How the special theory of relativity is applied in non-inertial frame of reference?

Answer 1

Relativity was a concept in physics before Einstein came along. Galileo wrote about "The Basic Principle of Relativity". So "Special Relativity" refers to relativity applied to the special case of light beams (and spacetime)--the study of which Galileo was not equipped to tackle, since in his time the speed of light had not even been determined (although later it became clear that there had been some data laying around).

Einstein's General Theory of Relativity attempted to wrap up this "relativity" thing by including the theory of space curvature being the reason for gravity.

Einstein could just as well have called them "Relativity" and "Relativity The Sequel".

Answer 2

Relativity is either of two theories of physics developed by the German-born American physicist Albert Einstein. Those theories are (1) the special theory of relativity, which was published in 1905; and (2) the general theory of relativity, announced in 1915. Einstein's theories explain the behavior of matter, energy, and even time and space. They are two of the "foundation blocks" upon which modern physics is built.

The theories of relativity describe events so strange that people find it difficult to understand how they could possibly occur. For example, one person can observe that two events happen at the same time, while another person observes that they occur at different times. A clock can appear to one observer to be running at a given rate, yet seem to another observer to run at a different rate. Two observers can measure the length of the same rod correctly but obtain different results. Matter can turn into energy, and energy can turn into matter.

Galilean relativity

In developing his theories, Einstein used ideas from a principle of relativity developed by the Italian astronomer and physicist Galileo. That principle is now known as Galilean relativity.

Undetectable motion. Galileo presented the main idea behind Galilean relativity in the Dialogue Concerning the Two Chief World Systems (1632). In this work, a character named Salvatius describes two scenarios involving a ship's cabin. In both scenarios, two friends are in the cabin, along with butterflies and other small flying animals, fish swimming in a bowl, a bottle from which drops of water fall into another container, and a ball. The cabin is below deck, so neither person can see outside.

In the first scenario, the ship is at rest. The animals move about naturally, and the two friends throw the ball to each other and jump about. The friends observe that the flying animals fly with equal speed to all sides of the cabin, the fish swim in all directions, and the drops of water fall straight downward. When one friend throws the ball to the other, the effort required for the throw does not depend on the direction of the throw. When either person jumps forward, the effort required for the jump does not depend on the direction of the jump.

In the second scenario, the ship is traveling at a constant velocity. That is, both the speed and direction of the ship are unchanging. All the events that occurred in the first scenario happen again: The small creatures fly and swim, the water drips, and the two friends throw the ball and jump. The motion of the ship has no effect on any of these events. Salvatius explains why this is so: All the objects in the cabin, including the living things, share in the motion of the ship.

Because the ship's motion has no effect on the events in the cabin, neither friend can tell by observing those events whether the ship is at rest or moving. This is the main idea behind Galilean relativity.

Strictly speaking, an actual ship would not travel at a constant velocity. For example, the ship would travel in a curve because Earth's surface-including the surface of the water-is curved. The ship would also curve due to Earth's rotation on its axis and its revolution around the sun. During periods of a few seconds, however, the ship's velocity could be almost perfectly constant.

Inertial frames. Physicists would refer to the cabin as an inertial «ihn UR shuhl» frame. This term comes from the fact that, in the cabin, the principle of inertia would apply relative to the cabin. Inertia is a body's resistance to a change in its motion. A body that is at rest tends to remain at rest due to inertia. A moving body tends to maintain its velocity. For example, the fishbowl would be at rest relative to the cabin. Due to inertia, the bowl would tend to remain at rest relative to the cabin.

But suppose the ship suddenly gained speed, causing the bowl to slide. The friends in the cabin would observe that the principle of inertia no longer applied relative to the cabin. The cabin would no longer be moving at a constant velocity, so it would no longer be an inertial frame. Because the cabin was accelerating(gaining speed) it would be an accelerating frame of reference.

The principle of inertia is also known as Newton's first law of motion. It is one of three laws of motion discovered by the English scientist Isaac newton. Those laws were published in 1687 in Philosophiae naturalis principia mathematica (Mathematical Principles of Natural Philosophy), a work usually called simply Principia or Principia mathematica.

Until the late 1800's, most scientists thought that all natural events could be explained by Newton's laws. So the principle of Galilean relativity could be stated as: "The laws of nature are the same in all inertial frames," where the laws of nature were understood to be Newton's laws of motion and any laws based on them.

Galilean transformations. Certain kinds of calculations involving Galilean relativity are an important part of the background of Einstein's theories. Such calculations are known as Galilean transformations. They show how an event occurring in one inertial frame would appear to an observer in another inertial frame.

Galilean transformations apply a principle that is based on Newton's first law: Any frame of reference that is moving at a constant velocity relative to an inertial frame is also an inertial frame.

Suppose, for example, two jet aircraft, Jet A and Jet B, are flying in the same direction. Jet A is traveling 30 kilometers per hour (kph) faster than Jet B. A flight attendant in Jet A is walking at a speed of 5 kph in Jet A's direction of flight. A Galilean transformation will give the speed of the flight attendant relative to Jet B. The transformation will be an addition: 30 kph + 5 kph = 35 kph.

Now, suppose the attendant turns around and walks at a speed of 5 kph in the opposite direction. The Galilean transformation will be a subtraction: 30 kph - 5 kph = 25 kph.

The Michelson-Morley experiment. In 1887, an experiment conducted by two American physicists showed that there was something incorrect about Galileo's principle of relativity. The physicists, Albert A. Michelson and Edward W. Morley, performed their experiment on light rays.

The Michelson-Morley experiment can be traced back to a theory produced in 1864 by the Scottish scientist James Clerk Maxwell. Part of this theory describes the relationship between electric and magnetic fields. An electric field is an influence that an electrically charged object creates in the region around it. Electrically charged objects can act through their electric fields to attract or repel one another. Similarly, a magnetic field is an influence that a magnet or an electric current creates in the region around it. And similarly, magnets and objects that carry current can act through their magnetic fields to attract or repel one another.

Maxwell developed equations showing that electric and magnetic fields can combine in ways that create waves. The equations also indicate that these electromagnetic waves travel at the speed of light. Maxwell said that light itself consists of electromagnetic waves-a statement later proved to be true. He also said that other kinds of electromagnetic waves exist. The German physicist Heinrich Hertz discovered such waves-now known as radio waves-between 1886 and 1888.

Physicists reasoned that, if light consisted of waves, the waves had to travel through some substance, just as water waves travel through water. They called the substance ether, and they imagined that it filled all space. Although the ether could transmit waves, they said, it could not move from place to place. The ether's immovability made it a special inertial frame.

Maxwell's equations indicate that light moves at a particular speed, represented by the letter c. The value of c is now known to be 186,282 miles (299,792 kilometers) per second. Maxwell assumed that c was the speed of light relative to the ether. According to this assumption, light would travel faster or slower than c in an inertial frame moving relative to the ether.

Physicists also reasoned that Earth moved through the ether as the planet spun on its axis and circled the sun. Thus, any object on Earth's surface-including Michelson and Morley's laboratory-moved relative to the ether. The speed of light relative to the lab would therefore be different for light rays moving in different directions relative to the lab. And one could use Galilean transformations to calculate the speed of various rays relative to the lab.

For example, suppose the lab moved through the ether at a speed of 150 kilometers per second (kps). Imagine that a ray of light were emitted in the direction of the lab's movement. A Galilean transformation would show that the expected speed of the light relative to the lab would be c - 150 kps.

Now, imagine that a light ray were emitted in the opposite direction. The expected speed of the light relative to the lab would be c + 150 kps.

Michelson and Morley conducted their experiment to measure expected differences in the speed of light relative to their laboratory. Although light travels extremely rapidly, their experiment could measure tiny differences in speed. Surprisingly, Michelson and Morley found no difference at all. This result was a great puzzle. Physicists tried without success to determine how light could act in a manner consistent with both Galilean relativity and the Michelson-Morley experiment.

Special relativity

Einstein noted that there was no evidence for the existence of the ether. He therefore eliminated the ether from consideration. He argued that Maxwell's equations mean that the speed of light must be the same in all inertial frames. Therefore, Galileo's principle cannot be absolutely correct.

Accordingly, Einstein introduced a new principle, the special principle of relativity. This principle has two parts: (1) There is no ether, and the speed of light is the same for all observers, whatever their relative motion. (2) The laws of nature are the same in all inertial frames, where the laws are understood to include those described by Maxwell.

Einstein based his special theory of relativity on this principle. The theory solved the puzzle of the Michelson-Morley experiment. It also made dramatic new predictions that were verified by later experiments.

Lorentz transformations. Special relativity uses equations known as Lorentz transformations to describe how an event occurring in one inertial frame would appear to an observer in another inertial frame. The equations are named for the Dutch physicist Hendrik A. Lorentz, who first wrote them down in 1895. Lorentz developed the equations in an attempt to understand the Michelson-Morley experiment.

In the complex mathematics of special relativity, time and space are not absolutely separate. Instead, physicists refer to a single entity, space-time. This entity is a combination of the dimension of time and the three dimensions of space-length, width, and height. Thus, space-time is four-dimensional.

Time dilation. The Lorentz transformations show that a number of strange effects can occur. One of these is known as time dilation (dilation means widening).

For an example of this effect, consider two spaceships, A and B. The ships are moving relative to each other at a speed close to c. There is a clock in each ship. Both clocks keep time accurately, and people in both ships can see both clocks. Strangely, the people in the two ships will read the clocks differently. The people in Spaceship A will observe that the clock in Spaceship B is running more slowly than the clock in Spaceship A. But the people in Spaceship B will observe that the clock in Spaceship A is running more slowly than the clock in Spaceship B.

Time dilation actually occurs at all relative velocities. But at everyday velocities, even the most sensitive instruments cannot detect it. Thus, people are not aware of time dilation as they go about their normal activities.

However, time dilation is important in the study of cosmic rays, high-energy particles that travel through space. Some cosmic rays that originate in outer space collide with atoms at the top of Earth's atmosphere. The collisions create a variety of particles, including muons. The muons travel at almost the speed of light. In addition, they are radioactive-that is, they break apart as they travel.

Each muon can be considered to be its own reference frame. Physicists have measured how quickly muons break apart in terms of the passage of time in their reference frames. They break apart so rapidly that one might conclude that hardly any of them could ever reach Earth's surface. However, due to time dilation, the muons break apart much more slowly relative to the reference frame of Earth. As a result, many of them reach the surface.

Lorentz-Fitzgerald contraction. Another strange effect of special relativity is the Lorentz-Fitzgerald contraction, or simply the Fitzgerald contraction. Lorentz proposed that contraction occurred as an effect of the Lorentz transformations. In 1889, the Irish physicist George F. Fitzgerald had made a similar proposal.

For an example of the Lorentz-Fitzgerald contraction, again consider the two spaceships. The people in Spaceship A will observe that Spaceship B and all the objects in it have become shorter in the direction of Spaceship B's motion relative to Spaceship A. But they will observe no change in the size of Spaceship B or any of the objects as measured from top to bottom or from side to side.

This effect, like time dilation, also occurs in reverse: The people in Spaceship B will observe that Spaceship A and all the objects in it have shrunk in the direction of Spaceship A's motion relative to Spaceship B. The Lorentz-Fitzgerald contraction also occurs at all relative velocities.

Mass-energy relationship. One of the most famous effects of special relativity is the relation between mass and energy: E = mc-squared (E = mc2). Mass can be thought of as the amount of matter in an object. The equation says that an object at rest has an energy E equal to its mass m times the speed of light c multiplied by itself, or squared.

The speed of light is so high that the conversion of a tiny quantity of mass releases a tremendous amount of energy. For example, the complete conversion of an object with a mass of 1 gram would release 90 trillion joules of energy. This quantity is roughly equal to the energy released in the explosion of 22,000 tons (20,000 metric tons) of TNT.

The conversion of mass creates energy in the sun and other stars. It also produces the heat energy that is converted to electric energy in nuclear power plants. In addition, mass-to-energy conversion is responsible for the tremendous destructive force of nuclear weapons.

General relativity

Einstein developed the general theory of relativity to modify Newton's law of gravitation so that it would agree with special relativity. The key disagreement lay in descriptions of how objects exert forces on one another.

In special relativity, nothing can travel between two points faster than the speed of light. This principle applies to forces as well as rays of light.

Consider, for example, an atom of the simplest form of hydrogen. This atom consists of a single electron in orbit around a single proton. The electron carries a negative electric charge, while the proton is positively charged. The position of the proton determines the motion of the electron. It does so by exerting a force of attraction on the electron-an application of the familiar principle "opposite charges attract."

The proton exerts the force by means of electromagnetic waves that can be thought of as light rays. The proton emits(sends out) a ray, which the electron then absorbs. Thus, the electron's motion depends on what the position of the proton was when the proton emitted the ray.

In the Principia, Newton had given the law of gravity as F = m1m2 ÷ d2, where F is the gravitational force between two objects, m1 and m2 are the masses of the objects, and d2 is the distance between them squared. This law explained the motion of the planets. According to the law, a planet's motion depends on the position of the sun and the other planets. All these objects influence one another by means of gravitational force.

But Newton's law says that the force between two objects is transmitted instantaneously, no matter how far apart the objects are. That is, the law describes a gravitational action at a distance. This description disagrees with special relativity, which says that there is no action at a distance.

Principle of equivalence. To eliminate action at a distance from Newton's laws, Einstein began with an observation that he called the principle of equivalence. According to this principle, an object's gravitational mass equals its inertial mass.

Gravitational mass helps determine the force of gravity on an object. The masses m1 and m2 in Newton's law of gravity are gravitational masses.

Inertial mass is a measure of an object's inertia. Inertial mass is given in the equation for Newton's second law of motion: F = ma, where F is the force exerted on an object, mis the inertial mass of the object, and a is the acceleration of the object. This equation applies, for example, when you push an object across the floor. If your force is greater than the force of friction between the object and the floor and any other force that is working against you, the object will go faster and faster. The amount of acceleration will depend on the mass of the object and on your force minus the opposing forces.

The Hungarian physicist Lorand Eotvos had verified the principle of equivalence experimentally in 1889. Einstein saw that the principle reveals a close connection between the way an object moves through space-time and the gravitational force that acts on the object. He recognized that gravity is therefore related to the structure of space-time.

A "thought experiment." To describe how he would work to eliminate action at a distance, Einstein offered an example called a "thought experiment": First, consider an elevator that is falling freely toward Earth's surface. Suppose a person in the elevator drops a rock. The rock will fall with the person, and so it will merely hover in the air beside the person.

Now, imagine that the elevator is in outer space-so far from any planet or star that almost no gravitational force is present. The person drops the rock and, again, the rock hovers beside the person.

Einstein said that the "thought experiment" reveals a general truth: A person in free fall cannot determine by observation within his or her reference frame that gravitation is present. Thus, gravitation must be a characteristic of the space-time in which the observer is falling.

Nowadays, the principle that underlies Einstein's example is familiar in the phenomenon of weightlessness. Astronauts in the space shuttle are so close to Earth that the planet's gravity acts on them. But, like the rock in the elevator, the shuttle and its passengers are in free fall. Therefore, their experience is the same as it would be if there were no gravity at all.

Distortions in space-time. Einstein translated this principle into mathematical terms in his general theory of relativity. In this theory, matter and energy distort(change the shape of) space-time, and the distortion is experienced as gravity. A more common-but less precise-way of explaining the distortion is "Mass curves space."

Einstein suggested that astronomers could make certain observations to test the general theory of relativity. The most dramatic of these would be a bending of light rays by the sun's gravitation. In relativity, mass and energy are equivalent; and, because light carries energy, it also is affected by gravity. The light-bending effect is small, but Einstein calculated that it could be observed during a solar eclipse. In 1919, the British astronomer Arthur S. Eddington observed it, thereby making Einstein world-famous.

Gravitational waves. General relativity indicates that gravitational waves transmit gravitational force, just as electromagnetic waves transmit electric and magnetic forces. Scientists have observed gravitational waves indirectly in a pair of neutron stars that orbit each other. Neutron stars are the smallest and densest stars known. A neutron star measures only about 12 miles (20 kilometers) across, but has more mass then the sun.

By observing the pair of stars for several years, the scientists determined that the stars' orbit is becoming smaller. Calculations involving equations of general relativity show that the orbit is shrinking because the stars are emitting gravitational waves.

Most gravitational waves produce such small distortions of space-time that they are impossible to detect directly. However, collisions between neutron stars and even more compact objects called black holes create tremendous distortions. Physicists are building observatories to detect the resulting waves directly.

An observatory known as the Laser Interferometer Gravitational-Wave Observatory (LIGO) has three facilities-two in Hanford, Washington, and one in Livingston, Louisiana. Each facility is designed to detect gravitational waves by sensing their effect on two metal tubes that are 2 1/2 miles (4 kilometers) long. The tubes are built along the ground, and they are connected to each other in the shape of an L. When a gravitational wave passes through them, it changes their lengths by an amount much smaller than an atomic nucleus. A laser system detects changes in the lengths.

Answer 3

A well-established theory in modern physics, according to which neither time, distances, or mass are "absolute"; their value for a specific object depend on the observer. This is a fairly complex topic; for more details, better read the introductory article in the Wikipedia.

Answer 4

In non-inertial frames of reference, the special theory of relativity can still be applied by using mathematical transformations to correct for the non-inertial acceleration. This allows observers in non-inertial frames to account for relativistic effects such as time dilation and length contraction, ensuring consistent physical laws across different frames of reference.

Answer 5

The law of special relativity holds within an inertial frame. If the frame is accelerating, we need general theory of relativity to describe the laws in the frame.