life. Time is strange, it is certainly the most precious thing, and nobody knows what happens in the future. Nothing is written, maybe nothing happens by chance, so enjoy every moment and, as Doc said, “the future is not written but it can be created”.

Let’s create a good one, then!

Another concept behind time travel is wormholes which theoretically connect two points in the universe. Entry from one end of the wormhole would almost instantly lead to exit from the other end, although these two points may be billions of light-years apart. It is not as simple as that. The problem is knowing how to get through it. These “tunnels” first appeared as a solution to Einstein’s field equations. They are also called Einstein-Rosen bridges, named after Einstein and his then assistant Nathan Rosen.

The wormhole is constructed in such a way that one end could remain nearly motionless, while the other could move at almost the speed of light. Entering this wormhole would allow you to go back in time. A wormhole is basically a shortcut through space-time, so sending a pulse of light into it could allow for faster-than-light communication. Furthermore, according to the theory, the wormhole connects at least two points to a single throat, through which we could transfer matter to another point in space-time, or even to another dimension or universe according to some scholars, but this would require further theoretical and experimental research.

We can imagine space-time as something “pierced”: A hypothesis that involves the possibility of passing from one area of space-time to another, connecting parallel universes. This type of event could be very useful for controlling space-time. We can only live in the present, moving every second into the future, but slowly and without the slightest chance of returning to our past. A condition, impossible for the moment, is to be able to travel in space at the speed of light: It is the only way to slow down our clock and then, once back on Earth, find ourselves in the future. According to Kip Thorne and Michael Morris, CalTech physicists, the “wormhole” could be used to jump from one point in the universe to another, thus exceeding the speed of light. Does a “wormhole” connect two distant points in the universe or does it cross two parallel dimensions? We will have to find out.

In addition to the discovery of gravitational waves, an important demonstration of Einstein’s general relativity is the set of Lunar Laser Ranging experiments. On the moon, there is a mirror. More than one, actually. The astronauts of the various Apollo missions have placed them on different points of the lunar surface. These mirrors, called retroreflectors, have a very useful feature. From any angle you look at them, the mirror sends back the observer’s image. The reflectors have a diameter of 3.8 cm and are inserted into circular holes in the panel for thermal control. They were brought to the surface of the Moon 40 years ago and, over time, allowed us to know the orbit of the Moon with the accuracy of a few centimeters, to carry out multiple accurate tests of Einstein’s Theory of General Relativity and important measurements of the internal structure of the moon.

The experiment (Lunar Laser Ranging, LLR) is conceptually very simple: It consists in sending a light signal (a monochromatic laser beam) from the Earth to the Moon’s retroreflectors, measuring how long it takes to go back. A simple physics measurement. By knowing the speed of light and time, we can plot the distance which will be given by the speed of light multiplied by the travel time, divided by 2 as it is a round trip. One of the most interesting questions that the LLR tries to answer precisely is this: Is there a difference in the way the Earth and the Moon are attracted to the Sun? The signals are sent using a ruby laser that sends pulses of tens of nanoseconds, with energies of the order of ten joules per pulse and repetition times of 3 to 30 seconds. Each laser shot sends about 10^{17 }photons to the Moon, but only one returning photon can be detected. For this reason, the laser is sent thousands of times. The return radius takes about 2 and a half seconds to return to the starting point; since the laser emits monochromatic light, the returning photons can be identified. But they are obviously too weak to be seen by the human eye. An instrument called a photomultiplier is used to amplify the signal. These measurements have led to relevant scientific results, such as the understanding that the Moon probably has a liquid core. It was also possible to verify the Theory of Relativity with remarkable accuracy: The Moon moves exactly as Einstein’s equations predict. Mirrors on the Moon tend to degrade over time due to dust and the absence of the atmosphere. One day, they will have to be replaced with new mirrors to improve laser light-gathering efficiency. And then, we will need to take a new trip to the Moon.

*This article was originally published on **EE Times Europe**.*

*Maurizio Di Paolo Emilio** holds a Ph.D. in Physics and is a telecommunication engineer and journalist. He has worked on various international projects in the field of gravitational wave research. He collaborates with research institutions to design data acquisition and control systems for space applications. He is the author of several books published by Springer, as well as numerous scientific and technical publications on electronics design.*