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Motion of an object with constant proper acceleration in special relativity
Hyperbolic motion is the motion of an object with constant proper acceleration in special relativity. It is called hyperbolic motion because the equation
Hyperbolic motion (relativity)
Hyperbolic_motion_(relativity)
Isometric automorphisms of a hyperbolic space
In geometry, hyperbolic motions are isometric automorphisms of a hyperbolic space. Under composition of mappings, the hyperbolic motions form a continuous
Hyperbolic_motion
Measured time difference as explained by relativity theory
be expressed as a logarithmic function or, equivalently, as an inverse hyperbolic function: τ ( t ) = c g ln ( g t c + 1 + ( g t c ) 2 ) = c g arsinh
Time_dilation
Tool from special relativity
spacetime. In special relativity, a uniform acceleration results in hyperbolic motion, for which a uniformly accelerating frame of reference in which it
Rindler_coordinates
Geometric mean and hyperbolic angle as coordinates in quadrant I
In mathematics, hyperbolic coordinates are a method of locating points in quadrant I of the Cartesian plane { ( x , y ) : x > 0 , y > 0 } = Q {\displaystyle
Hyperbolic_coordinates
Recoil force on accelerating charged particle
The ALD equations are known to be zero for constant acceleration or hyperbolic motion in Minkowski spacetime diagram. The subject of whether in such condition
Abraham–Lorentz_force
Transformation of a geometric space preserving structure
establish the idea of distance in hyperbolic geometry when he wrote Elements of Non-Euclidean Geometry. He explains: By a motion or displacement in the general
Motion_(geometry)
Velocity differential over time, as described in Minkowski spacetime
Well known special cases are hyperbolic motion for constant longitudinal proper acceleration or uniform circular motion. Eventually, it is also possible
Acceleration (special relativity)
Acceleration_(special_relativity)
Concept in astrodynamics
the eccentricity. However, with a hyperbolic orbit other parameters may be more useful in understanding a body's motion. The following table lists the main
Hyperbolic_trajectory
Thought experiment in special relativity
acceleration indicated by a comoving accelerometer. This leads to hyperbolic motion, in which the observer continuously changes momentary inertial frames
Bell's_spaceship_paradox
Coordinates system in an accelerating, "at rest" setting
such as three-acceleration, four-acceleration, proper acceleration, hyperbolic motion etc. are defined and related to each other.) A fundamental property
Proper reference frame (flat spacetime)
Proper_reference_frame_(flat_spacetime)
Concept in special relativity, governing a body's dynamics at high speeds
description of the case of constant proper acceleration which he called hyperbolic motion. When subsequent authors such as Paul Ehrenfest (1909) tried to incorporate
Born_rigidity
Type of non-Euclidean geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate
Hyperbolic_geometry
Special class of linear fractional transformations
have been used to study the force field of an electric charge in hyperbolic motion. The inversion can also be taken to be multiplicative inversion of
Special conformal transformation
Special_conformal_transformation
Hypothetical travel between stars or planetary systems
in that field, undergoing hyperbolic motion. As part of this, distances between objects in the direction of the ship's motion will gradually contract until
Interstellar_travel
Change in the position of an object
Velocity is then interpreted as rapidity, the hyperbolic angle φ {\displaystyle \varphi } for which the hyperbolic tangent function tanh φ = v ÷ c {\displaystyle
Motion
Kinematic prediction of quantum field theory for an accelerating observer
out a hyperbola in Minkowski space, therefore this type of motion is called hyperbolic motion. The coordinate ρ {\displaystyle \rho } is related to the
Unruh_effect
case spiralling outwards (2), the hyperbolic case (4) is a mixed scenario which depends on the direction of motion. Given a manifold that splits into
Smooth_dynamics
Astronomical object not orbiting the Sun
weakly hyperbolic and will not be of interstellar origin. The planets and most satellites of the Solar System revolve in an almost circular motion – called
Hyperbolic_asteroid
Four-vector that is analogous to classical acceleration
constant four-acceleration is a Minkowski-circle i.e. hyperbola (see hyperbolic motion) The scalar product of a particle's four-velocity and its four-acceleration
Four-acceleration
Argument of the hyperbolic functions
In geometry, hyperbolic angle is a real number determined by the area of the corresponding hyperbolic sector of xy = 1 in Quadrant I of the Cartesian plane
Hyperbolic_angle
Type of spacecraft
} The time in the rest frame relates to the proper time by the hyperbolic motion equation: t ′ = c a sinh ( a t c ) . {\displaystyle t'={\frac {c}{a}}\sinh
Relativistic_rocket
Measure in astrodynamics
{\displaystyle C_{3}=0} A spacecraft that is leaving the central body on a hyperbolic trajectory has more than enough energy to escape: C 3 = μ | a | > 0 {\displaystyle
Characteristic_energy
Two geometries based on axioms closely related to those specifying Euclidean geometry
forms associated with metric geometry. In the former case, one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries
Non-Euclidean_geometry
Upper-half plane model of hyperbolic non-Euclidean geometry
way of representing the hyperbolic plane using points in the familiar Euclidean plane. Specifically, each point in the hyperbolic plane is represented using
Poincaré_half-plane_model
Laws describing planetary orbits
In astronomy, Kepler's laws of planetary motion give good approximations for the orbits of planets around the Sun. They were published by Johannes Kepler
Kepler's laws of planetary motion
Kepler's_laws_of_planetary_motion
Motion of launched objects due to gravity
In physics, projectile motion describes the motion of an object that is launched into the air and moves under the influence of gravity alone, with air
Projectile_motion
Field of classical mechanics concerned with the motion of spacecraft
Leonhard Euler in 1744, whose work was in turn generalised to elliptical and hyperbolic orbits by Johann Lambert in 1761–1777. Another milestone in orbit determination
Orbital_mechanics
Linear map that preserves areas
is. For this reason it is natural to think of the squeeze mapping as a hyperbolic rotation, as did Émile Borel in 1914, by analogy with circular rotations
Squeeze_mapping
Development of linear transformations forming the Lorentz group
hyperbolic motion and its transformation, Gustav Herglotz (1909–10) classified the one-parameter Lorentz transformations as loxodromic, hyperbolic, parabolic
History of Lorentz transformations
History_of_Lorentz_transformations
Curved path of an object around a point
comets, meteoroids, and even space debris. A comet in a parabolic or hyperbolic orbit about a barycenter is not gravitationally bound to the star and
Orbit
Acceleration (special relativity), Proper reference frame (flat spacetime), Hyperbolic motion, Rindler coordinates, Born coordinates). Paradoxes relying on insufficient
Criticism of the theory of relativity
Criticism_of_the_theory_of_relativity
Austrian physicist (1886–1965)
Gustav Herglotz in 1909, in particular for hyperbolic motion (Kottler-Møller metric) and uniform circular motion. In (1916a), he also discussed the conformal
Friedrich_Kottler
Geometric figure
t,\sinh t).} This parameter t is the hyperbolic angle, which is the argument of the hyperbolic functions. One finds an early expression of the
Unit_hyperbola
Parameters that define a specific orbit
elliptical orbits, undefined for parabolic trajectories, and negative for hyperbolic trajectories, which can hinder its usability when working with different
Orbital_elements
Transport of a substance by bulk motion
cycle. The advection equation is a first-order hyperbolic partial differential equation that governs the motion of a conserved scalar field as it is advected
Advection
(1909) and Sommerfeld (1910), with Born introducing the expression "hyperbolic motion". He noted that uniform acceleration can be used as an approximation
History_of_special_relativity
Amount by which an orbit deviates from a perfect circle
Circular orbit: e = 0 Elliptic orbit: 0 < e < 1 Parabolic trajectory: e = 1 Hyperbolic trajectory: e > 1 The eccentricity e is given by e = 1 + 2 E L 2
Orbital_eccentricity
Mathematical equation describing the motion of a rocket
or ideal rocket equation is a mathematical equation that describes the motion of vehicles that follow the basic principle of a rocket: a device that can
Tsiolkovsky_rocket_equation
Measure of relativistic velocity
Mathematically, rapidity can be defined as the hyperbolic angle that differentiates two frames of reference in relative motion, each frame being associated with distance
Rapidity
Astrodynamic equation
(parabolic or hyperbolic orbit): the motion is either away from the central body, or towards it. if the energy is negative: the motion can be first away
Orbit_equation
Quaternion of norm 1 (unit quaternion)
saw the modelling power of hyperbolic versors operating on the split-complex number plane, and in 1891 he introduced hyperbolic quaternions to extend the
Versor
American theoretical physicist
reference under constant acceleration is sometimes described as in hyperbolic motion. In 1936 Page and Adams presented in Physical Review their analysis
Leigh_Page
Orbital mechanics term
orbits ( 0 ≤ e < 1 {\displaystyle 0\leq e<1} ). The hyperbolic Kepler equation is used for hyperbolic trajectories ( e > 1 {\displaystyle e>1} ). The radial
Kepler's_equation
Comets that may not be orbiting the Sun
This is a list of parabolic and hyperbolic comets in the Solar System. Many of these comets may come from the Oort cloud, or perhaps even have interstellar
List of parabolic and hyperbolic comets
List_of_parabolic_and_hyperbolic_comets
Branch of astronomy
are parabolic and hyperbolic. This is useful for calculating the behaviour of planets and comets and such (parabolic and hyperbolic orbits are conic section
Celestial_mechanics
Chronology of development from 1905
(1943) who discussed a case involving constant proper acceleration (hyperbolic motion), describing the accompanied accelerated frame and its homogeneous
History_of_the_twin_paradox
Iranian mathematician (1977–2017)
professor of mathematics at Stanford University. Her research focused on hyperbolic geometry, dynamical systems, complex analysis, and topology. In 2014,
Maryam_Mirzakhani
Continuous probability distribution
function of the logistic distribution is also a scaled version of the hyperbolic tangent. F ( x ; μ , s ) = 1 1 + e − ( x − μ ) / s = 1 2 + 1 2 tanh
Logistic_distribution
Continuous probability distribution
The generalised hyperbolic distribution (GH) is a continuous probability distribution defined as the normal variance-mean mixture where the mixing distribution
Generalised hyperbolic distribution
Generalised_hyperbolic_distribution
Type of orbit
an unbound orbit that is exactly on the border between elliptical and hyperbolic. When moving away from the source it is called an escape orbit, otherwise
Parabolic_trajectory
Fractal named after mathematician Benoit Mandelbrot
known as density of hyperbolicity, is one of the most important open problems in complex dynamics. Hypothetical non-hyperbolic components of the Mandelbrot
Mandelbrot_set
German-born theoretical physicist (1879–1955)
them, he outlined a theory of the photoelectric effect, explained Brownian motion, introduced his special theory of relativity, and demonstrated that if the
Albert_Einstein
Trajectory of Earth around the Sun
the size of the orbit). As seen from Earth, the planet's orbital prograde motion makes the Sun appear to move with respect to other stars at a rate of about
Earth's_orbit
Propulsive maneuver used to arrive at the Moon
Finally, the spacecraft enters the Moon's sphere of influence, making a hyperbolic lunar swingby. In some cases it is possible to design a TLI to target
Trans-lunar_injection
Ratio in relativity
or "map time". For unidirectional motion, each of these is also simply related to a traveling object's hyperbolic velocity angle or rapidity η by η =
Proper_velocity
Various meanings of the terms
In special relativity, a time axis determined by a rapidity of motion is hyperbolic-orthogonal to a space axis of simultaneous events, also determined
Orthogonality
Paths of particles in the Schwarzschild solution to Einstein's field equations
geodesics describe the motion of test particles in the gravitational field of a central fixed mass M , {\textstyle M,} that is, motion in the Schwarzschild
Schwarzschild_geodesics
Moment in time used as a reference point in astronomy
quantities specified in this way is to calculate other relevant parameters of motion, in order to predict future positions and velocities. The applied tools
Epoch_(astronomy)
Dutch graphic artist (1898–1972)
reflection, symmetry, perspective, truncated and stellated polyhedra, hyperbolic geometry, and tessellations. Although Escher believed he had no mathematical
M._C._Escher
Celestial orbit whose trajectory is a conic section in the orbital plane
Two-body problem Kepler problem Kepler's laws of planetary motion Elliptic orbit Hyperbolic trajectory Parabolic trajectory Radial trajectory Orbit modeling
Kepler_orbit
Angular speed required for a body to complete one orbit
In orbital mechanics, mean motion (represented by n) is the angular speed required for a body to complete one orbit, assuming constant speed in a circular
Mean_motion
Archimedean spiral Cornu spiral Cotes' spiral Fermat's spiral Galileo's spiral Hyperbolic spiral Lituus Logarithmic spiral Nielsen's spiral Golden spiral Syntractrix
List_of_mathematical_shapes
Parametrizes complex structures on a surface
{\displaystyle S} to itself. It can be viewed as a moduli space for marked hyperbolic structure on the surface, and this endows it with a natural topology for
Teichmüller_space
Specifies the orbit of an object in space
ray from (0, 0) to (x, y), having the same sign as y. For parabolic and hyperbolic trajectories the mean anomaly is not defined, because they don't have
Mean_anomaly
Relation between sides of a right triangle
where cosh is the hyperbolic cosine. This formula is a special form of the hyperbolic law of cosines that applies to all hyperbolic triangles: cosh
Pythagorean_theorem
Navigation and surveillance technique
TOAs are multiple and known. When MLAT is used for navigation (as in hyperbolic navigation), the waves are transmitted by the stations and received by
Pseudo-range_multilateration
Abstract coordinate system
ISBN 978-0-8218-3900-3. Arlan Ramsay; Robert D. Richtmyer (1995). Introduction to Hyperbolic Geometry. Springer. p. 11. ISBN 0-387-94339-0. geometry axiom coordinate
Frame_of_reference
Type of curve in hyperbolic geometry
In hyperbolic geometry, a hypercycle, hypercircle or equidistant curve is a curve whose points have the same orthogonal distance from a given straight
Hypercycle_(geometry)
Framework of distances and directions
type of geometry that does not include the parallel postulate, called hyperbolic geometry. In this geometry, an infinite number of parallel lines pass
Space
Motion problem in classical mechanics
classical mechanics, the two-body problem is used to calculate and predict the motion of two massive bodies that are orbiting each other in space. The problem
Two-body_problem
Concept in celestial mechanics
parabolic trajectories rv2 is constant and equal to 2μ. For elliptic and hyperbolic orbits magnitude of μ = 2 times the magnitude of a times the magnitude
Standard gravitational parameter
Standard_gravitational_parameter
Transfer manoeuvre between two orbits
orbit Hyperbolic orbit Radial orbit Decaying orbit Equations Dynamical friction Escape velocity Kepler's equation Kepler's laws of planetary motion Orbital
Hohmann_transfer_orbit
Problem in physics and celestial mechanics
determined first, then the equation of motion resolved. This differential equation has elliptic, or parabolic or hyperbolic solutions. It is incorrect to think
N-body_problem
Theory of interwoven space and time by Albert Einstein
the sides of the cube that are perpendicular to the direction of motion appear hyperbolic in shape. The cube is not rotated. Rather, light from the rear
Special_relativity
Plane curve: conic section
cone Hyperbolic cylinder Hyperbolic paraboloid Hyperboloid of one sheet Hyperboloid of two sheets Elliptic cone Hyperbolic cylinder Hyperbolic paraboloid
Hyperbola
Concept in celestial mechanics
than the escape speed at its current distance. In contrast if it is on a hyperbolic trajectory its speed will always be higher than the escape speed at its
Escape_velocity
Topics referred to by the same term
in hyperbolic geometry, the angle at one vertex of a right hyperbolic triangle that has two hyperparallel sides Axial parallelism, a type of motion characteristic
Parallelism
Interstellar object that passed near Earth in 2017
October. A two-week observation arc had verified a strongly hyperbolic trajectory. It has a hyperbolic excess velocity (velocity at infinity, v ∞ {\displaystyle
1I/ʻOumuamua
Kepler orbit with an eccentricity of less than one
\!} is the length of the semi-major axis. The velocity equation for a hyperbolic trajectory has either ( + 1 a ) {\displaystyle (+{1 \over {a}})} , or
Elliptic_orbit
Physics problem related to laws of motion and gravity
27030. Hulkower, Neal D. (1980). "Central Configurations and Hyperbolic-Elliptic Motion in the Three-Body Problem". Celestial Mechanics. 21 (1): 37–41
Three-body_problem
Orbit around Earth between 160 and 2000 km
Archived from the original on 26 June 2018. Retrieved 13 July 2018. LEO: Mean Motion > 11.25 & Eccentricity < 0.25 Muciaccia, Andrea (2021). Fragmentations in
Low_Earth_orbit
Periodic, three-dimensional orbit
first computed in 1998 by M.A. Andreu, who introduced a new model for the motion of a spacecraft in the Earth-Moon-Sun system, which was called Quasi-Bicircular
Halo_orbit
Time an astronomical object takes to complete one orbit around another object
bodies, G is the gravitational constant. In a parabolic or hyperbolic trajectory, the motion is not periodic, and the duration of the full trajectory is
Orbital_period
HyperPhysics Hyperbolic coordinates Hyperbolic equilibrium point Hyperbolic motion (relativity) Hyperbolic orthogonality Hyperbolic quaternion Hyperbolic set Hyperbolic
Index_of_physics_articles_(H)
Concept in gravitational orbital mechanics
the work is being done. For any Keplerian orbit (elliptic, parabolic, hyperbolic, or radial), the vis-viva equation is as follows: v 2 = G M ( 2 r − 1
Vis-viva_equation
Spacecraft end-of-life orbit
orbits Types General Box Circular Elliptical / Highly elliptical Horseshoe Hyperbolic trajectory Inclined / Non-inclined Kepler Lagrange point Osculating Parabolic
Graveyard_orbit
Distinguished surfaces of dynamic trajectories
flow is hyperbolic (saddle-type) in any frame. Since Newton's equation for particle motion and the Navier–Stokes equations for fluid motion are well
Lagrangian_coherent_structure
Motion of a certain space that preserves at least one point
geometry. Any rotation is a motion of a certain space that preserves at least one point. It can describe, for example, the motion of a rigid body around a
Rotation_(mathematics)
Vector quantity in celestial mechanics
connects the two bodies. The proof starts with the two body equation of motion, derived from Newton's law of universal gravitation: r ¨ + G m 1 r 2 r r
Specific_angular_momentum
Method to calculate trajectory calculations for spacecraft
example, when departing a planet, the spacecraft follows a hyperbolic escape trajectory with hyperbolic excess velocity v ∞ {\displaystyle \mathbf {v} _{\infty
Patched_conic_approximation
cases: the bodies move away from each other or towards each other. Radial hyperbolic trajectory: a non-periodic orbit where the relative speed of the two objects
Radial_trajectory
Local and global geometry of the universe
than 180°; such 3-dimensional space is locally modeled by a region of a hyperbolic space H3. Curved geometries are in the domain of non-Euclidean geometry
Shape_of_the_universe
the body's orbit inclination were zero. It is usually denoted ϖ. For the motion of a planet around the Sun, this position is called longitude of perihelion
Longitude_of_periapsis
American retired actress (born 1924)
acclaim in TV productions were of such a high level that "one slightly hyperbolic primordial TV critic dubbed her 'the Helen Hayes of television.'" Saint
Eva_Marie_Saint
Type of spacecraft maneuver
{{2V_{\text{esc}}}{\Delta v}}}.} Similar effects happen in closed and hyperbolic orbits. If the vehicle travels at velocity v at the start of a burn that
Oberth_effect
Circular orbit above Earth's Equator and following the direction of Earth's rotation
gravity, and the flattening of the Earth at its poles causes a precession motion of the orbital plane of any geostationary object, with an orbital period
Geostationary_orbit
Speed at which a body orbits around the barycenter of a system
periapsis their distance from the focus increases without bound. See radial hyperbolic trajectory If the total energy is zero, (Ek = Ep): the orbit is a parabola
Orbital_speed
Interpolation method in computer graphics
algorithm for hyperbolic interpolation that is used in GPUs as a perspective correct alternative to linear interpolation. Both the linear and hyperbolic variants
Gouraud_shading
In finance, economics, and decision theory, hyperbolic absolute risk aversion (HARA) refers to a type of risk aversion that is particularly convenient
Hyperbolic absolute risk aversion
Hyperbolic_absolute_risk_aversion
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