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The geodetic effect (also known as geodetic precession, de Sitter precession or de Sitter effect) represents the effect of the curvature of
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
, predicted by
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, on a vector carried along with an orbiting body. For example, the vector could be the angular momentum of a gyroscope orbiting the Earth, as carried out by the
Gravity Probe B
Gravity Probe B (GP-B) was a satellite-based experiment to test two unverified predictions of general relativity: the geodetic effect and frame-dragging. This was to be accomplished by measuring, very precisely, tiny changes in the direction of ...
experiment. The geodetic effect was first predicted by
Willem de Sitter in 1916, who provided relativistic corrections to the Earth–Moon system's motion. De Sitter's work was extended in 1918 by
Jan Schouten and in 1920 by
Adriaan Fokker
Adriaan Daniël Fokker (; 17 August 1887 – 24 September 1972) was a Dutch physicist. He worked in the fields of special relativity and statistical mechanics. He was the inventor of the Fokker organ, a 31-tone equal-tempered (31-TET) organ. ...
.
It can also be applied to a particular secular
precession
Precession is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the first Euler angle, whereas the third Euler angle defines the rotation itself. In othe ...
of astronomical orbits, equivalent to the rotation of the
Laplace–Runge–Lenz vector
In classical mechanics, the Laplace–Runge–Lenz (LRL) vector is a vector used chiefly to describe the shape and orientation of the orbit of one astronomical body around another, such as a binary star or a planet revolving around a star. For ...
.
The term geodetic effect has two slightly different meanings as the moving body may be spinning or non-spinning. Non-spinning bodies move in
geodesic
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
s, whereas spinning bodies move in slightly different orbits.
The difference between de Sitter precession and
Lense–Thirring precession
In general relativity, Lense–Thirring precession or the Lense–Thirring effect (; named after Josef Lense and Hans Thirring) is a Theory of relativity, relativistic correction to the precession of a gyroscope near a large rotating mass such as ...
(frame dragging) is that the de Sitter effect is due simply to the presence of a central mass, whereas Lense–Thirring precession is due to the rotation of the central mass. The total precession is calculated by combining the de Sitter precession with the Lense–Thirring precession.
Experimental confirmation
The geodetic effect was verified to a precision of better than 0.5% percent by
Gravity Probe B
Gravity Probe B (GP-B) was a satellite-based experiment to test two unverified predictions of general relativity: the geodetic effect and frame-dragging. This was to be accomplished by measuring, very precisely, tiny changes in the direction of ...
, an experiment which measures the tilting of the spin axis of
gyroscope
A gyroscope (from Ancient Greek γῦρος ''gŷros'', "round" and σκοπέω ''skopéō'', "to look") is a device used for measuring or maintaining orientation and angular velocity. It is a spinning wheel or disc in which the axis of rota ...
s in orbit about the Earth. The first results were announced on April 14, 2007 at the meeting of the
American Physical Society
The American Physical Society (APS) is a not-for-profit membership organization of professionals in physics and related disciplines, comprising nearly fifty divisions, sections, and other units. Its mission is the advancement and diffusion of k ...
.
Formulae
To derive the precession, assume the system is in a rotating
Schwarzschild metric
In Einstein's theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild solution) is an
exact solution to the Einstein field equations that describes the gravitational field outside a spherical mass, on the assumpti ...
. The nonrotating metric is
:
where ''c'' = ''G'' = 1.
We introduce a rotating coordinate system, with an angular velocity
, such that a satellite in a circular orbit in the θ = π/2 plane remains at rest. This gives us
:
In this coordinate system, an observer at radial position ''r'' sees a vector positioned at ''r'' as rotating with angular frequency ω. This observer, however, sees a vector positioned at some other value of ''r'' as rotating at a different rate, due to relativistic time dilation. Transforming the Schwarzschild metric into the rotating frame, and assuming that
is a constant, we find
:
with
. For a body orbiting in the θ = π/2 plane, we will have β = 1, and the body's world-line will maintain constant spatial coordinates for all time. Now, the metric is in the
canonical form
In mathematics and computer science, a canonical, normal, or standard form of a mathematical object is a standard way of presenting that object as a mathematical expression. Often, it is one which provides the simplest representation of an obje ...
:
From this canonical form, we can easily determine the rotational rate of a gyroscope in proper time
:
where the last equality is true only for free falling observers for which
there is no acceleration, and thus
. This leads to
:
Solving this equation for ω yields
:
This is essentially
Kepler's law of periods, which happens to be relativistically exact when expressed in terms of the time coordinate ''t'' of this particular rotating coordinate system. In the rotating frame, the satellite remains at rest, but an observer aboard the satellite sees the gyroscope's angular momentum vector precessing at the rate ω. This observer also sees the distant stars as rotating, but they rotate at a slightly different rate due to time dilation. Let τ be the gyroscope's
proper time
In relativity, proper time (from Latin, meaning ''own time'') along a timelike world line is defined as the time as measured by a clock following that line. It is thus independent of coordinates, and is a Lorentz scalar. The proper time interval b ...
. Then
:
The −2''m''/''r'' term is interpreted as the gravitational time dilation, while the additional −''m''/''r'' is due to the rotation of this frame of reference. Let α' be the accumulated precession in the rotating frame. Since
, the precession over the course of one orbit, relative to the distant stars, is given by:
:
With a first-order
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
we find
:
Thomas precession
One can attempt to break down the de Sitter precession into a
kinematic
Kinematics is a subfield of physics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause them to move. Kinematics, as a fie ...
effect called
Thomas precession
In physics, the Thomas precession, named after Llewellyn Thomas, is a relativistic correction that applies to the spin of an elementary particle or the rotation of a macroscopic gyroscope and relates the angular velocity of the spin of a pa ...
combined with a geometric effect caused by gravitationally curved spacetime. At least one author does describe it this way, but others state that "The Thomas precession comes into play for a gyroscope on the surface of the Earth ..., but not for a gyroscope in a freely moving satellite." An objection to the former interpretation is that the Thomas precession required has the wrong sign. The
Fermi-Walker transport equation[Misner, Thorne, and Wheeler, Gravitation, p. 165, pp. 175-176, pp. 1117-1121] gives both the geodetic effect and Thomas precession and describes the transport of the spin 4-vector for accelerated motion in curved spacetime. The spin 4-vector is orthogonal to the velocity 4-vector. Fermi-Walker transport preserves this relation. If there is no acceleration, Fermi-Walker transport is just parallel transport along a geodesic and gives the spin precession due to the geodetic effect. For the acceleration due to uniform circular motion in flat Minkowski spacetime, Fermi Walker transport gives the Thomas precession.
See also
*
Frame-dragging
Frame-dragging is an effect on spacetime, predicted by Albert Einstein's general theory of relativity, that is due to non-static stationary distributions of mass–energy. A stationary field is one that is in a steady state, but the masses cau ...
*
Geodesics in general relativity
In general relativity, a geodesic generalizes the notion of a "straight line" to curved spacetime. Importantly, the world line of a particle free from all external, non-gravitational forces is a particular type of geodesic. In other words, a fre ...
*
Gravity well
The Hill sphere of an astronomical body is the region in which it dominates the attraction of satellites. To be retained by a planet, a moon must have an orbit that lies within the planet's Hill sphere. That moon would, in turn, have a Hill sp ...
*
Timeline of gravitational physics and relativity
The following is a timeline of gravitational physics and general relativity.
Before 1500
* 3rd century BC - Aristarchus of Samos proposes heliocentric model, measures the distance to the Moon and its size
1500s
* 1543 – Nicolaus Copernicus pla ...
Notes
References
*
Wolfgang Rindler
Wolfgang Rindler (18 May 1924 – 8 February 2019) was a physicist working in the field of general relativity where he is known for introducing the term "event horizon", Rindler coordinates, and (in collaboration with Roger Penrose) for the use of ...
(2006) Relativity: special, general, and cosmological (2nd Ed.), Oxford University Press,
External links
* Gravity Probe B websites a
NASAan
Stanford University
{{Relativity
General relativity