Spacetime

Figure 1-1. Each location in spacetime is marked by four numbers defined by a frame of reference: the position in space, and the time (which can be visualized as the reading of a clock located at each position in space). The 'observer' synchronizes the clocks according to their own reference frame.
Figure 1–4. Hand-colored transparency presented by Minkowski in his 1908 Raum und Zeit lecture
Figure 2-2. Galilean diagram of two frames of reference in standard configuration
Figure 2–3. (a) Galilean diagram of two frames of reference in standard configuration, (b) spacetime diagram of two frames of reference, (c) spacetime diagram showing the path of a reflected light pulse
Figure 2–4. The light cone centered on an event divides the rest of spacetime into the future, the past, and "elsewhere"
Figure 2–5. Light cone in 2D space plus a time dimension
Figure 2–6. Animation illustrating relativity of simultaneity
Figure 2–7. (a) Families of invariant hyperbolae, (b) Hyperboloids of two sheets and one sheet
Figure 2–8. The invariant hyperbola comprises the points that can be reached from the origin in a fixed proper time by clocks traveling at different speeds
Figure 2–9. In this spacetime diagram, the 1 m length of the moving rod, as measured in the primed frame, is the foreshortened distance OC when projected onto the unprimed frame.
Figure 2-11. Spacetime explanation of the twin paradox
Figure 3–2. Relativistic composition of velocities
Figure 3-3. Spacetime diagrams illustrating time dilation and length contraction
Figure 3–5. Derivation of Lorentz Transformation
Figure 3–7. Transverse Doppler effect scenarios
Figure 3–8. Relativistic spacetime momentum vector
Figure 3–9. Energy and momentum of light in different inertial frames
Figure 3-10. Relativistic conservation of momentum
Figure 4–2. Plot of the three basic Hyperbolic functions: hyperbolic sine ([[:File:Hyperbolic Sine.svg|sinh]]), hyperbolic cosine ([[:File:Hyperbolic Cosine.svg|cosh]]) and hyperbolic tangent ([[:File:Hyperbolic Tangent.svg|tanh]]). Sinh is red, cosh is blue and tanh is green.
Figure 4-4. Dewan–Beran–Bell spaceship paradox
Figure 4–5. The curved lines represent the world lines of two observers A and B who accelerate in the same direction with the same constant magnitude acceleration. At A' and B', the observers stop accelerating. The dashed lines are lines of simultaneity for either observer before acceleration begins and after acceleration stops.
Figure 4–6. Accelerated relativistic observer with horizon. Another well-drawn illustration of the same topic may be viewed [[:File:ConstantAcceleration02.jpg|here]].
Figure 5–1. Tidal effects.
Figure 5–2. Equivalence principle
Figure 5–3. Einstein's argument suggesting gravitational redshift
Figure 5-5. Contravariant components of the stress–energy tensor
Figure 5–7. Origin of gravitomagnetism
Figure 5–9. (A) Cavendish experiment, (B) Kreuzer experiment
Figure 5-11. Gravity Probe B confirmed the existence of gravitomagnetism

Mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold.

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Theory of relativity

The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively.

A diagram of the Michelson–Morley experiment
The Kennedy–Thorndike experiment shown with interference fringes.

It introduced concepts including 4-dimensional spacetime as a unified entity of space and time, relativity of simultaneity, kinematic and gravitational time dilation, and length contraction.

Lorentz transformation

Lorentz boost of an electric charge, the charge is at rest in one frame or the other.

In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former.

Dimension

[[File:Dimension levels.svg|thumb | 236px | The first four spatial dimensions, represented in a two-dimensional picture. 1. Two points can be connected to create a line segment.

From left to right: the square, the cube and the tesseract. The two-dimensional (2D) square is bounded by one-dimensional (1D) lines; the three-dimensional (3D) cube by two-dimensional areas; and the four-dimensional (4D) tesseract by three-dimensional volumes. For display on a two-dimensional surface such as a screen, the 3D cube and 4D tesseract require projection.

The four dimensions (4D) of spacetime consist of events that are not absolutely defined spatially and temporally, but rather are known relative to the motion of an observer.

Time

Continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future.

The flow of sand in an hourglass can be used to measure the passage of time. It also concretely represents the present as being between the past and the future.
Horizontal sundial in Taganrog
An old kitchen clock
A contemporary quartz watch, 2007
Chip-scale atomic clocks, such as this one unveiled in 2004, are expected to greatly improve GPS location.
Scale of time in Jain texts shown logarithmically
Time's mortal aspect is personified in this bronze statue by Charles van der Stappen.
Two-dimensional space depicted in three-dimensional spacetime. The past and future light cones are absolute, the "present" is a relative concept different for observers in relative motion.
Relativity of simultaneity: Event B is simultaneous with A in the green reference frame, but it occurred before in the blue frame, and occurs later in the red frame.
Views of spacetime along the world line of a rapidly accelerating observer in a relativistic universe. The events ("dots") that pass the two diagonal lines in the bottom half of the image (the past light cone of the observer in the origin) are the events visible to the observer.
Philosopher and psychologist William James

The physical nature of time is addressed by general relativity with respect to events in spacetime.

Minkowski space

Hermann Minkowski (1864–1909) found that the theory of special relativity, introduced by his former student Albert Einstein, could be best understood as a four-dimensional space, since known as the Minkowski spacetime.
Subdivision of Minkowski spacetime with respect to an event in four disjoint sets. The light cone, the absolute future, the absolute past, and elsewhere. The terminology is from.
Linear functionals (1-forms) α, β and their sum σ and vectors u, v, w, in 3d Euclidean space. The number of (1-form) hyperplanes intersected by a vector equals the inner product.
Red circular arc is geodesic in Poincaré disk model; it projects to the brown geodesic on the green hyperboloid.

In mathematical physics, Minkowski space (or Minkowski spacetime) is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inertial frame of reference in which they are recorded.

Special relativity

1) The laws of physics are invariant (that is, identical) in all inertial frames of reference (that is, frames of reference with no acceleration).

Albert Einstein around 1905, the year his "Annus Mirabilis papers" were published. These included Zur Elektrodynamik bewegter Körper, the paper founding special relativity.
Figure 2–1. The primed system is in motion relative to the unprimed system with constant velocity v only along the x-axis, from the perspective of an observer stationary in the unprimed system. By the principle of relativity, an observer stationary in the primed system will view a likewise construction except that the velocity they record will be −v. The changing of the speed of propagation of interaction from infinite in non-relativistic mechanics to a finite value will require a modification of the transformation equations mapping events in one frame to another.
Figure 4–1. The three events (A, B, C) are simultaneous in the reference frame of some observer O. In a reference frame moving at v = 0.3c, as measured by O, the events occur in the order C, B, A. In a reference frame moving at v = −0.5c with respect to O, the events occur in the order A, B, C. The white lines, the lines of simultaneity, move from the past to the future in the respective frames (green coordinate axes), highlighting events residing on them. They are the locus of all events occurring at the same time in the respective frame. The gray area is the light cone with respect to the origin of all considered frames.
Figure 4–3. Light cone
Figure 5–1. Highly simplified diagram of Fizeau's 1851 experiment.
Figure 5–2. Illustration of stellar aberration
Figure 5–3. Transverse Doppler effect for two scenarios: (a) receiver moving in a circle around the source; (b) source moving in a circle around the receiver.
Figure 5–4. Comparison of the measured length contraction of a cube versus its visual appearance.
Figure 5-5. Galaxy M87 streams out a black-hole-powered jet of electrons and other sub-atomic particles traveling at nearly the speed of light.
Figure 10–1. Orthogonality and rotation of coordinate systems compared between left: Euclidean space through circular angle φ, right: in Minkowski spacetime through hyperbolic angle φ (red lines labelled c denote the worldlines of a light signal, a vector is orthogonal to itself if it lies on this line).
Figure 10–2. Three-dimensional dual-cone.

Rather than an invariant time interval between two events, there is an invariant spacetime interval.

General relativity

Geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics.

According to general relativity, objects in a gravitational field behave similarly to objects within an accelerating enclosure. For example, an observer will see a ball fall the same way in a rocket (left) as it does on Earth (right), provided that the acceleration of the rocket is equal to 9.8 m/s2 (the acceleration due to gravity at the surface of the Earth).
Light cone
Schematic representation of the gravitational redshift of a light wave escaping from the surface of a massive body
Deflection of light (sent out from the location shown in blue) near a compact body (shown in gray)
Ring of test particles deformed by a passing (linearized, amplified for better visibility) gravitational wave
Newtonian (red) vs. Einsteinian orbit (blue) of a lone planet orbiting a star. The influence of other planets is ignored.
Orbital decay for PSR 1913+16: time shift (in s), tracked over 30 years (2006).
Orbital decay for PSR J0737−3039: time shift (in s), tracked over 16 years (2021).
Einstein cross: four images of the same astronomical object, produced by a gravitational lens
Artist's impression of the space-borne gravitational wave detector LISA
Simulation based on the equations of general relativity: a star collapsing to form a black hole while emitting gravitational waves
This blue horseshoe is a distant galaxy that has been magnified and warped into a nearly complete ring by the strong gravitational pull of the massive foreground luminous red galaxy.
Penrose–Carter diagram of an infinite Minkowski universe
The ergosphere of a rotating black hole, which plays a key role when it comes to extracting energy from such a black hole
Projection of a Calabi–Yau manifold, one of the ways of compactifying the extra dimensions posited by string theory
Simple spin network of the type used in loop quantum gravity
Observation of gravitational waves from binary black hole merger GW150914

General relativity generalizes special relativity and refines Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time or four-dimensional spacetime.

Manifold

Topological space that locally resembles Euclidean space near each point.

The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface.
The surface of the Earth requires (at least) two charts to include every point. Here the globe is decomposed into charts around the North and South Poles.
Figure 1: The four charts each map part of the circle to an open interval, and together cover the whole circle.
Four manifolds from algebraic curves: circles, parabola,  hyperbola,  cubic.
The chart maps the part of the sphere with positive z coordinate to a disc.
A finite cylinder is a manifold with boundary.
Möbius strip
The Klein bottle immersed in three-dimensional space
A Morin surface, an immersion used in sphere eversion

Symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics, while four-dimensional Lorentzian manifolds model spacetime in general relativity.

Physics

Various examples of physical phenomena
Ancient Egyptian astronomy is evident in monuments like the ceiling of Senemut's tomb from the Eighteenth Dynasty of Egypt.
Ibn al-Haytham (c. 965–c. 1040), Book of Optics Book I, [6.85], [6.86]. Book II, [3.80] describes his camera obscura experiments.
The basic way a pinhole camera works
Galileo Galilei showed a modern appreciation for the proper relationship between mathematics, theoretical physics, and experimental physics.
Sir Isaac Newton (1643–1727), whose laws of motion and universal gravitation were major milestones in classical physics
Max Planck (1858–1947), the originator of the theory of quantum mechanics
Albert Einstein (1879–1955), whose work on the photoelectric effect and the theory of relativity led to a revolution in 20th century physics
The basic domains of physics
Solvay Conference of 1927, with prominent physicists such as Albert Einstein, Werner Heisenberg, Max Planck, Hendrik Lorentz, Niels Bohr, Marie Curie, Erwin Schrödinger and Paul Dirac
This parabola-shaped lava flow illustrates the application of mathematics in physics—in this case, Galileo's law of falling bodies.
Mathematics and ontology are used in physics. Physics is used in chemistry and cosmology.
The distinction between mathematics and physics is clear-cut, but not always obvious, especially in mathematical physics.
Classical physics implemented in an acoustic engineering model of sound reflecting from an acoustic diffuser
Archimedes' screw, a simple machine for lifting
Experiment using a laser
The astronaut and Earth are both in free fall.
Lightning is an electric current.
Physics involves modeling the natural world with theory, usually quantitative. Here, the path of a particle is modeled with the mathematics of calculus to explain its behavior: the purview of the branch of physics known as mechanics.
A simulated event in the CMS detector of the Large Hadron Collider, featuring a possible appearance of the Higgs boson.
Velocity-distribution data of a gas of rubidium atoms, confirming the discovery of a new phase of matter, the Bose–Einstein condensate
The deepest visible-light image of the universe, the Hubble Ultra-Deep Field
Feynman diagram signed by R. P. Feynman.
A typical phenomenon described by physics: a magnet levitating above a superconductor demonstrates the Meissner effect.

Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force.

Motion

Phenomenon in which an object changes its position with respect to time.

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Motion applies to various physical systems: objects, bodies, matter particles, matter fields, radiation, radiation fields, radiation particles, curvature, and space-time.