Undergraduate
  1. Classical mechanics  1.1. dynamics  1.1.1. Motion in one dimension  1.1.2. Motion in two dimensions  1.1.3. projectile motion  1.1.4. Relative speed  1.1.5. Uniform circular motion  1.1.6. Non-uniform circular motion  1.1.7. Reference frames and transformations  1.2. Newton's Laws of Motion  1.2.1. First law of motion  1.2.2. Second law of motion  1.2.3. Third Law of Motion  1.2.4. Applications of Newton's Laws  1.2.5. Friction and its types  1.2.6. Free Body Diagram  1.2.7. Constraints and pseudo-forces  1.3. Work and Energy  1.3.1. work done by the force  1.3.2. Work–energy theorem  1.3.3. Kinetic energy  1.3.4. Potential energy in classical mechanics  1.3.5. Conservative and non-conservative forces  1.3.6. Energy Conservation  1.3.7. Power and efficiency  1.4. Speed and collisions  1.4.1. Linear momentum  1.4.2. Conservation of momentum  1.4.3. Impulse and impact force  1.4.4. Elastic and inelastic collision  1.4.5. Center of mass and speed  1.4.6. Rocket Propulsion  1.5. Rotational motion  1.5.1. Torque and Angular Momentum  1.5.2. Moment of inertia  1.5.3. Rotational Kinematics  1.5.4. Rotational Energy  1.5.5. Rolling Motion  1.5.6. Precession and gyroscopic motion  1.6. Gravitational force  1.6.1. Newton's law of universal gravitation  1.6.2. Gravitational potential energy  1.6.3. Orbital mechanics  1.6.4. Kepler's laws of planetary motion  1.6.5. Gravitational field and potential  1.7. Fluid mechanics  1.7.1. Pressure and Pascal's Principle  1.7.2. Buoyancy and Archimedes' principle  1.7.3. Fluid Dynamics and Bernoulli's Principle  1.7.4. Viscosity and Poiseuille's law  1.7.5. Surface tension and capillarity  1.8. Oscillations and waves  1.8.1. Simple Harmonic Motion  1.8.2. Damped and driven oscillations  1.8.3. Coupled oscillations and common modes  1.8.4. Wave properties and types  1.8.5. Sound Waves and the Doppler Effect  1.8.6. Wave interference and superposition
  2. Electromagnetism  2.1. Electrostatics  2.1.1. Electric charge and properties  2.1.2. Coulomb's law  2.1.3. Electric field and electric potential  2.1.4. Gauss's Law  2.1.5. Capacitance and Dielectric  2.1.6. Electric dipole and potential energy  2.2. Electric circuits  2.2.1. Current and Resistance  2.2.2. Ohm's law  2.2.3. Kirchhoff's Laws  2.2.4. RC Circuit  2.2.5. AC Circuits and Reactance  2.2.6. Electric power and energy  2.3. Magnetism  2.3.1. Magnetic Fields and Forces  2.3.2. Ampere's Law  2.3.3. Magnetic dipole moment  2.3.4. Biot-Savart law  2.3.5. Magnetic Materials and Hysteresis  2.4. Electromagnetic induction  2.4.1. Faraday's Law  2.4.2. Lenz's law  2.4.3. Self and mutual induction  2.4.4. Transformers and Inductive Circuits  2.5. Maxwell's equations  2.5.1. Gauss's law for electricity  2.5.2. Gauss's law for magnetism  2.5.3. Faraday's law of induction  2.5.4. Ampere-Maxwell law  2.5.5. Electromagnetic waves
  3. Thermodynamics  3.1. Laws of Thermodynamics  3.1.1. Zeroth law of thermodynamics  3.1.2. First law of thermodynamics  3.1.3. Second Law  3.1.4. Third Law  3.2. Heat and work  3.2.1. Heat transfer (conduction, convection, radiation)  3.2.2. Thermal expansion  3.2.3. Heat engines and refrigerators  3.2.4. Carnot cycle and efficiency  3.3. Statistical mechanics  3.3.1. Maxwell–Boltzmann distribution  3.3.2. Entropy and Probability  3.3.3. Partition function  3.3.4. Fermi–Dirac and Bose–Einstein statistics
  4. Optics  4.1. Geometrical Optics  4.1.1. Reflection and Refraction  4.1.2. Mirrors and lenses  4.1.3. Optical Instruments  4.1.4. Fermat's principle  4.2. Wave optics  4.2.1. Interference in wave optics  4.2.2. Diffraction  4.2.3. Polarization  4.2.4. Coherence and holography
  5. Quantum mechanics  5.1. Wave–particle duality  5.1.1. Blackbody radiation in wave–particle duality in quantum mechanics  5.1.2. Photoelectric effect  5.1.3. Compton scattering  5.1.4. De Broglie wavelength  5.2. Schrödinger Equation  5.2.1. Time-independent Schrödinger equation  5.2.2. Particle in a box  5.2.3. Quantum Tunneling  5.2.4. Potential wells and obstacles  5.3. Quantum States  5.3.1. Wave function  5.3.2. Quantum operators  5.3.3. Heisenberg uncertainty principle  5.3.4. Angular momentum and spin
  6. Relativity  6.1. Special relativity  6.1.1. Lorentz transformations  6.1.2. Time expansion and length contraction  6.1.3. Relativistic energy and momentum  6.2. General relativity  6.2.1. Principle of Equivalence  6.2.2. Schwarzschild metric  6.2.3. Gravitational waves  6.2.4. Black holes and event horizons
  7. Solid state physics  7.1. Crystal structure  7.1.1. Bravais lattices  7.1.2. X-ray diffraction  7.1.3. Band theory  7.1.4. Phonons and lattice vibrations  7.2. Electrical and Magnetic Properties  7.2.1. Conductors, Semiconductors and Insulators  7.2.2. Superconductivity  7.2.3. Hall effect
  8. Nuclear and particle physics  8.1. Atomic Structure  8.1.1. Atomic Model  8.1.2. Nuclear binding energy  8.2. Radioactivity  8.2.1. Alpha, beta, gamma decay  8.2.2. Half life  8.2.3. Nuclear Fission and Fusion  8.3. Particle physics  8.3.1. Standard Model  8.3.2. Quarks and Leptons  8.3.3. antimatter  8.3.4. Fundamental interactions
  9. Astrophysics and cosmology  9.1. Stellar evolution  9.1.1. Star formation  9.1.2. Black holes and neutron stars  9.1.3. White dwarfs and supernovae  9.2. Cosmology  9.2.1. The Big Bang Theory  9.2.2. Dark matter and dark energy  9.2.3. Cosmic microwave background

Undergraduate → Relativity → General relativity ↓


Principle of Equivalence


The principle of equivalence is one of the cornerstones of Albert Einstein's theory of general relativity. Although it may seem abstract at first glance, this principle is a profound insight into how gravity works in the universe. In its simplest form, this principle states that the effects of gravity are locally indistinguishable from acceleration. In other words, being inside an accelerating vehicle feels the same as being in a gravitational field.

Understanding through thought experiments

To understand the principle of equivalence, let's start with some thought experiments. These are mental exercises that help us understand complex ideas without the need for physical experiments or equipment.

Example 1: Elevator

Imagine you are inside a sealed elevator that is floating in space, far from any planets or stars. In this scenario, you, the elevator, and everything inside are in free fall under gravity, but because you are in space and not near any massive objects, you feel no gravitational pull. This is a condition known as "weightlessness."

Now, if the elevator is pulled upward by a cable with a constant acceleration equal to the acceleration due to gravity on Earth, you will feel a force pressing you against the floor. This force is indistinguishable from the force of gravity you feel when you stand on Earth. Inside the elevator, you cannot tell whether the force you feel is due to the Earth pulling you down or due to the elevator accelerating upward.

Lift Free fall or gravity?

This example illustrates the main idea of the equivalence principle: locally (i.e., in a small region of space and time), there is no experiment by which you can tell the difference between a uniform gravitational field and an acceleration that is constant in your reference frame.

Example 2: Spacecraft

Consider a spacecraft accelerating at a constant rate through space. Inside, the astronauts will feel the same effects as they feel from gravity on Earth. Dropped objects will fall to the floor in the same way as they would under Earth's gravity. The upward acceleration of the spacecraft mimics the downward acceleration due to gravity.

vehicle

If the spacecraft were sitting on the surface of a planet, the force felt by the astronauts would be due to the planet's gravitational pull. Yet inside a spacecraft moving in space, the force arises from acceleration. According to the principle of equivalence, the two situations are indistinguishable from inside the spacecraft.

Mathematical formulation

The principle of equivalence also has a mathematical aspect. Traditionally, the force of gravity has been contained in Newton's law of universal gravitation:

F = G * (m1 * m2) / r^2

Where F is the gravitational force, G is the gravitational constant, m1 and m2 are the combined masses, and r is the distance between the centers of the two masses.

On the other hand, according to Newton's second law of motion:

F = m * a

where m is mass and a is acceleration. In the context of general relativity and the principle of equivalence, the laws of physics in an accelerated frame of reference (such as an accelerating elevator or spacecraft) are the same as those in a frame of reference in a gravitational field. Thus, gravity is not a force between masses, but rather an effect of masses on the geometry of space-time.

Experimental verification

Over time, various experiments have confirmed the principle of equivalence:

Eötvös experiment

Hungarian physicist Loránd Eötvös performed experiments to show that different substances fall at the same rate in a gravitational field. This was in part to demonstrate the equivalence between inertial mass (resistance to acceleration) and gravitational mass (response to gravity).

Microscope mission

The Microscope satellite mission, launched by the French Space Agency, improved the accuracy of tests on the principle of equivalence. It used two test masses to assess the difference in their trajectories due to gravity, proving the principle with extraordinary accuracy.

Implications and real-world applications

The principle of equivalence has profound implications:

Relativity and GPS

One of the most practical applications is in the field of Global Positioning Systems (GPS). These systems operate on precise time determination algorithms. Due to the equivalence principle and the effects of general relativity, clocks on satellites must include corrections for time dilation effects due to their relative speed and distance in the Earth's gravitational field.

Satellite Time dilation effect Earth

Gravitational time dilation

Time appears to slow down in strong gravitational fields due to the principle of equivalence. This fact has been confirmed by experiments with highly accurate atomic clocks placed at various heights. These deviations, although small, are measurable and have been observed repeatedly.

Conclusion

The equivalence principle challenges our everyday intuition, yet it beautifully describes a universe where the fabric of space-time itself is shaped by mass and energy. It underlies the revolutionary framework of general relativity – which furthers the relativity introduced by Galileo and Newton – and represents one of the most successful theories in modern physics.


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Principle of Equivalence

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