Relativity

1. Geometry's Truth Lies in Physical Correspondence

The concept “true” does not tally with the assertions of pure geometry, because by the word “true” we are eventually in the habit of designating always the correspondence with a “real” object.

Geometry and Reality. Euclidean geometry, with its axioms and theorems, is not inherently "true" in an absolute sense. Its validity stems from its correspondence to real-world objects and measurements. Geometry becomes a branch of physics when its propositions are tested against the behavior of rigid bodies and the measurement of distances.

Ruler and Compass. The "truth" of a geometrical proposition is understood as its validity for constructions with a ruler and compass. This conviction is based on incomplete experience, and the general theory of relativity later reveals the limitations of this "truth."

Example: The straight line, a fundamental concept in geometry, is associated with a natural object. Three points on a rigid body lie on a straight line when the middle point minimizes the sum of the distances to the other two. This physical interpretation grounds geometry in the observable world.

2. Reference Frames Dictate Motion's Description

There is no such thing as an independently existing trajectory (lit. “path-curve”1), but only a trajectory relative to a particular body of reference.

Relative Motion. The description of motion is always relative to a chosen reference frame. There is no absolute motion or trajectory independent of an observer's perspective. The path of an object, whether a straight line or a curve, depends on the reference frame from which it is observed.

Example: A stone dropped from a moving train appears to fall in a straight line to a passenger, but to a stationary observer on the ground, it follows a parabolic path. Both descriptions are valid, each relative to its respective reference frame.

Coordinate Systems. The concept of a "body of reference" can be formalized using coordinate systems. The position of an event is specified by its coordinates relative to a rigid body. The Cartesian system, with its three perpendicular axes, provides a numerical way to describe positions in space.

3. Classical Mechanics Relies on Inertial Frames

A system of co-ordinates of which the state of motion is such that the law of inertia holds relative to it is called a “Galileian system of co-ordinates.”

Law of Inertia. Classical mechanics, as formulated by Galilei and Newton, is based on the law of inertia: a body remains at rest or in uniform motion unless acted upon by a force. This law holds only in specific reference frames called Galileian or inertial frames.

Fixed Stars. The fixed stars serve as an approximate inertial frame. A coordinate system attached to the Earth is not inertial because the stars appear to move in circles relative to it, violating the law of inertia.

Galileian Frames. The laws of mechanics are valid only for Galileian coordinate systems. These systems are in uniform motion relative to each other. This restriction highlights a limitation of classical mechanics.

4. Relativity Extends Equivalence to All Frames

If, relative to K, K′ is a uniformly moving co-ordinate system devoid of rotation, then natural phenomena run their course with respect to K 'according to exactly the same general laws as with respect to K.

Principle of Relativity. The principle of relativity states that the laws of nature are the same for all observers in uniform motion. This means that no experiment can distinguish between a state of rest and a state of uniform motion.

Anisotropy Absence. The absence of anisotropic properties in terrestrial physical space supports the principle of relativity. Experiments have failed to detect any physical non-equivalence of different directions, which would be expected if the Earth were moving through a privileged reference frame.

Generalization. The general principle of relativity extends this equivalence to all bodies of reference, regardless of their state of motion. This generalization requires a new understanding of space and time.

5. Time's Relativity Defies Absolute Simultaneity

Events which are simultaneous with reference to the embankment are not simultaneous with respect to the train, and vice versa (relativity of simultaneity).

Simultaneity is Relative. The concept of simultaneity is not absolute but depends on the observer's reference frame. Two events that are simultaneous in one frame may not be simultaneous in another frame moving relative to the first.

Lightning Strikes. Imagine lightning striking two points, A and B, on a railway embankment. An observer at the midpoint between A and B on the embankment sees the flashes simultaneously. However, an observer on a train moving relative to the embankment will see one flash before the other.

Reference Frame Dependence. Every reference frame has its own particular time. A statement of time is meaningless without specifying the reference frame to which it refers. This challenges the classical assumption of absolute time.

6. Lorentz Transformation Bridges Reference Frames

This system of equations is known as the “Lorentz transformation.”

Incompatibility Resolution. The apparent conflict between the constancy of the speed of light and the principle of relativity is resolved by the Lorentz transformation. This set of equations relates space and time coordinates between different reference frames.

Transformation Equations. The Lorentz transformation equations are:

• x′ = (x - vt) / √(1 - v²/c²)
• y′ = y
• z′ = z
• t′ = (t - vx/c²) / √(1 - v²/c²)

Velocity of Light. The Lorentz transformation ensures that the speed of light is the same in all reference frames. If a light signal travels along the x-axis with velocity c in frame K, it will also travel with velocity c in frame K'.

7. Mass and Energy Are Fundamentally Equivalent

The inertial mass of a system of bodies can even be regarded as a measure of its energy.

Unified Conservation Law. The special theory of relativity unites the conservation of energy and the conservation of mass into a single law. Mass is not constant but varies with the energy of the body.

Mass Increase. If a body absorbs energy E, its inertial mass increases by an amount E/c². This relationship is expressed by the famous equation E = mc².

Energy Expression. The energy of a body is given by mc²/√(1 - v²/c²), where mc² represents the energy possessed by the body before it absorbs any additional energy. This term is significant even when the body is at rest.

8. General Relativity Explains Gravitation as Curvature

The earth produces in its surroundings a gravitational field, which acts on the stone and produces its motion of fall.

Gravitational Field. General relativity describes gravity not as a force but as a curvature of space-time caused by mass and energy. Objects move along geodesics, which are the shortest paths in this curved space-time.

Equivalence Principle. The equivalence principle states that the effects of gravity are indistinguishable from the effects of acceleration. This is illustrated by the thought experiment of an observer in an accelerating chest, who experiences the same effects as if they were in a gravitational field.

Inertial and Gravitational Mass. The equality of inertial and gravitational mass is a consequence of the equivalence principle. This law implies that the same property of a body manifests itself as inertia or as weight, depending on the circumstances.

9. Gravity's Influence on Light Reveals Space-Time

From this we conclude, that, in general, rays of light are propagated curvilinearly in gravitational fields.

Curvature of Light. General relativity predicts that light rays are bent by gravitational fields. This effect is most noticeable when light passes close to massive objects like the sun.

Experimental Verification. The bending of light was confirmed during a solar eclipse in 1919. Stars near the sun appeared to be displaced from their usual positions, as predicted by the theory.

Variable Velocity of Light. The curvature of light rays implies that the velocity of light varies with position in a gravitational field. This challenges the constancy of the speed of light assumed in the special theory of relativity.

10. The Universe's Structure is Tied to Matter's Density

According to the general theory of relativity, the geometrical properties of space are not independent, but they are determined by matter.

Matter and Geometry. The general theory of relativity links the geometry of space to the distribution of matter. The presence of matter curves space-time, influencing the motion of objects and the propagation of light.

Non-Euclidean Universe. The influence of matter on space-time excludes the possibility of exact Euclidean geometry in the universe. The universe may be quasi-Euclidean, but calculations suggest that the average density of matter would necessarily be zero in such a case.

Spherical Universe. If the universe has a non-zero average density of matter, it must be spherical or elliptical. The theory provides a connection between the space-expanse of the universe and the average density of matter within it.

11. Space-Time is a Construct Shaped by Experience

Physical objects are not in space, but these objects are spatially extended. In this way the concept “empty space” loses its meaning.

Space-Time as a Construct. Space and time are not independent entities but constructs shaped by our experiences and observations. The concept of "empty space" loses its meaning, as physical objects are spatially extended.

Objective Reality. The objective world is composed of events localized in space and time. Physical descriptions involve statements about the space-time coincidence of two events.

Gaussian Coordinates. The description of the time-space continuum using Gaussian coordinates replaces the description with a body of reference. This approach is not tied down to the Euclidean character of the continuum.

Last updated: May 30, 2025