THEORY OF RELATIVITY BY MUNEEB MAZHAR BY MUNEEB MAZHAR IMPORTANT PERSONS IN THEORY OF RELATIVITY Albert Einstein Albert Einstein (1879-1955) was a german physicist who developed a lot of concepts in physics, like theory of relativity, photoelectric effect, gravitational waves, EPR paradox, theory of Brownian motion, etc. He received the 1921 Nobel Prize in Physics by his discovery of photoelectric effect. Einstein left Germany over December 1932 and he went to the United States because of Nazism in Germany. Einstein divided the theory of relativity in two parts, special theory of relativity and general theory of relativity, he published the special theory of relativity after completing his phD in Physics, in 1905 and the general theory of relativity in 1916. 1 BY MUNEEB MAZHAR Hendrik Antoon Lorentz Hendrik Antoon Lorentz ( 1853-1928) was a dutch physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery of theoretical explanation of the Zeeman effect. He also helped Albert Einstein for creating a mathematical formula called “ The Lorentz factor” in special theory of relativity. He is also known for Lorentz ether theory, Lorentz force, Lorentzian metric, RayleighLorentz pendulum, etc. He was also a professor of Physics at the University of Leiden. 2 BY MUNEEB MAZHAR Karl Schwarzschild Karl Schwarzschild (1873-1916) was a german physicist and astronomer. Schwarzschild provided the first exact solution to the Einstein field equations of general relativity. He studied at Strasbourg and Munich, completing his doctarate in Physics in 1896 for a work on Henri Poincaré’s theories. In 1914 he joined the german army because of the outbreak of World War I, when he was 40 years old. He died at the age of 42 with a painful autoimmune skin disease called pemphigus. Schwarzschild helped Einstein to create the formula of gravitational time dilation, in that formula he applied a concept called Schwarzschild radius. Sr= Schwarzschild radius : 2GM C² 3 BY MUNEEB MAZHAR INTRODUCTION TO THEORY OF RELATIVITY Theory of relativity is divided in two parts, special relativity and general relativity, before going on these concepts, it is important to know some basic concepts for understanding it clearly. BASIC CONCEPTS OF THEORY OF RELATIVITY ● Frame of reference : a system of geometric axes in relation to which measurements of size, position, or motion can be made. In easy words, it is the position where the observer is observing a particular event. There are two types of reference : - Inertial frame of reference : when a body is moving at constant velocity, it means it doesn’t accelerate or decelerate and the net force applied on it is zero, it is in an inertial frame of reference. If a body isn’t moving, it’s on complete rest with zero net force applied on it, it is also an inertial frame of reference. - Non inertial frame of reference : when a body isn’t moving at constant velocity, it accelerates and decelerates, it is called non inertial frame of reference. ● Relative : something having, or standing in, some relation or connection to something else. ● Speed of light : speed of light is a physical constant which is 299792458 m/s. In astronomy it is a unit to calculate the distance between two objects. For example, there is a star which is 20 light year far from the 4 BY MUNEEB MAZHAR Earth, it means the light which travels 299792458 m/s, at this speed it will take 20 years to go to that star from Earth. If that star suddenly breaks, we will perceive after 20 years that the star is broken, because the light takes time to come to Earth. ● Simultaneous : happening or existing at the same time, events that happen at the same time. ● Spacetime : spacetime is any mathematical model which fuses the three dimensions of space and the one dimension of time into a single four dimensional manifold. In summary, the space is 4D, it is 4D because the time is also considered a dimension. So the position of an object in space is (t, x, y, z). 5 BY MUNEEB MAZHAR SPECIAL THEORY OF RELATIVITY Special theory of relativity is based on objects moving in an inertial frame of reference. The theory is called special because Einstein didn’t take in account the variable of gravity. There are two postulates in special theory of relativity : ● The principle of relativity : the laws of physics are the same in every inertial reference frame. ● The constancy of speed of light : observers in all inertial systems measure the same value for the speed of light in a vacuum. Characteristics of special theory of relativity : ● Speed of light is constant in any frame of reference and for all observers. 6 BY MUNEEB MAZHAR In this picture, there are two situations and in both situations there are two persons, person A and person B. 1st situation : - The person A is hitting an arrow to the diana at a speed of 200 km/h but he is also on a train which is moving at 100 km/h. The arrow of person A will hit the diana at the speed of 300 km/h, because the speed of the train and the arrow are plus. Speed of arrow = 100 km/h + 200 km/h = 300 km/h - The person B is hitting the arrow but, he is stationary, he isn’t moving in a frame of reference, that's why the arrow will hit the diana at 200 km/h. 2nd situation : - The person A has a torch which has a speed c and he is also in a train which is moving at the speed of light, c, the light will hit the diana at the speed of c, the speeds won’t pluss, because the light is constant in any frame of reference for all observers. - The person B has a torch which has a speed of c and it will hit the diana at the speed of c. Conclusion : speed of light is constant in any frame of reference and is the same for all observers. 7 BY MUNEEB MAZHAR ● Motion is relative, it depends on the frame of reference of the observer Example : In this picture, there are two observers, observer A who is on Earth and observer B who is on moon. - Observer A (Earth) : the observer A on Earth is looking at the moon and he says, “The Earth isn’t moving, it’s on rest and the moon is moving”. - Observer B (Moon) : The observer B on the moon is looking to Earth and he says, “ The Moon isn’t moving, it’s on rest and the Earth is moving”. Conclusion : the motion is relative, it depends on the frame of reference of the observer. 8 BY MUNEEB MAZHAR ● Nothing is simultaneous, it depends on the frame of reference of the observer. The relativity of simultaneity is a concept which explain, if two events happen at the same time, it means they are simultaneous, they will be simultaneous if the observer is in stationary frame of reference, it means in rest, if the observer is in a moving frame of reference, it won’t be simultaneous. The both answers will be valid because it depends on the frame of reference. Example : In this picture, there is a car moving at 50% light of speed, 0,5c, and there is a person in a stationary frame of reference. There are also two lightning bolts which fall on the trees at the same time. But the lightning bolts have a different behaviour for the two observers. - For car : for the car, the lightning bolts fall at different times, one later then another. 9 BY MUNEEB MAZHAR - For person : for the person, the lightning bolts were simultaneous, it means they fall at the same time. Link : understanding the relativity of simultaneity ● Time is relative, it depends on the frame of reference, if a person is moving near the speed of light, time will pass slower than the other person who is in a stationary frame of reference.This concept is called time dilation. Isaac Newton said time is absolute, in all the universe there is only one clock, the time is the same in all the universe, but he was wrong, time is relative. Example : There are two brothers in this situation, and they are twins, one who is in the rocket that travels near the speed of light, it means in a moving frame of reference,time will run slower and for the brother on Earth it will run faster because he is in a stationary 10 BY MUNEEB MAZHAR frame of reference. When the brother in the rocket will come back to Earth after years or months, he will be younger than the brother on Earth. This concept is called Twin paradox. To calculate the difference in time we use the Lorentz factor. Conclusion : Time is relative, it depends on the frame of reference and the speed of the object. CONCEPTS OF SPECIAL THEORY OF RELATIVITY ● Mass-energy equivalence :Mass-energy equivalence is the relation between mass and energy. The equation E = mc² means, energy can convert in a lot of mass and mass can convert in less energy. It also means that energy has got mass. This equation is also used in nuclear reactors to obtain energy from mass. Example : In this picture, there are two clocks, the first clock isn’t moving and the second clock is moving. Which of the following clocks will have more mass ? 11 BY MUNEEB MAZHAR 1st clock (not moving) : The mass of this clock is less than the other, because it isn’t moving and it doesn't have any energy in it. The extra mass comes from energies. Total mass = m (clock) + m (extra) 2nd clock (moving) : The mass of this clock is more because energy has also got mass, this clock has got kinetic energy, from the gears in it, potential energy, from the spring and thermal energy from the moving parts. The extra mass comes from all energies. To calculate the extra mass we have to clear the equation, e = mc², where m = e/c². Total mass = m (clock) + m (extra) Calculate m (extra) = E (kinetic) + E (potential) + E (thermal) c² ● Lorentz factor : mathematical formula used in theory of relativity. ● Time dilation : time dilation is a concept in relation to speed and time. If a person is moving near the speed of light, time will run slower than the person in a stationary frame of reference, time will run faster. In time 12 BY MUNEEB MAZHAR dilation there are two clocks, one in a moving frame of reference and another in a stationary frame of reference. The Lorentz factor is used to calculate time dilation. ● Length contraction : length contraction is a concept which explains length is relative, if there is a rocket in a moving frame of reference and it is moving with a lot of speed, like near the speed of light, the length will get shorter and if it reaches the speed of light it will disappear. The persons inside the rocket don’t feel the changes but an observer can see it. Lorentz factor is used to calculate the length contraction. 13 BY MUNEEB MAZHAR ● Relativistic mass : mass has two meanings in special relativity : rest mass or invariant mass is an invariant quantity which is the same for all observers in all reference frames, while relativistic mass depends on the velocity of the object in the moving frame of reference. If there is an object moving near the speed of light, its mass increases, and if it is moving with speed of light, its mass will be infinite. 14 BY MUNEEB MAZHAR General example : In this situation, there are persons on a spaceship and another person on Earth.The original length of the spaceship, in a stationary frame of reference is 20m, the original mass is 200 kg and its velocity at 80% of the speed of light. People in the spaceship have passed 30 days. Spaceship : - Length (L₀) = 20m - Mass (m₀) = 200 kg - v = 0.8c - Time (t₀) = 30 days a) How many days have passed on Earth? Δt′ = = Δt √ 1− v² c² 30 √1−0.64 = = 30 √ 30 √0.36 1−( 0.8c c )² = 30 0.6 = 30 √1−0.8² = 50 days 15 BY MUNEEB MAZHAR Solution : 30 days of spaceship are 50 days on Earth. In the second step, you have to cut the c of the numerator with the c of the denominator and you get 0.8². This data can also apply to years, like 30 years in a spaceship are 50 years on Earth. The rule applied in first step is : a² b² = ( ab ) ² b) Which is the length of the spaceship at 80% of the speed of light? l′ = l √ 1− v² c² √ = 20 1 − ( 0.8c c )² = 20√1 − 0.8² = 20√1 − 0.64 = 20√0.36 = 20 × 0.6 = 12m solution : 12m is the length of the spaceship at 80% of the speed of light. c) Which is the mass of the spaceship at 80% of the speed of light? 16 BY MUNEEB MAZHAR m= = m₀ √ 1− v² c² 200 √1−0.64 = = 200 √ 1−( 0.8c c )² 200 √0.36 = 200 0.6 = 200 √1−0.8² = 333.33kg Solution : mass of the spaceship at 80% speed of light is 333.33kg. GENERAL THEORY OF RELATIVITY General relativity is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. General relativity provides the unified description of gravity as a geometric property of space and time. Spacetime is bent when there is matter on it, the more matter will bend the more spacetime. CONCEPTS OF GENERAL RELATIVITY ● Gravitational time dilation : Gravitational time dilation is a concept that tell us that gravity also slow down time, if there is a lot of gravity, time will run slower, for example if there is a person travelling near a black hole, time will run very slow near the black hole and on Earth the person will get 17 BY MUNEEB MAZHAR older, every hour near the balck hole can be years on Earth. The Schwarzschild factor is used to calculate gravitational time dilation. Δt′ = the change in time in the gravitationally inf luenced ref erence f rame Δt = the change in time of a stationary ref erence f rame. G = the gravitational constant c = speed of light M = the mass of the object being approached r = the distant f rom the center of the mass Example : Calculate the gravitational time dilation between Earth and Sun. 18 BY MUNEEB MAZHAR Sun : - Mass = 1.9891 × 10³⁰kg - Radius = 6.96342 × 10⁸m Earth : - Mass = 5.97219 × 10²⁴kg - Radius = 6.371 × 10⁶m General information : - c = 299792458m/s - G = 6.6742 × 10 −11 N m²/kg² γ Sun/Earth = time of Sun/time of Earth = √ = 1− √ 2GM Earth R Earth c² 1− / √ 1− 2GM Sun R Sun c² = 2×(6.6742×10 −11 )×(5.97219×10²⁴) (6.371×10⁶)×(299792458)² = 1, 00002120 / √ 1− 2×(6.6742×10 −11 )×(1.9891×10³⁰) (6.96342×10⁸)×(299792458)² γ − 1 = 1.00002120 − 1 = 2.120 × 10 −6 ( × 60 × 60 × 24) = 0.183 sec/day 1s × 1día 0.183s = 5.5 days 1sec/5.5 days ● Gravitational waves : gravitational waves are waves in spacetime that generate with the close up of two black holes, when two black holes get near, they move with a lot of speed and they generate waves called gravitational waves. Gravitational waves transport energy as gravitational radiation, a form of radiant energy similar to electromagnetic energy. On 11 February 2016, the LIGO (Laser Interferometer Gravitational Wave observatory) collaboration announced the first observation of gravitational waves. 19 BY MUNEEB MAZHAR ● Gravitational lensing : gravitational lensing is a concept that explains the position of the object, like stars, planets, galaxies, etc. For example people see stars in the sky and they know the position, but this is not true. Stars that we see in the sky aren’t in the same position that we perceive. The Sun has created a bend in spacetime, if there is a star behind the Sun, the light of the star will hit the bend and it will change its position. 20 BY MUNEEB MAZHAR ● Einstein field equation : Einstein field equation relates the geometry of spacetime with the distribution of matter within it. It describes the fundamental interaction of gravitation as a result of spacetime being bent by matter and energy. In this equation, Einstein used a concept called tensor, tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors are also called multivariable matrices, they are used a lot in computation, for example, in programming languages. This equation describes the whole general relativity. ● Rμν = Ricci Curvature tensor, this tensor tells how the curvature of spacetime changes place to place, R is the scalar curvature. ● g μν = This is a metric tensor, it uses to know the distance between two points given by curvature of spacetime. 21 BY MUNEEB MAZHAR ● Λ g μν = This is the cosmological constant opposes gravity, which is multiplying the tensor. ● T μν = This is the stress energy tensor, it tells the distribution of mass. ● c = speed of light, c = 299792458m/s ● G = Newton gravitational constant, G = 6.6742 × 10 −11 N m²/kg² 22
