Basic ideas of A. Einstein's theory of relativity

The revolutionary physicist used his imagination rather than complex mathematics to come up with his most famous and elegant equation. Einstein is known for predicting strange but true phenomena, such as astronauts in space aging slower than people on Earth and the shapes of solid objects changing at high speeds.

But what's interesting is that if you pick up a copy of Einstein's original 1905 paper on relativity, it's fairly easy to decipher. The text is simple and clear, and the equations are mostly algebraic - any high school student can understand them.

This is because complex mathematics was never Einstein's strong point. He loved to think visually, to perform experiments in his imagination and to think through them until the physical ideas and principles became crystal clear.

This is where Einstein's thought experiments began when he was just 16 years old, and how they eventually led him to the most revolutionary equation in modern physics.

By this point in Einstein's life, his poorly concealed disdain for his German roots and Germany's authoritarian teaching methods had already taken its toll, and he had been kicked out of high school, so he moved to Zurich in hopes of attending the Swiss Federal Institute of Technology (ETH).

But first, Einstein decided to spend a year of preparation at a school in the neighboring town of Aarau. At this point, he soon found himself wondering what it would be like to run next to a beam of light.

Einstein had already learned in physics class what a beam of light was: a set of oscillating electric and magnetic fields moving at 300,000 kilometers per second, the measured speed of light. If he ran nearby at the same speed, Einstein realized, he could see many oscillating electric and magnetic fields next to him, as if frozen in space.

But this was impossible. First, stationary fields would violate Maxwell's equations, the mathematical laws that underlie everything physicists knew about electricity, magnetism, and light. These laws were (and still are) quite strict: any waves in these fields must travel at the speed of light and cannot stand still, no exceptions.

Worse, stationary fields did not fit with the principle of relativity, which had been known to physicists since the days of Galileo and Newton in the 17th century. Essentially, the principle of relativity says that the laws of physics cannot depend on how fast you are moving: you can only measure the speed of one object relative to another.

But when Einstein applied this principle to his thought experiment, a contradiction arose: relativity dictated that anything he could see when moving near a beam of light, including stationary fields, must be something mundane that physicists could create in the laboratory. But no one has ever observed this.

This problem would haunt Einstein for another 10 years, as he studied and worked at ETH and moved on to the Swiss capital of Bern, where he would become an examiner at the Swiss patent office. It is there that he will resolve the paradox once and for all.

1904: Measuring light from a moving train

It wasn't easy. Einstein tried every solution he could think of, but nothing worked. Almost in despair, he began to think about a simple, yet radical solution. Perhaps Maxwell's equations worked for everything, he thought, but the speed of light had always been constant.

In other words, when you see a beam of light fly by, it doesn't matter whether its source is moving towards you, away from you, away from you, or anywhere else, and it doesn't matter how fast its source is moving. The speed of light that you measure will always be 300,000 kilometers per second. Among other things, this meant that Einstein would never see stationary oscillating fields, since he would never be able to catch a beam of light.

This was the only way Einstein saw to reconcile Maxwell's equations with the principle of relativity. At first glance, however, this solution had its own fatal flaw. He later explained it with another thought experiment: imagine a beam that is fired along a railway embankment while a train passes by in the same direction at, say, 3000 kilometers per second.

Someone standing near the embankment would have to measure the speed of the light beam and get the standard number of 300,000 kilometers per second. But someone on a train will see light moving at 297,000 kilometers per second. If the speed of light is not constant, Maxwell's equation inside the carriage should look different, Einstein concluded, and then the principle of relativity would be violated.

This apparent contradiction gave Einstein pause for almost a year. But then, one fine morning in May 1905, he was walking to work with his best friend Michel Besso, an engineer he had known since his student days in Zurich. The two men talked about Einstein's dilemma, as they always did. And suddenly Einstein saw the solution. He worked on it all night, and when they met the next morning, Einstein said to Besso: “Thank you. I completely solved the problem."

May 1905: Lightning strikes a moving train

Einstein's revelation was that observers in relative motion perceive time differently: it is quite possible for two events to occur simultaneously from the point of view of one observer, but at different times from the point of view of another. And both observers will be right.

Einstein later illustrated his point with another thought experiment. Imagine that an observer is again standing next to the railway and a train is rushing past him. At the moment when the central point of the train passes the observer, lightning strikes each end of the train. Since lightning strikes at the same distance from the observer, their light enters his eyes at the same time. It would be fair to say that lightning strikes simultaneously.

Meanwhile, another observer sits exactly in the center of the train. From his point of view, the light from two lightning strikes travels the same distance and the speed of light will be the same in any direction. But because the train is moving, the light coming from the rear lightning has to travel a greater distance, so it arrives at the observer a few moments later than the light from the beginning. Since the light pulses arrive at different times, we can conclude that the lightning strikes are not simultaneous - one occurs faster.

Einstein realized that it is precisely this simultaneity that is relative. And once you accept this, the strange effects we now associate with relativity are resolved using simple algebra.

Einstein feverishly wrote down his thoughts and submitted his work for publication. The title was “On the Electrodynamics of Moving Bodies,” and it reflected Einstein’s attempt to connect Maxwell’s equations with the principle of relativity. Besso received special thanks.

September 1905: mass and energy

This first work, however, was not the last. Einstein was obsessed with relativity until the summer of 1905, and in September he submitted a second paper for publication, this time in retrospect.

It was based on another thought experiment. Imagine an object at rest, he said. Now imagine that it simultaneously emits two identical pulses of light in opposite directions. The object will remain in place, but since each pulse carries away a certain amount of energy, the energy contained in the object will decrease.

Now, Einstein wrote, what would this process look like to a moving observer? From his point of view, the object will simply continue to move in a straight line while the two pulses fly away. But even if the speed of the two pulses remains the same - the speed of light - their energies will be different. An impulse that moves forward in the direction of travel will have higher energy than one that moves in the opposite direction.

Adding a little algebra, Einstein showed that for this to be consistent, the object must not only lose energy when sending out light pulses, but also mass. Or mass and energy should be interchangeable. Einstein wrote down an equation that connects them. And it became the most famous equation in the history of science: E = mc 2.

One of the pearls of scientific thought in the tiara of human knowledge with which we entered the 21st century is the General Theory of Relativity (hereinafter referred to as GTR). This theory has been confirmed by countless experiments; I will say more, there is not a single experiment where our observations would differ even a little bit, even a tiny bit, from the predictions of the General Theory of Relativity. Within the limits of its applicability, of course.

Today I want to tell you what kind of beast this General Theory of Relativity is. Why is it so difficult and why In fact she's so simple. As you already understand, the explanation will go on your fingers™, therefore, I ask you not to judge too harshly for very free interpretations and not entirely correct allegories. I want anyone to read this explanation humanitarian, without any knowledge of differential calculus and surface integration, was able to understand the basics of general relativity. After all, historically, this is one of the first scientific theories that begin to move away from the usual everyday human experience. With Newtonian mechanics everything is simple; three fingers are enough to explain it - here is the force, here is the mass, here is the acceleration. Here is an apple falling on your head (has everyone seen how apples fall?), here is the acceleration of its free fall, here are the forces acting on it.

With general relativity, not everything is so simple - space curvature, gravitational time dilation, black holes - all this should cause (and does!) a lot of vague suspicions in an unprepared person - are you messing with my ears, dude? What are the curvatures of space? Who saw these distortions, where do they come from, how can something like this even be imagined?

Let's try to figure it out.

As can be understood from the name of the General Theory of Relativity, its essence is that in general, everything in the world is relative. Joke. Not really though.

The speed of light is the quantity relative to which all other things in the world are relative. Any reference frames are equal, no matter where they move, no matter what they do, even spinning in place, even moving with acceleration (which is a serious blow to the guts of Newton and Galileo, who thought that only uniformly and rectilinearly moving frames of reference can be relative and equal, and even then, only within the framework of elementary mechanics) - all the same, you can always find clever trick(scientifically this is called coordinate transformation), with the help of which it will be possible to painlessly move from one frame of reference to another, practically without losing anything along the way.

A postulate helped Einstein reach such a conclusion (let me remind you - a logical statement taken on faith without proof due to its obviousness) "on the equality of gravity and acceleration". (attention, there is a strong simplification of the formulations here, but in general terms everything is correct - the equivalence of the effects of uniformly accelerated motion and gravity is at the very heart of General Relativity).

Prove this postulate, or at least mentally to taste quite simple. Welcome to the Einstein Elevator.

The idea of ​​this thought experiment is that if you were locked in an elevator without windows and doors, then there is not the slightest, absolutely not a single way to know what situation you are in: either the elevator continues to stand as it stood at the ground floor level, and you (and all other contents of the elevator) the usual force of attraction acts, i.e. the force of gravity of the Earth, or the entire planet Earth was removed from under your feet, and the elevator began to rise upward, with an acceleration equal to the acceleration of free fall g=9.8m/s 2 .

No matter what you do, no matter what experiments you carry out, no matter what measurements of surrounding objects and phenomena you make, it is impossible to distinguish between these two situations, and in the first and second cases, all processes in the elevator will take place exactly the same.

The reader with an asterisk (*) probably knows one tricky way out of this difficulty. Tidal forces. If the elevator is very (very, very) large, 300 kilometers across, it is theoretically possible to distinguish gravity from acceleration by measuring the force of gravity (or the magnitude of acceleration, we don’t yet know which is which) at different ends of the elevator. Such a huge elevator will be slightly compressed by tidal forces in the cross section and slightly stretched by them in the longitudinal plane. But these are already tricks. If the elevator is small enough, you won't be able to detect any tidal forces. So let's not talk about sad things.

In total, in a fairly small elevator we can assume that gravity and acceleration are the same thing. It would seem that the idea is obvious, and even trivial. What is so new or complicated here, you say, this should be clear to a child! Yes, in principle, nothing complicated. It was not Einstein who invented this; such things were known much earlier.

Einstein decided to find out how a beam of light would behave in such an elevator. But this idea had very far-reaching consequences, which no one seriously thought about until 1907. I mean, to be honest, many people thought about it, but only one decided to get so deeply involved.

Let's imagine that we shine a flashlight on Einstein in our mental elevator. A ray of light flew out of one wall of the elevator, from point 0) and flew parallel to the floor towards the opposite wall. While the elevator is standing still, it is logical to assume that the light beam will hit the opposite wall exactly opposite the starting point 0), i.e. will arrive at point 1). The rays of light travel in a straight line, everyone went to school, they all learned this at school, and so did young Albertik.

It’s easy to guess that if the elevator went up, then during the time the beam was flying across the cabin, it would have time to move a little upward.
And if the elevator moves with uniform acceleration, then the beam will hit the wall at point 2), that is when viewed from the side the light will seem to move as if in a parabola.

Well, it's clear that In fact there is no parabola. The beam flew straight and still does. It’s just that while it was flying in its straight line, the elevator managed to go up a little, so here we are Seems that the beam moved in a parabola.

Everything is exaggerated and exaggerated, of course. A thought experiment, why our light flies slowly, and elevators move quickly. There is still nothing particularly cool here, all this should also be understandable to any schoolchild. You can conduct a similar experiment at home. You just need to find “very slow beams” and good, fast elevators.

But Einstein was truly a genius. Today many people scold him, like he’s a nobody and nothing at all, he sat in his patent office, weaved his Jewish conspiracies and stole ideas from real physicists. Most of those who say this do not understand at all who Einstein is and what he did for science and humanity.

Einstein said - since “gravity and acceleration are equivalent” (I repeat once again, he didn’t say exactly that, I’m deliberately exaggerating and simplifying), it means that in the presence of a gravitational field (for example, near the planet Earth), light will also fly not in a straight line, but along a curve . Gravity will bend the light beam.

Which in itself was an absolute heresy for that time. Any peasant should know that photons are massless particles. This means that light “doesn’t weigh” anything. Therefore, light should not care about gravity; it should not be “attracted” by the Earth, as stones, balls and mountains are attracted. If anyone remembers Newton's formula, gravity is inversely proportional to the square of the distance between bodies and directly proportional to their masses. If a ray of light has no mass (and light really has none), then there should be no attraction! Here contemporaries began to look sideways at Einstein with suspicion.

And he, the infection, went even further. He says we won’t break the heads of the peasants. Let's believe the ancient Greeks (hello, ancient Greeks!), let the light spread as before strictly in a straight line. Let's better assume that the space itself around the Earth (and any body with mass) bends. And not just three-dimensional space, but four-dimensional space-time.

Those. The light flew in a straight line and still does. Only this straight line is now drawn not on a plane, but lies on a sort of crumpled towel. And in 3D too. And it is the close presence of the mass that crumples this towel. Well, more precisely the presence of energy-momentum, to be absolutely precise.

All to him - “Albertik, you’re driving, stop with opium as soon as possible! Because LSD has not yet been invented, and you definitely wouldn’t come up with such a thing on your sober head! What a bent space, what are you talking about?”

And Einstein was like, “I’ll show you again!”

Locked yourself in your white tower (in the patent office, that is) and let’s adjust the mathematics to the ideas. I pushed for 10 years until I gave birth to this:

More precisely, this is the quintessence of what he gave birth to. In the more detailed version there are 10 independent formulas, and in the full version there are two pages of mathematical symbols in small print.

If you decide to take a real course in General Relativity, the introductory part ends here and then two semesters of studying the harsh language must follow. And to prepare to study this math, you need at least three more years of higher mathematics, given that you graduated from high school and are already familiar with differential and integral calculus.

Hand on heart, the matan there is not so much complicated as tedious. Tensor calculus in pseudo-Riemannian space is not a very confusing topic to understand. This is not quantum chromodynamics, or, God forbid, not string theory. Everything is clear here, everything is logical. Here's a Riemann space, here's a manifold without breaks or folds, here's a metric tensor, here's a non-degenerate matrix, write out formulas for yourself, and balance the indices, making sure that covariant and contravariant representations of vectors on both sides of the equation correspond to each other. It is not difficult. It's long and tedious.

But let's not go to such lengths and return to to our fingers™. In our opinion, in a simple way, Einstein’s formula means approximately the following. To the left of the equal sign in the formula are the Einstein tensor plus the covariant metric tensor and the cosmological constant (Λ). This lambda is essentially dark energy which we still have today we don't know anything, but we love and respect. And Einstein doesn’t even know about it yet. It has its own interesting story, worthy of a whole separate post.

In a nutshell, everything to the left of the equal sign shows how the geometry of space changes, i.e. how it bends and twists under the influence of gravity.

And on the right, in addition to the usual constants like π , speed of light c and gravitational constant G there is a letter T- energy-momentum tensor. In Lammer terms, we can consider that this is the configuration of how mass is distributed in space (more precisely, energy, because what mass or energy is the same emtse square) in order to create gravity and bend space with it in order to correspond to the left side of the equation.

That, in principle, is the whole General Theory of Relativity on your fingers™.

A hundred years ago, in 1915, a young Swiss scientist, who at that time had already made revolutionary discoveries in physics, proposed a fundamentally new understanding of gravity.

In 1915, Einstein published the general theory of relativity, which characterizes gravity as a fundamental property of spacetime. He presented a series of equations that described the effect of the curvature of spacetime on the energy and motion of the matter and radiation present in it.

A hundred years later, the general theory of relativity (GTR) became the basis for the construction of modern science, it withstood all the tests with which scientists attacked it.

But until recently it was impossible to conduct experiments under extreme conditions to test the theory's stability.

It's amazing how strong the theory of relativity has proven to be in 100 years. We still use what Einstein wrote!

Clifford Will, theoretical physicist, University of Florida

Scientists now have the technology to search for physics beyond general relativity.

A New Look at Gravity

The general theory of relativity describes gravity not as a force (as it appears in Newtonian physics), but as a curvature of space-time due to the mass of objects. The Earth revolves around the Sun not because the star attracts it, but because the Sun deforms space-time. If you put a heavy bowling ball on a stretched blanket, the blanket will change shape - gravity affects space in much the same way.

Einstein's theory predicted some crazy discoveries. For example, the possibility of the existence of black holes, which bend space-time to such an extent that nothing can escape from inside, not even light. Based on the theory, evidence was found for the generally accepted opinion today that the Universe is expanding and accelerating.

General relativity has been confirmed by numerous observations. Einstein himself used general relativity to calculate the orbit of Mercury, whose motion cannot be described by Newton's laws. Einstein predicted the existence of objects so massive that they bend light. This is a gravitational lensing phenomenon that astronomers often encounter. For example, the search for exoplanets relies on the effect of subtle changes in radiation bent by the gravitational field of the star around which the planet orbits.

Testing Einstein's theory

General relativity works well for ordinary gravity, as shown by experiments carried out on Earth and observations of the planets of the solar system. But it has never been tested under conditions of extremely strong fields in spaces lying on the boundaries of physics.

The most promising way to test the theory under such conditions is by observing changes in spacetime called gravitational waves. They appear as a result of large events, the merger of two massive bodies, such as black holes, or especially dense objects - neutron stars.

A cosmic fireworks display of this magnitude would only reflect the smallest ripples in space-time. For example, if two black holes collided and merged somewhere in our Galaxy, gravitational waves could stretch and compress the distance between objects located a meter apart on Earth by one thousandth the diameter of an atomic nucleus.

Experiments have appeared that can record changes in space-time due to such events.

There is a good chance of detecting gravitational waves in the next two years.

Clifford Will

The Laser Interferometer Gravitational-Wave Observatory (LIGO), with observatories near Richland, Washington, and Livingston, Louisiana, uses a laser to detect minute distortions in dual L-shaped detectors. As spacetime ripples pass through the detectors, they stretch and compress space, causing the detector to change dimensions. And LIGO can measure them.

LIGO began a series of launches in 2002, but failed to achieve results. Improvements were made in 2010, and the organization's successor, Advanced LIGO, should be operational again this year. Many of the planned experiments are aimed at searching for gravitational waves.

Another way to test the theory of relativity is to look at the properties of gravitational waves. For example, they can be polarized, like light passing through polarized glasses. The theory of relativity predicts the features of such an effect, and any deviations from the calculations may become a reason to doubt the theory.

Unified theory

Clifford Will believes that the discovery of gravitational waves will only strengthen Einstein's theory:

I think we must continue to search for evidence of general relativity in order to be sure that it is correct.

Why are these experiments needed at all?

One of the most important and elusive tasks of modern physics is the search for a theory that will connect together Einstein’s research, that is, the science of the macrocosm, and quantum mechanics, the reality of the smallest objects.

Advances in this area, quantum gravity, may require changes to general relativity. It is possible that quantum gravity experiments would require so much energy that they would be impossible to carry out. “But who knows,” says Will, “maybe there is an effect in the quantum universe that is insignificant, but searchable.”

The general theory of relativity, along with the special theory of relativity, is the brilliant work of Albert Einstein, who at the beginning of the 20th century changed the way physicists looked at the world. A hundred years later, general relativity is the fundamental and most important theory of physics in the world, and together with quantum mechanics claims to be one of the two cornerstones of the “theory of everything.” The general theory of relativity describes gravity as a consequence of the curvature of space-time (united in general relativity into one whole) under the influence of mass. Thanks to general relativity, scientists have derived many constants, tested a bunch of unexplained phenomena and come up with things like black holes, dark matter and dark energy, the expansion of the Universe, the Big Bang and much more. GTR also vetoed exceeding the speed of light, thereby literally trapping us in our surroundings (the Solar System), but left a loophole in the form of wormholes - short possible paths through space-time.

Einstein's theory of relativity has always seemed abstract and incomprehensible to me. Let's try to describe Einstein's theory of relativity in simple words. Imagine being outside in heavy rain with the wind blowing at your back. If you start running fast, raindrops will not fall on your back. The drops will be slower or not reach your back at all, this is a scientifically proven fact, and you can check it yourself in a rainstorm. Now imagine if you turned around and ran against the wind with the rain, the drops would hit your clothes and face harder than if you just stood.

Scientists previously thought that light acted like rain in windy weather. They thought that if the Earth moved around the Sun, and the Sun moved around the galaxy, then it would be possible to measure the speed of their movement in space. In their opinion, all they have to do is measure the speed of light and how it changes relative to two bodies.

Scientists did it and found something very strange. The speed of light was the same, no matter what, no matter how the bodies moved and no matter in which direction the measurements were taken.

It was very strange. If we take the situation with a rainstorm, then under normal circumstances the raindrops will affect you more or less depending on your movements. Agree, it would be very strange if a rainstorm blew at your back with equal force, both when running and when stopping.

Scientists have discovered that light does not have the same properties as raindrops or anything else in the universe. No matter how fast you move, and no matter what direction you are heading, the speed of light will always be the same. This is very confusing and only Albert Einstein was able to shed light on this injustice.

Einstein and another scientist, Hendrik Lorentz, figured out that there was only one way to explain how all this could be. This is only possible if time slows down.

Imagine what would happen if time slowed down for you, and you didn't know that you were moving slower. You will feel like everything else is happening faster., everything around you will move, like in a movie in fast forward.

So now let's imagine that you are again in a windy downpour. How is it possible that rain will affect you the same even if you are running? It turns out that if you were trying to run away from the rain, then your time would slow down and the rain would speed up. Raindrops would hit your back at the same speed. Scientists call this time dilation. No matter how fast you move, your time slows down, at least for the speed of light this expression is true.

Duality of dimensions

Another thing that Einstein and Lorentz figured out was that two people under different circumstances can get different calculated values ​​and the strangest thing is that they will both be right. This is another side effect of light always moving at the same speed.

Let's do a thought experiment

Imagine that you are standing in the center of your room and you have installed a lamp right in the middle of the room. Now imagine that the speed of light is very slow and you can see how it travels, imagine that you turn on a lamp.

As soon as you turn on the lamp, the light will begin to spread out and illuminate. Since both walls are at the same distance, the light will reach both walls at the same time.

Now imagine that there is a large window in your room, and a friend of yours drives by. He will see something else. To him, it will look like your room is moving to the right and when you turn on the lamp, he will see the left wall moving towards the light. and the right wall moves away from the light. He will see that the light first hit the left wall, and then the right. It will seem to him that the light did not illuminate both walls at the same time.

According to Einstein's theory of relativity, both points of view will be right. From your point of view, light hits both walls at the same time. From your friend's point of view, this is not so. There is nothing wrong.

This is why scientists say that “simultaneity is relative.” If you measure two things that are supposed to happen at the same time, then someone moving at a different speed or in a different direction will not be able to measure them in the same way as you.

This seems very strange to us, because the speed of light is instantaneous for us, and we move very slowly in comparison. Since the speed of light is so high, we do not notice the speed of light until we carry out special experiments.

The faster an object moves, the shorter and smaller it is

Another very strange side effect that the speed of light does not change. At the speed of light, moving things become shorter.

Again, let's imagine that the speed of light is very slow. Imagine that you are traveling on a train and you have installed a lamp in the middle of the carriage. Now imagine that you turn on a lamp, like in a room.

The light will spread and simultaneously reach the walls in front and behind the car. This way you can even measure the length of the carriage by measuring how long it took the light to reach both sides.

Let's do the calculations:

Let's imagine that it takes 1 second to travel 10 meters and it takes 1 second for the light to spread from the lamp to the wall of the carriage. This means that the lamp is located 10 meters from both sides of the car. Since 10 + 10 = 20, this means the length of the car is 20 meters.

Now let's imagine that your friend is on the street, watching a train pass by. Remember that he sees things differently. The rear wall of the carriage moves towards the lamp, and the front wall moves away from it. This way, the light will not touch the front and back of the wall of the car at the same time. The light will reach the back first and then the front.

Thus, if you and your friend measure the speed of light propagation from the lamp to the walls, you will get different values, but from a scientific point of view, both calculations will be correct. Only for you, according to the measurements, the length of the carriage will be the same size, but for a friend, the length of the carriage will be less.

Remember, it's all about how and under what conditions you take measurements. If you were inside a rocket moving at the speed of light, you would not feel anything unusual, unlike the people on the ground measuring your movement. You wouldn't be able to realize that time was moving slower for you, or that the front and back of the ship had suddenly become closer to each other.

At the same time, if you were flying on a rocket, it would seem to you as if all the planets and stars were flying past you at the speed of light. In this case, if you try to measure their time and size, then logically for them time should slow down and their sizes should decrease, right?

All this was very strange and incomprehensible, but Einstein proposed a solution and combined all these phenomena into one theory of relativity.