Modern Physics Chapter 2. Theory of Relativity
Recommended Article : 【Physics】 Physics Table of Contents
4. Special Theory of Relativity
5. General Theory of Relativity
1. Newtonian Paradigm (Newtonian paradigm)
⑴ Newton’s Laws 1, 2, and 3 describe the relationships among distance, time, force, and energy.
① Newton’s defined time.
② Einstein’s defined time.
⑵ Newtonian Paradigm
① Belief that all phenomena can be interpreted with Newton’s laws.
② Mechanistic worldview.
③ (Note) Laplace’s demons, monsters, witches, goblins.
⑶ By the late 1800s, phenomena that could not be explained by Newtonian mechanics emerged.
① Introduction of quantum mechanics: Newtonian mechanics is not applicable on a very small scale (ℓ ≪ 1).
② Introduction of the theory of relativity: Newtonian mechanics is not applicable on a very large scale (v ≫ 1).
2. Principle of Relativity
⑴ Principle of Relativity
① Principle of Relativity 1: Physical quantities can vary depending on the reference frame (relativity).
② Principle of Relativity 2: Physical laws themselves do not change depending on the reference frame (absolute).
⑵ Theory of Relativity
① Definition: Application of the principle of relativity to specific coordinate transformations.
② Theory 1: Galilean relativity
○ Principle: Principle of relativity.
○ Coordinate transformation: Vector addition.
○ Coordinate space: Euclidean space.
○ Conditions: Inertial frame, v ≪ c.
○ Related theory: Newtonian mechanics.
③ Theory 2: Special theory of relativity
○ Principle: Principle of relativity, principle of the constancy of the speed of light.
○ Coordinate transformation: Lorentz transformation (4-vector linear transformation).
○ Coordinate space: Minkowski space.
○ Conditions: Inertial frame, GM / r = Φ ≪ c2.
○ Related theory: Maxwell’s electromagnetism.
④ Theory 3: General theory of relativity
○ Principle: Principle of relativity, equivalence principle, Mach’s principle.
○ Coordinate transformation: Metric transformation (2nd-order tensor transformation).
○ Coordinate space: Riemann space.
○ Conditions: Non-inertial frame, differentiable, h → 0 (i.e., neglecting quantum mechanics).
○ Related theory: Cosmology.
3. Galilean Relativity
⑴ Characteristics
① Very familiar concepts to us.
② The law of addition of velocities holds.
③ Inertial frame: A frame with zero acceleration.
⑵ Coordinate Transformations
① Let observer X start from (0, 0, 0) and move with a velocity of (u, 0, 0).
② Let there be a point P(x, 0, 0).
③ According to a stationary observer Y, P is still at (x, 0, 0).
④ Observer X initially perceived P at coordinates (x, 0, 0).
⑤ After moving a time t, observer X perceives P at coordinates (x - ut, 0, 0).
⑶ Newtonian mechanics satisfies the principle of relativity.
⑷ Inertial and Accelerated Frames
① Inertial frame: A coordinate system defined relative to an observer at rest or moving with constant velocity.
② Accelerated frame: A coordinate system defined relative to an observer undergoing accelerated motion.
③ Practical distinction: Inertial frames have no inertial forces. Accelerated frames experience inertial forces.
○ Observer in an elevator accelerating upward by a: The observer perceives an additional inertial force downward of ma.
○ Observer in an elevator accelerating downward by a: The observer perceives an additional inertial force upward of ma.
○ Observer moving in uniform circular motion: The observer perceives a centrifugal force equal to ma.
④ The acceleration experienced by an observer is not relative, as it implies the presence of inertial forces.
⑸ Finite Speed of Light
① Evidence 1: Jupiter’s Moon (eclipse)
○ Giovanni Cassini (1625-1712)
○ Shorter eclipse time when Earth is closer to Jupiter.
○ Longer eclipse time when Earth is farther from Jupiter.
○ Rømer’s argument
○ Difference in eclipse time attributed to difference in light travel time.
○ Conclusion: Light speed is finite.
② Evidence 2: Toothed-wheel experiment
○ In reality, the wheel must rotate very fast to not obstruct the reflected light.
4. Special Theory of Relativity
⑴ Einstein’s Idea
① 1st: Around 1870, the concept began to be used.
② 2nd: For trains, a method of synchronizing clocks is needed to know departure and arrival times.
③ 3rd: No method existed at the time to synchronize clocks in different regions.
④ 4th: Many patents for clock synchronization were filed with the Swiss Patent Office.
⑤ 5th: Einstein was working at the Swiss Patent Office at the time.
⑥ 6th: Einstein realized that the signal takes more time to reach him when he looks at a clock to know the current time.
⑦ 7th: In other words, Einstein understood that wristwatches and biological clocks cannot show the same time.
⑧ 8th: Furthermore, Einstein believed that time is not absolute.
⑵ Law of the Invariance of the Speed of Light
① Maxwell’s equations
○ Maxwell discovered that electrical phenomena are described by the wave equation of electromagnetic waves.
○ Discovered that the speed of electromagnetic waves is similar to the known speed of light (3 × 10^8 m/s).
○ Concluded that light is an electromagnetic wave.
② Ether theory
○ All waves require a medium to propagate.
○ Light is an electromagnetic wave, thus it must have a medium called ether.
○ When light moves through ether, its speed varies according to the speed of ether.
Figure. 1. Example of Ether Flow
③ Einstein’s Idea
○ Electrical phenomena occur whether an object is moving or stationary.
○ Precisely 3 × 10^8 m/s velocity of electromagnetic waves is required for electrical phenomena to occur ( ∵ Maxwell’s equations).
○ Electrical phenomena occur even in a moving frame.
○ Therefore, the speed of light is constant.
④ Evidence: Michelson-Morley Interference Experiment
Figure. 2. Michelson-Morley Interference Experiment
○ Michelson interferometer is a precise instrument to observe interference patterns.
○ Improved the precision of the Michelson-Morley experiment over 50 years.
○ Michelson, Studies in Optics (1881)
○ Shankland, et al., Rev. Mod. Phys. 27, 167 (1955)
○ Even though the same light starts, there is a time difference when it arrives after reflection in the mirror.
○ Assume the speed of ether is v = (vx, vy).
○ Calculating the time difference.
○ Result 1: The interference pattern does not shift when the Michelson-Morley interferometer is rotated, refuting the ether theory.
○ For a time difference equivalent to one more wavelength, constructive interference should occur.
○ Result 2: Since the ether theory is refuted, the speed of light is constant.
○ Michelson and Morley concluded that they couldn’t measure the speed of ether due to the experiment’s low precision.
○ One of the few cases where a Nobel Prize was awarded for the absence of results.
⑶ Special Theory of Relativity
① Einstein published his first paper at the age of 26 in 1905.
○ First paper: “On the Electrodynamics of Moving Bodies.”
② Assumptions
○ Assumption 1: Physical laws are the same in any inertial frame (including Maxwell’s equations).
○ Assumption 2: The speed of light in a vacuum is the same in any inertial frame.
○ Assumption 3: The observer’s coordinate system must be inertial with respect to the observed object.
③ Application 1: Simultaneity and Heterochrony of Events
○ Even if events happen simultaneously, they appear differently depending on the observer’s motion state.
Figure. 3. Simultaneity and Heterochrony of Events
○ While A sees light emitted from P and R reaching Q simultaneously, the same occurs when seen by B.
○ The relativity of simultaneity means different observers can see the sequence of different events differently.
○ However, this doesn’t mean facts about a single event are viewed differently.
Figure. 4. Caution Regarding Simultaneity of Events
④ Application 2: Relative Velocity
○ Situation: A and B moving at constant velocities.
○ Conclusion: The magnitude of the relative velocity seen by A looking at B and by B looking at A is the same.
○ Very important concept.
⑤ Application 3: Time Dilation
Figure. 5. Time Dilation Modeling
○ Event: Light starts at the initial point, reflects at the mirror, and returns to the initial point.
○ Time Δt0 measured by the observer in the spaceship.
○ Time Δt measured by the observer on the ground.
○ Summary: An observer sees a moving object as if it is moving slowly.
○ Example 1: Muons generated in the stratosphere have a half-life of 2.2 × 10^-6 seconds at rest.
○ Example 2: Inside the stratosphere with v = 0.999c, its half-life is 1.1 × 10^-3 seconds.
⑥ Application 4: Length Contraction
Figure. 6. Length Contraction Modeling
○ Thought experiment: Relative velocity v is the same for the observer in the spaceship and the observer on the ground.
○ Derived length contraction: Using the constant relative velocity.
○ Δt : Time taken by the moving object in the observer’s frame of reference. Measured as a delayed time.
○ L0 : Distance between two points P and Q measured in the observer’s frame of reference.
○ Δt0 : Time taken by the observer at rest to measure the moving observer’s time.
○ L : Distance between two points P and Q measured in the moving observer’s frame of reference.
○ Time is important from the moving observer’s viewpoint, so the time measured by the moving observer becomes the “proper time.”
○ Distance between P and Q is important from the moving observer’s viewpoint, so the distance measured by the moving observer becomes the “proper distance.”
○ Events at P or Q for the moving observer do not have simultaneity from the observer’s perspective.
○ Thus, even if the moving observer reaches Q, the observer appears not to have reached Q.
○ (Comment) It’s important to properly understand the meanings of each symbol.
○ (Comment) Personally, considering the essence of the theory of relativity, it’s inappropriate to label a certain measured quantity by an intrinsic physical quantity.
○ Summary : When observing a moving object, the object appears shortened.
○ Example 1: From the perspective of a muon, to travel from the stratosphere to the Earth’s surface within 2.2 × 10^-6 seconds, the distance must be shortened.
⑷ Lorentz Coordinate Transformation
① γ factor : Greater than 1. γ increases rapidly when v > 0.9c.
② Derivation of Lorentz Transformation
Figure. 7. Model of Derivation of Lorentz Transformation
○ Situation
○ Light emitted from the origin reaches point (x, y, z) after time t.
○ An observer at rest at the origin sees the light reach point (x, y, z) after time t has passed.
○ An observer S’ moving from the origin sees the light reach point (x’, y’, z’) after time t has passed.
○ Additional assumption : Light is assumed to be emitted along the x-axis.
○ Note
○ In the same inertial frame as observer S, S is the stationary observer, and S’ is the moving object.
○ In the same inertial frame as observer S’, S is the moving object, and S’ is the stationary observer.
○ The way S’ perceives (x, y, z) is the same as how a stationary observer perceives a moving object.
○ Application of the Law of Invariant Speed
○ Derivation of the x’ relation : According to Galilean relativity, x’exp = x - vt, and x’exp appears shortened by a factor of γ to observer S’.
○ Fusion of the Law of Invariant Speed and the x’ relation : To observer S’, observer S takes longer to observe light.
○ Formulation : For β = v / c
⑸ Lorentz Velocity Transformation
① Situation
○ Particle moves with velocity v in the S coordinate system.
○ S’ moves with velocity u (along the x-axis) relative to S.
○ What is the velocity of the particle observed in S’?
② Derivation
③ Conclusion
④ Relativistic Velocity Addition
○ Speed observed from the ground when an object is shot with velocity v2 in the direction of a rocket moving with velocity v1.
○ Approximation : If v1, v2 ≪ c
○ The statement applies to different reference frames, not the same one.
○ Situation : In frame A, objects B and C move towards each other with velocities 0.8c and 0.9c respectively.
○ Speed of approach in frame A : 1.7c
○ Speed of C as seen by B : Speed of Q as seen by P is defined as vPQ
⑹ Doppler Effect of Light
① Vertical Doppler Effect : As the light source approaches the observer, the wavelength shortens due to the source itself approaching. Time dilation effect.
② Horizontal Doppler Effect : As the light source approaches from the side.
⑺ Mass-Energy Equivalence Formula
① Relativistic Mass
○ Electron’s mass is measured differently based on its velocity.
○ (Note) Mass remains constant in classical mechanics.
○ Reason : Mass is energy.
Figure. 8. Experiments by Kaufmann, Bucherer, Lorentz
(Abraham, Lorentz, Bucherer’s experiment)
② Rest Mass-Energy
○ Concept : Thought experiment when energy is emitted from matter.
Figure. 9. Einstein’s Box
○ Situation : Energy emitted from the left end of the box → Box with mass M moves → Energy meets the right end.
○ (Comment) Even for photons, which have no mass, there is momentum due to factors like radiation pressure.
○ Equation derivation for v : Using conservation of momentum. For energy’s momentum, like for photons, p = E / c.
○ Equation derivation for Δt : Time taken for energy emitted from the left end of the box to meet the right end.
○ Equation derivation for Δx
○ Assuming energy can be exchanged for mass m (Key Point)
○ Conclusion : E = mc^2
○ Limitation 1: Equation for Δt is approximate.
○ Limitation 2: Doesn’t consider special relativistic considerations for time and space.
○ Significance 1: Implies that mass is energy.
○ Significance 2: Conclusion of energy being proportional to mc^2 is also supported by dimensional analysis.
○ Application 1: Calculation of radiation pressure
○ E : Energy of a photon
○ p : Momentum of a photon
○ c : Speed of light
○ F : Force on a plate with area A
○ Δt : Suitable time interval
○ P : Radiation pressure
③ Kinetic Energy : Work done to accelerate an object with force F to velocity v.
○ Derivation
○ Reference
⑤ Total Energy of an Object
⑥ Relativistic Momentum
○ Relativistic momentum : p = γm0v
○ Relation with relativistic energy : E^2 = p^2c^2 + (m0c^2)^2
○ Massless objects (e.g., photons) : m0 = 0, E = pc
○ Rest mass-energy : p = 0, E = m0c^2
⑦ Example 1: Atomic Bomb
⑧ Example 2: Solar Fusion : Prior to Einstein’s theory, the source of the Sun’s energy was unknown.
⑨ Example 3: New particles produced in particle accelerators
⑽ Minkowski Spacetime Diagram : Visualization of spacetime for two inertial frames
① Stationary Case
Figure. 10. World Line Diagram for the Stationary Case
○ Consider the 1-dimensional case first.
○ Define the horizontal axis as x and the vertical axis as ct.
○ An object stationary at x = 1 is represented by a red line as time progresses.
○ Since x’ = 0, this red line becomes the ct’ axis.
② Creating the Time Axis
Figure. 11. Definition of Time Axis for World Line Diagram of Moving Object
○ Object moving at speed v is represented by a green solid line.
○ Object moving at speed -v is represented by a green dashed line.
○ Angle θ between ct’ axis and ct axis is given by tan θ = v / c.
○ Light moving at v = -c can be represented by blue color.
○ In this case, angle θ between ct’ axis and ct axis is 45°.
③ Creating the Spacetime Axis
Figure. 12. Definition of Spacetime Axis for World Line Diagram of Moving Object
○ Coordinates on the x-axis are all at t = 0.
○ Therefore, coordinates on the x’ axis are all at t’ = 0.
○ For ct’ to be 0, ct - βx = 0, which means ct = βx.
○ Thus, it forms a straight line with tan θ = β.
④ Creating Coordinate Scales
○ Distance between two events (x1, y1, z1, t1) and (x2, y2, z2, t2) is defined using Lorentz Transformation.
⑾ Phenomenon : When moving close to the speed of light
① Everything starts to appear curved.
○ Length in the direction of motion appears shortened.
○ Length in the perpendicular direction remains unchanged.
② Even colors change.
○ Approaching stars show blue shift, receding stars show red shift.
○ If only the Doppler effect is considered, the front would appear black.
○ When both Doppler and contraction effects are considered, the front appears white.
⑿ Summary : Relativistic Dynamics
① 4-Vector : xμ ≡ (ct, x, y, z) ≡ (x0, x1, x2, x3)
② 4-Velocity Vector : uμ ≡ (γc, γ v )
③ 4-Momentum : pμ ≡ muμ ≡ (γmc, γm v )
④ Lorentz Scalar
○ Spacetime Interval : xμ 2 ≡ (ct)2 - x2 - y2 - z2 = (ct’)2 - x’2 - y’2 - z’2
○ Momentum : pμ 2 ≡ (γmc)2 - (γmv)2 = m2c2 ⇔ (E/c)2 = m2c2 + p2
⑤ Equation of Motion : m d v / dt = f ⇔ dpμ / dτ = fμ
5. General Theory of Relativity
⑴ Overview
① Einstein pondered why gravity exists.
② Gravity warps spacetime, affecting mass.
⑵ Key Principles
① Principle of Relativity
② Equivalence Principle : Gravity and inertia are indistinguishable.
Figure. 13. Equivalence Principle
③ Mach’s Principle
○ Definition : The distribution of matter in the universe affects local object motion.
○ Implies that matter distribution causes changes in spacetime structure.
⑶ Einstein’s Insight
① Considered the motion of planets orbiting the Sun.
② Sun’s mass influences planet motion, making planets feel the Sun.
③ Sun imparts information to planets.
④ Sun transmits information at the speed of light for some reason.
○ Only massless entities can transmit signals at the speed of light.
⑤ Density of transmitted information from the Sun at distance r is given by
Information Flux = Total Information / 4πr2
⑥ Gravity is also information, inversely proportional to distance squared.
⑦ Gravity is an interaction, leading to Newton’s law of universal gravitation.
⑷ Content
① Announced by Einstein in 1915.
② Overview : Spacetime curvature determined by energy-matter distribution; objects move along geodesics.
③ Key Concept 1. Einstein Field Equations (Cosmological Equations)
○ Left side represents space, right side represents mass and energy.
○ Sometimes denoted as Gμν on the left side.
○ Einstein added the Λgμν term assuming the universe doesn’t contract due to gravity.
○ Λ is the cosmological constant.
○ Later admitted introducing Λ was a mistake.
○ Subsequent discovery of cosmic expansion validated the introduction of Λ.
④ Key Concept 2. Geodesic Equation : Determines straight-line paths.
○ 0 means absence of external forces.
○ Includes inertial motion (including gravity effect).
⑤ Key Concept 3. Friedmann Equations
○ Applies homogeneity and isotropy to Einstein’s field equations.
○ FLRW metric.
○ Tμν = (ρ, -p, -p, -p)
⑥ Einstein’s theory proved for the first time by the 1919 Eddington solar eclipse observation.
⑸ Example 1. Twin Paradox
① Characters : Twin A who travels in space, Twin B who remains on Earth.
② Situation
○ When A and B are 30 years old, A travels to a distant planet and returns to Earth.
○ Rocket’s velocity is v = 0.6 c.
③ Perspective of B
○ B is stationary, A is in motion relative to B : Time dilation occurs for A.
○ When B is 60 years old, A is 54 years old (due to time delay).
④ Perspective of A
○ A is stationary, B is in motion relative to A : Time dilation occurs for B.
○ When A is 60 years old, B is 54 years old (due to time delay).
⑤ Paradox : When A returns to Earth, is A older or is B older?
⑥ Solution : When B is 60 years old, A is 54 years old.
⑦ Reason : A experienced high acceleration, B experienced only gravity acceleration → Time flows slower for A.
⑹ Example 2. Artificial Satellites
① Space can determine position if distances from three reference points are known.
② Determining spatial position using satellites requires an additional satellite for time, totaling four.
③ GPS satellites require adjustments due to Earth’s gravity, as per general relativity.
④ Special relativistic effect : Clocks slow down by 7 μs/day.
⑤ General relativistic effect : Clocks speed up by 45 μs/day.
Figure. 14. GPS Correction According to Relativity
⑺ Example 3. Perihelion Shift of Mercury
Figure. 15. Perihelion Shift of Mercury
① According to Newtonian mechanics, Mercury should undergo a 574” per century shift due to solar motion.
② The observed value differs by 43” from the calculated value.
③ General relativity accounts for the 43” shift through spacetime distortion.
⑻ Example 4. Gravitational Lensing : Space distortion caused by massive objects like black holes.
Figure. 16. Gravitational Lensing Effect
① Einstein’s Prediction
Figure. 17. Einstein’s Prediction
○ During a solar eclipse on May 29, 1919, stars near the Sun were photographed.
○ Result : Stars closer to the Sun showed larger position shifts.
○ Conclusion : Spacetime is greatly curved near the Sun.
② Einstein’s Circle and Einstein’s Cross
⑼ Example 5. Gravitational Redshift Theory
① Type 1. Redshift due to General Doppler Effect
○ vs : Source (e.g., sound) velocity
○ vd : Observer’s velocity
○ v0 : Wave (e.g., sound) velocity
○ f0 : Wave frequency
○ f : Observed frequency
② Type 2. Redshift due to Cosmological Doppler Effect
○ 1st. Universe is expanding.
○ 2nd. As space expands, light wavelengths increase.
○ 3rd. Visible light shifts towards the red.
③ Type 3. Redshift due to Relativistic Doppler Effect
○ 1st. Gravity warps spacetime → increases space → shifts wavelengths.
○ 2nd. Increased space causes longer wavelengths.
○ 3rd. Visible light shifts towards the red.
④ Light from larger stars is observed to have longer wavelengths than expected.
⑽ Example 6. First Black Hole Observation
Figure. 18. First Black Hole Observation
Figure. 19. Black Hole at the Center of Our Galaxy
① General relativity predicts that very massive objects can create spacetime curvature so strong that even light cannot escape.
② First observation of a black hole on April 10, 2019, 22:00.
○ Required telescopes equivalent to the size of the Earth to observe black holes.
○ Algorithmically combined images from telescopes worldwide to solve the problem.
⑾ Example 7. Gravitational Waves
Figure. 20. Gravitational Waves from Rotating Objects
① Gravitational waves : When massive objects vibrate, they create spacetime distortions that propagate as waves.
② According to general relativity, gravity also carries information, suggesting the existence of gravity-carrying particles.
③ Until recently, it was believed that three of the fundamental forces of physics had particles that carry the forces, excluding gravity.
④ In 2015, the LIGO observatory successfully measured gravitational waves for the first time.
○ On February 11, 2016, results of gravitational wave measurements were announced.
○ Measured the gravitational fluctuations caused by the merger of two massive celestial bodies.
Input : 2019.04.16 00:09