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Modern Physics Chapter 2. Theory of Relativity

Recommended Article : 【Physics】 Physics Table of Contents


1. Newtonian Paradigm

2. Principle of Relativity

3. Galilean Relativity

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

Principle of Relativity 1: Physical quantities can vary depending on the reference frame (relativity).

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.

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.

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.

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

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.

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.

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

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.

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.

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.

Δt > Δt0

○ (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.

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

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

Example 1: Atomic Bomb

Example 2: Solar Fusion : Prior to Einstein’s theory, the source of the Sun’s energy was unknown.

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.

Practice of Minkowski Spacetime

Summary of Minkowski Spacetime

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 : 2 ≡ (ct)2 - x2 - y2 - z2 = (ct’)2 - x’2 - y’2 - z’2
○ Momentum : 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.

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.

Key Concept 2. Geodesic Equation : Determines straight-line paths.

○ 0 means absence of external forces.

○ Includes inertial motion (including gravity effect).

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.

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.

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

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.

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

Example 5. Gravitational Redshift Theory

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

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.

○ Calculation method (ref 1, ref 2)

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.

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.

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

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