basic concept
Principle of Relativity (Relativity)
Relativity is the basic theory about space-time and gravity, which was mainly founded by Einstein and divided into special relativity (special relativity) and general relativity (general relativity). The basic assumption of relativity is the principle of relativity, that is, the laws of physics have nothing to do with the choice of reference frame. The difference between special relativity and general relativity is that the former discusses the physical laws between reference frames (inertial reference frames) with uniform linear motion, while the latter is extended to reference frames with acceleration (non-inertial reference frames), which is widely used in gravitational fields under the assumption of equivalence principle. Relativity and quantum mechanics are
Two basic pillars of modern physics. Classical mechanics, which laid the foundation of classical physics, is not suitable for high-speed moving objects and microscopic fields. Relativity solves the problem of high-speed motion; Quantum mechanics solves problems under microscopic subatomic conditions. Relativity subverts the common sense concepts of the universe and nature, and puts forward new concepts such as relativity of time and space, four-dimensional time and space, and curved space.
The most famous inference of special relativity is the mass-energy formula, which can be used to calculate the energy released during the nuclear reaction, leading to the birth of the atomic bomb. The gravitational lens and black hole predicted by general relativity are also confirmed by astronomical observations.
Demonstration process
In addition to quantum theory, an article entitled "On Electrodynamics of Moving Objects" published by Einstein in 1905 triggered another revolution in physics in the 20th century. This paper studies the influence of object motion on optical phenomena, which is another difficult problem faced by classical physics at that time.
Einstein put forward two basic principles as the basis for discussing the optical phenomena of moving objects. The first is called the principle of relativity. That is to say, if the coordinate system K' moves at a constant speed relative to the coordinate system K without rotating, it is impossible to distinguish which coordinate system is K and which coordinate system is K' in any physical experiment made relative to these two coordinate systems. The second principle is called the principle that the speed of light is constant, which means that the speed of light c (in vacuum) is constant, and it does not depend on the moving speed of the luminous object.
On the surface, the constant speed of light seems to conflict with the principle of relativity. Because according to the classical law of mechanical speed synthesis, the speed of light should be different for the two coordinate systems, k' and k, which move at a relatively uniform speed. Einstein thought that if we want to admit that these two principles do not conflict, we must re-analyze the physical concepts of time and space.
Einstein found that if the principle of light speed invariance and the principle of relativity are recognized to be compatible, then both hypotheses must be abandoned. At this time, the simultaneous events of one clock are not necessarily simultaneous for another clock, and they are relative at the same time. In two coordinate systems with relative motion, the values obtained by measuring the distance between two specific points are no longer equal. Distance is also relative.
If an event in the K coordinate system can be determined by three spatial coordinates X, Y, Z and a time coordinate T, and the same event in the K coordinate system is determined by X', Y', Z' and T', Einstein found that X', Y', Z' and T' can be solved by a set of equations. The relative velocity of the two coordinate systems and the speed of light c are the only parameters of the equation. This equation was first derived by Lorentz, so it is called Lorentz transformation.
Using Lorentz transformation, it is easy to prove that the clock will slow down because of movement, the ruler will be shorter when it is moving than when it is at rest, and the sum of speeds satisfies a new law. The principle of relativity is also expressed as a clear mathematical condition, that is, under the Lorentz transformation, the space-time variables X', Y', Z' and T' with apostrophes will replace the space-time variables X, Y, Z and T, and any expression of natural laws will still take the same form as before. What people call the universal law of nature is covariant for Lorentz transformation. This is very important for us to explore the universal laws of nature.
Besides, in classical physics, time is absolute. It has always played an independent role different from the three spatial coordinates. Einstein's theory of relativity involves time and space. It is believed that the real world of physics is composed of various events, and each event is described by four numbers. These four numbers are its space-time coordinates T and X, Y and Z, which form a four-dimensional continuous space, usually called Minkowski four-dimensional space. In relativity, it is natural to examine the real world of physics in a four-dimensional way. Another important result caused by special relativity is about the relationship between mass and energy. Before Einstein, physicists always thought that mass and energy were completely different and were separately conserved quantities. Einstein found that in the theory of relativity, mass and energy are inseparable, and the two conservation laws are combined into one. He gave a famous formula of mass and energy: e = MC 2, where c is the speed of light. So quality can be regarded as a measure of its energy. Calculations show that tiny masses contain enormous energy. This wonderful formula has laid a theoretical foundation for mankind to obtain huge energy, make atomic bombs and hydrogen bombs, and use atomic energy to generate electricity.
Most physicists, including Lorenz, the founder of relativistic transformation relation, find it hard to accept these new concepts introduced by Einstein. The obstacle of the old way of thinking made this new physical theory not familiar to physicists until a generation later. Even in 1922, when the science prize was awarded to Einstein by the Royal Swedish Academy, it only said, "Because of his contribution to theoretical physics and because he discovered the law of photoelectric effect." Not a word about relativity.
Einstein further established the general theory of relativity in 19 15. The principle of relativity in a narrow sense is limited to two coordinate systems with uniform motion, while the principle of relativity in a broad sense cancels the restriction of uniform motion. He introduced an equivalence principle, arguing that it is impossible for us to distinguish between gravitational effect and non-uniform motion, that is, non-uniform motion and gravity are equivalent. He further analyzed the phenomenon that light will be bent by gravity when passing near a planet, and thought that the concept of gravity itself was completely unnecessary. It can be considered that the mass of the planet makes the space around it curved, and the light takes the shortest path. Based on these discussions, Einstein derived a set of equations, which can determine the curved space geometry caused by the existence of matter. Using this equation, Einstein calculated the displacement of the perihelion of Mercury, which was completely consistent with the experimental observation, and solved a long-term unexplained problem, which made Einstein excited. In his letter to Erenfest, he wrote that this equation gives the correct value of perihelion. You can imagine how happy I am! For days, I was so happy that I didn't know what to do. "
1915165438+1On October 25th, Einstein submitted a paper entitled "Equation of Gravitation" to the Prussian Academy of Sciences in Berlin, which fully discussed the general theory of relativity. In this article, he not only explained the mystery of the perihelion motion of Mercury's orbit found in astronomical observation, but also predicted that the starlight would deflect after passing through the sun, and the deflection angle was twice that predicted by Newton's theory. The first world war delayed the determination of this value. 19 19 The total solar eclipse on May 25th provided people with the first observation opportunity after the war. Eddington, an Englishman, went to principe island on the west coast of Africa and made this observation. 165438+1On October 6th, Thomson solemnly announced at the joint meeting of the Royal Society and the Royal Astronomical Society that Einstein, not Newton, had proved this result. He praised "this is one of the greatest achievements in the history of human thought." Einstein discovered not an island, but a brand-new continent of scientific ideas. "The Times reported this important news under the title of" Revolution in Science ". The news spread all over the world, and Einstein became a world-famous celebrity. General relativity has also been elevated to a mythical sacred position.
Since then, people have shown more and more interest in the experimental test of general relativity. However, because the gravitational field of the solar system is very weak and the gravitational effect itself is very small, the theoretical result of general relativity deviates very little from Newton's gravitational theory, which makes the observation very difficult. Since 1970s, due to the progress of radio astronomy, the observation distance has far exceeded the solar system, and the accuracy of observation has been greatly improved. Especially in September of 1974, Taylor of MIT and his student Hall observed with a large radio telescope with a diameter of 305 meters and found a pulse binary star. It is a neutron star and its companion star revolve around each other under the action of gravity, with a period of only 0.323 days. The gravity on its surface is100000 times stronger than that on the surface of the sun. This is a laboratory where it is impossible to test the theory of gravity on earth or even in the solar system. After more than ten years of observation, they got a very good result, which is in line with the prediction of general relativity. Because of this great contribution, Taylor and Hall won the 1993 Nobel Prize in Physics.
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Narrow sense theory
The concept of special relativity
The philosophy of Mach and Hume had a great influence on Einstein. Mach believes that the measurement of space-time is related to the movement of matter. The concept of time and space is formed through experience. Absolute time and space, no matter what experience is based on, can't be grasped. More specifically, Hume said: Space and extension are just visible objects that are filled with space and distributed in a certain order. And time is always discovered through the perceptual changes of changeable objects. 1905, Einstein pointed out that Michelson and Morey's experiments actually showed that the whole concept of "ether" was redundant and the speed of light was constant. Newton's concept of absolute space-time is wrong. There is no absolutely static reference object, and the measurement of time varies with different reference frames. He put forward Lorentz transformation based on the principle of invariance of light speed and relativity. Created the special theory of relativity.
Special relativity is based on the theory of four-dimensional space-time view, so to understand the content of relativity, we must first have a general understanding of its space-time view. There are various multidimensional spaces in mathematics, but so far, the physical world we know is only four-dimensional, that is, three-dimensional space plus one-dimensional time. The high-dimensional space mentioned in modern microphysics is another meaning, only mathematical meaning, so I won't discuss it here.
Four-dimensional space-time is the lowest dimension that constitutes the real world, and our world happens to be four-dimensional. As for the high-dimensional real space, at least we can't perceive it yet. I mentioned an example in a post. When a ruler rotates in three-dimensional space (excluding time), its length remains unchanged, but when it rotates, all its coordinate values change and the coordinates are related. The significance of four-dimensional space-time lies in that time is the fourth coordinate, which is related to spatial coordinates, that is to say, space-time is a unified and inseparable whole, and they are a kind of "one change and one change" relationship.
Four-dimensional space-time is not limited to this. According to the relationship between mass and energy, mass and energy are actually the same thing. Mass (or energy) is not independent, but related to the state of motion. For example, the greater the speed, the greater the mass. In four-dimensional space-time, mass (or energy) is actually the fourth component of four-dimensional momentum, and momentum is a quantity that describes the motion of matter, so it is natural that mass is related to the state of motion. In four-dimensional space-time, momentum and energy are unified, which are called four vectors of energy momentum. In addition, four-dimensional velocity, four-dimensional acceleration, four-dimensional force and four-dimensional electromagnetic field equations are all defined in four-dimensional space-time. It is worth mentioning that the four-dimensional electromagnetic field equation is more perfect, which completely unifies electricity and magnetism, and the electric field and magnetic field are described by a unified electromagnetic field tensor. The physical laws of four-dimensional space-time are much more perfect than those of three-dimensional, which shows that our world is indeed four-dimensional. It can be said that at least it is much more perfect than Newtonian mechanics. At least because of its perfection, we can't doubt it.
In the theory of relativity, time and space constitute an inseparable whole-four-dimensional spacetime, and energy and momentum also constitute an inseparable whole-four-dimensional momentum. This shows that there may be a deep connection between some seemingly unrelated quantities in nature. When we talk about general relativity in the future, we will also see that there is also a profound relationship between the four vectors of space-time and energy momentum.
Principle of narrow sense theory
Matter moves forever in interaction, and there is no matter that does not move and there is no matter that does not move. Because matter moves in interaction, it is necessary to describe motion in the relationship of matter, and it is impossible to describe motion in isolation. In other words, motion must have a reference object, and this reference object is the frame of reference.
Galileo once pointed out that the motion of a moving ship is inseparable from the motion of a stationary ship. That is to say, when you are completely isolated from the outside world in a closed cabin, even if you have the most developed mind and the most advanced instruments, you can't perceive whether your ship is moving at a constant speed or at a standstill. There is no way to perceive speed because there is no reference. For example, we don't know the whole motion state of our whole universe, because the universe is closed. Einstein cited it as the first basic principle of special relativity: the principle of special relativity. Its content is: the inertial system is completely equivalent and indistinguishable.
The famous Michelson-Morey experiment completely negates the ether theory of light and draws the conclusion that light has nothing to do with the frame of reference. In other words, whether you stand on the ground or on a speeding train, the measured speed of light is the same. This is the second basic principle of special relativity: the principle of constant speed of light.
From these two basic principles, we can directly deduce all the contents of special relativity, such as coordinate transformation formula and velocity transformation formula. For example, the speed change is contrary to the traditional law, but it has been proved to be correct in practice. For example, the speed of the train is 10m/s, and the speed of a person on the train is also10m/s. People on the ground see that the speed of people in the car is not 20m/s, but (20- 10 (-65438). In general, this relativistic effect can be completely ignored, but it increases obviously when it approaches the speed of light. For example, the speed of a train is 0.99 times the speed of light, and the speed of a person is 0.99 times the speed of light. Then the ground observer's conclusion is not 1.98 times the speed of light, but 0.999949 times the speed of light. The people in the car didn't slow down when they saw the light coming from behind, which was also the speed of light for him. So in this sense, the speed of light cannot be surpassed, because no matter in which reference system, the speed of light is constant. Velocity transformation in particle physics has been proved by countless experiments and is impeccable. It is precisely because of this unique property of light that it is chosen as the only scale of four-dimensional space-time.
Narrow sense effect
According to the principle of relativity in a special sense, the inertial system is completely equivalent. So in the same inertial system, there is a unified time, which is called simultaneity. Relativity proves that there is no unified simultaneity in different inertial systems, that is, two events (time and space points) may be different in one inertial system, which is simultaneity. The time of the same physical process in the inertial system. In the future general relativity, we can know that in a non-inertial system, time and space are not unified, that is, in the same non-inertial system, there is no unified time, so the unified simultaneity cannot be established.
Relativity deduces the time progress relationship between different inertial systems, and finds that the inertial system of motion is slow in time progress, which is the so-called clock slow effect. Generally, it can be understood that a moving clock goes slower than a stationary clock, faster and faster, slower and slower, and almost stops when it approaches the speed of light.
The length of a ruler is the difference between the coordinate values of two endpoints obtained at the same time in an inertial system. Because of the relativity of "simultaneity", the length measured in different inertial systems is also different. Relativity proves that the ruler moving in the length direction of the ruler is shorter than the static ruler, which is called scale effect. When the speed approaches the speed of light, the scale shrinks to a point.
As can be seen from the above statement, the principle of slow clock and scale contraction is that the progress of time is relative. In other words, the timetable is related to the reference system. This fundamentally negates Newton's absolute view of time and space. Relativity holds that absolute time does not exist, but time is still an objective quantity. For example, in the ideal twin experiment to be discussed in the next issue, my brother is 15 years old after returning from the spaceship, and my brother may be 45 years old, indicating that time is relative, but my brother did live 15 years old, and my brother really thought that he lived 45 years old, which has nothing to do with the frame of reference, and time is "absolute". This shows that no matter what the motion state of an object is, the time it experiences is an objective quantity and absolute, which is called intrinsic time. That is to say, no matter what form you exercise, you think that your coffee drinking speed is normal, and your lifestyle has not been disrupted, but others may see that you have been drinking coffee for 100 years, and it takes only one second from putting down the cup to dying.
It took Einstein only a few weeks to establish the special theory of relativity, while it took ten years to establish the general theory of relativity to solve these two difficulties. In order to solve the first problem, Einstein simply canceled the special position of inertial system in theory and extended the principle of relativity to non-inertial system. Therefore, the first problem is transformed into the space-time structure problem of non-inertial system. The first obstacle encountered in a non-inertial system is inertial force. Through the in-depth study of inertial force, the famous principle of equality is put forward, and it is found that the problem of reference frame may be solved together with the problem of gravity. After many twists and turns, Einstein finally established a complete general theory of relativity. General relativity surprises all physicists, and gravity is far more complicated than imagined. So far, Einstein's field equation has only got a few definite solutions. Its beautiful mathematical form has amazed physicists so far. While the general theory of relativity has made great achievements, quantum mechanics founded and developed by Copenhagen School has also made a major breakthrough. However, physicists soon found that the two theories were incompatible, and at least one of them needed to be revised. This led to the famous debate: Einstein VS Copenhagen School. The debate has not stopped yet, but more and more physicists are more inclined to quantum theory. Einstein spent the last 30 years of his life trying to solve this problem, but he got nothing. However, his work points out the direction for physicists: to establish a hyperunified theory containing four forces. At present, the most promising candidates recognized by academic circles are superstring theory and ultramembrane theory.
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Paradox problem
Clock twinning paradox
After the birth of the theory of relativity, there is a very interesting and difficult problem-twin paradox. A pair of twins, A and B, are on the earth, and B travels in a rocket and returns to the earth after a long time. Einstein asserted from the theory of relativity that the two experienced different times, and B would be younger than A when they met again. Many people have doubts, thinking that A watches B exercise and B watches A exercise. Why can't A be smaller than B? Because the earth can be approximated as an inertial system, and B has to go through the process of acceleration and deceleration, and it is a reference system with variable acceleration, so it is really complicated to discuss. So this problem that Einstein has discussed clearly is mistaken by many people as contradictory relativity. It is much easier to discuss this problem with the concepts of Shi Kongtu and World Line, but it requires a lot of mathematical knowledge and formulas. Here, we just use language to describe the simplest situation. However, it is impossible to explain the details in more detail by language alone. If you are interested, you can refer to some books on relativity. Our conclusion is that B is younger than A in any frame of reference.
In order to simplify the problem, we will only discuss this situation. After a while, the rocket accelerated to sub-light speed. After flying for a period of time, turn around in a very short time, fly for a period of time, and slow down to meet the earth in a very short time. The purpose of this treatment is to ignore the effects of acceleration and deceleration. It is easy to discuss in the earth reference system that the rocket is always a moving clock, and B is younger than A when we meet again. In the rocket reference system, the earth is the moving clock in the process of uniform motion, and the time process is slower than that in the rocket, but the most critical place is the process of rocket rotation. In the process of U-turn, the earth crossed half a circle from the distance behind the rocket in a very short time and reached the distance in front of the rocket. This is a "superluminal" process. It's just that this superluminal and relativity are not contradictory. This superluminal can't transmit any information, and it's not superluminal in the real sense. Without this u-turn process, the rocket and the earth would not meet. Because there is no uniform time in different reference systems, it is impossible to compare their ages. Only when they meet can they be compared. After the rocket turns around, B can't accept the message from A directly, because it takes time to transmit it. The actual process that B saw was that during the U-turn, the earth's time schedule accelerated sharply. In B's view, A was younger than B at first, then aged quickly when he turned around, and aged slower than himself when he returned to China. When we meet again, we are still younger than A. In other words, there is no logical contradiction in the theory of relativity.
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Generalized theory
The concept of general relativity
When the theory of relativity came out, people saw the following conclusions: four-dimensional curved space-time, finite boundless universe, gravitational wave, gravitational lens, big bang cosmology, black hole, the main theme of 2 1 century, and so on. All this comes too suddenly, which makes people feel that the theory of relativity is mysterious. Therefore, in the early years of the advent of the theory of relativity, some people threatened that "only twelve people in the world understand the theory of relativity." Some people even say that "only two and a half people in the world understand the theory of relativity". What's more, the theory of relativity is compared with spiritualism and idealism. In fact, the theory of relativity is not mysterious. It is the most down-to-earth theory, a truth that has been tested thousands of times, and not unattainable.
The geometry applied by relativity is not ordinary Euclidean geometry, but Riemann geometry. I believe many people know non-Euclidean geometry, which can be divided into Roche geometry and Riemann geometry. Riemann unified three kinds of geometry from a higher angle, which is called Riemann geometry. Non-Euclidean geometry has many strange conclusions. The sum of the internal angles of a triangle is not 180 degrees, and the pi is not 3. 14. So when it was first put forward, it was ridiculed as the most useless theory. It was not until its application was found in spherical geometry that it was paid attention to.
If there is no matter in space and space-time is flat, then Euclidean geometry is enough. For example, the application in special relativity is four-dimensional pseudo-Euclidean space. Because there is an imaginary unit I in front of the time coordinate, a pseudo word is added. When matter exists in space, the interaction between matter and space-time bends space-time, which means using non-Euclidean geometry.
Relativity predicted the existence of gravitational waves, and found that both gravitational fields and gravitational waves travel at the speed of light, denying the distance effect of the law of universal gravitation. When light comes from a star and meets a massive celestial body, it will converge again, that is, we can observe the stars blocked by celestial bodies. Generally speaking, what you see is a ring called Einstein ring. When Einstein applied the field equation to the universe, he found that the universe was not stable, and it either expanded or contracted. Cosmology at that time believed that the universe was infinite, static and the stars were infinite. So he did not hesitate to modify the field equation, added the universe term, got the stable solution, and put forward the finite infinite universe model. Soon Hubble discovered the famous Hubble law and put forward the theory of cosmic expansion. Einstein regretted this and gave up the cosmological term, calling it the biggest mistake in his life. In later research, physicists were surprised to find that the universe was not only expanding, but also exploding. The very early universe was distributed in a very small area. Cosmologists need to study the content of particle physics to put forward a more comprehensive model of the evolution of the universe, and particle physicists need cosmologists' observations and theories to enrich and develop particle physics. In this way, the two most active branches of physics-particle physics and cosmology-are combined with each other. As the preface of high school physics says, it's like a strange python biting its tail. It is worth mentioning that although Einstein's static universe has been abandoned, its finite boundless universe model is one of the three possible fates of the future universe and the most promising. In recent years, the cosmological term has been revalued. The problem of black holes will be discussed in a future article. Although black holes and big bang are predictions of relativity, their contents have gone beyond the limitation of relativity and are closely combined with quantum mechanics and thermodynamics. I hope the future theory can find a breakthrough here.
Generalized theoretical formula
According to the general theory of relativity, "the motion of all substances in the universe can be described by curvature, and the gravitational field is actually a curved space-time", Einstein gave the famous Einstein field equation:
Where g is Newton's gravitational constant, which is called Einstein's gravitational field equation, also called Einstein's field equation. The equation is a second-order hyperbolic partial differential equation with elliptic constraints, with time and space as independent variables and measurement as dependent variables. It is famous for its complexity and beauty, but it is not perfect, and only approximate solutions can be obtained in calculation. Finally, people get the exact solution of true spherical symmetry-Schwartz solution. The field equation after adding the cosmological constant is:
Principle of generalized theory
Because the inertial system cannot be defined, Einstein extended the principle of relativity to the non-inertial system and put forward the first principle of general relativity: the principle of general relativity. Its content is that all frames of reference are equivalent when describing the laws of nature. This is very different from the principle of relativity in a narrow sense. In different reference systems, all physical laws are completely equivalent, and there is no difference in description. But in all reference frames, this is impossible. It can only be said that different frames of reference can also effectively describe the laws of nature. This requires us to find a better description method to meet this requirement. Through the special theory of relativity, it is easy to prove that the pi of a rotating disk is greater than 3. 14. So the general frame of reference should be described by Riemann geometry. The second principle is the principle that the speed of light is constant: the speed of light is constant in any reference system. The space-time point equivalent to light is fixed in four-dimensional space-time Space-time is straight, and light moves in a straight line at the speed of light in three-dimensional space. When space-time is curved, light moves along the curved space in three-dimensional space. It can be said that gravity can deflect light, but it cannot accelerate photons. The third principle is the most famous principle of reciprocity. There are two kinds of mass. Inertia mass is used to measure the inertia of an object, which was originally defined by Newton's second law. Gravitational mass is a measure of the gravitational charge of an object, which was originally defined by Newton's law of universal gravitation. These are two unrelated laws. Inertial mass is not equal to charge, and it is not even important so far. Then inertial mass and gravitational mass (gravitational charge) should have nothing to do with Newtonian mechanics. However, the difference between them cannot be discovered through the most sophisticated experiments. Inertia mass is strictly proportional to gravitational mass (it can be strictly equal if appropriate coefficients are selected). General relativity regards inertial mass and gravitational mass as the content of equivalence principle. Inertia mass is related to inertia force, and gravitational mass is related to gravity. In this way, the relationship between non-inertial system and gravity is established. Then a very small free-falling reference frame can be introduced at any point in the gravitational field. Because inertial mass is equal to gravitational mass, it is neither inertia nor gravity in this reference system, and all theories of special relativity can be used. When the initial conditions are the same, the orbits of particles with equal mass and different charges are different in the same electric field, but all particles have only one orbit in the same gravitational field. The principle of equivalence made Einstein realize that the gravitational field is probably not the outfield of space-time, but the geometric field, which is an attribute of space-time itself. Due to the existence of matter, the originally flat space-time has become a curved Riemannian space-time. At the beginning of the establishment of general relativity, there was a fourth principle, the law of inertia: an object that is not subjected to force (except gravity, because gravity is not real force) does inertial motion. In Riemann space-time, it moves along geodesic lines. Geodesic is a generalization of straight lines, the shortest (or longest) straight line between two points, and it is unique. For example, the geodesic of a sphere is an arc of a great circle cut by a plane passing through the center of the sphere and the sphere. But after the field equation of general relativity is established, this law can be deduced from the field equation, so the law of inertia becomes the law of inertia. It is worth mentioning that Galileo once thought that uniform circular motion was inertial motion, and uniform linear motion would always close into a circle. This is proposed to explain planetary motion. Naturally, he was criticized by Newtonian mechanics, but the theory of relativity revived it. Planets do move in inertia, but it is not a standard uniform motion.