Born rigidity is a concept in
special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The laws o ...
. It is one answer to the question of what, in special relativity, corresponds to the
rigid body
In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external force ...
of non-relativistic
classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...
.
The concept was introduced by
Max Born
Max Born (; 11 December 1882 – 5 January 1970) was a German physicist and mathematician who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics and supervised the work of a n ...
(1909),
[Born (1909b)] who gave a detailed description of the case of constant
proper acceleration
In relativity theory, proper acceleration is the physical acceleration (i.e., measurable acceleration as by an accelerometer) experienced by an object. It is thus acceleration relative to a free-fall, or inertial, observer who is momentarily at ...
which he called
hyperbolic motion
In geometry, hyperbolic motions are isometric automorphisms of a hyperbolic space. Under composition of mappings, the hyperbolic motions form a continuous group. This group is said to characterize the hyperbolic space. Such an approach to geomet ...
. When subsequent authors such as
Paul Ehrenfest
Paul Ehrenfest (18 January 1880 – 25 September 1933) was an Austrian theoretical physicist, who made major contributions to the field of statistical mechanics and its relations with quantum mechanics, including the theory of phase transition an ...
(1909) tried to incorporate rotational motions as well, it became clear that Born rigidity is a very restrictive sense of rigidity, leading to the Herglotz–Noether theorem, according to which there are severe restrictions on rotational Born rigid motions. It was formulated by
Gustav Herglotz
Gustav Herglotz (2 February 1881 – 22 March 1953) was a German Bohemian physicist best known for his works on the theory of relativity and seismology.
Biography
Gustav Ferdinand Joseph Wenzel Herglotz was born in Volary num. 28 to a public not ...
(1909, who classified all forms of rotational motions)
[Herglotz (1909)] and in a less general way by
Fritz Noether
Fritz Alexander Ernst Noether (7 October 1884 – 10 September 1941) was a Jewish German mathematician who emigrated from Nazi Germany to the Soviet Union. He was later executed by the NKVD.
Biography
Fritz Noether's father Max Noether ...
(1909).
[Noether (1909)] As a result, Born (1910)
[Born (1910)] and others gave alternative, less restrictive definitions of rigidity.
Definition
Born rigidity is satisfied if the
orthogonal
In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''.
By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
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 ...
distance between infinitesimally separated curves or
worldlines is constant,
or equivalently, if the length of the rigid body in momentary co-moving
inertial frame
In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration. ...
s measured by standard measuring rods (i.e. the
proper length
Proper length or rest length is the length of an object in the object's rest frame.
The measurement of lengths is more complicated in the theory of relativity than in classical mechanics. In classical mechanics, lengths are measured based on t ...
) is constant and is therefore subjected to
Lorentz contraction
Lorentz is a name derived from the Roman surname, Laurentius, which means "from Laurentum". It is the German form of Laurence. Notable people with the name include:
Given name
* Lorentz Aspen (born 1978), Norwegian heavy metal pianist and keyboa ...
in relatively moving frames.
[Gron (1981)] Born rigidity is a constraint on the motion of an extended body, achieved by careful application of forces to different parts of the body. A body rigid in itself would violate special relativity, as its
speed of sound
The speed of sound is the distance travelled per unit of time by a sound wave as it propagates through an elastic medium. At , the speed of sound in air is about , or one kilometre in or one mile in . It depends strongly on temperature as w ...
would be infinite.
A classification of all possible Born rigid motions can be obtained using the Herglotz–Noether theorem. This theorem states, that all
irrotational
In vector calculus, a conservative vector field is a vector field that is the gradient of some function. A conservative vector field has the property that its line integral is path independent; the choice of any path between two points does not c ...
Born rigid motions (
class A) consist of
hyperplane
In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...
s rigidly moving through spacetime, while any rotational Born rigid motion (
class B) must be
isometric Killing
Killing, Killings, or The Killing may refer to:
Arts, entertainment, and media Films
* ''Killing'' (film), a 2018 Japanese film
* ''The Killing'' (film), a 1956 film noir directed by Stanley Kubrick Television
* ''The Killing'' (Danish TV serie ...
motions. This implies that a Born rigid body only has three
degrees of freedom
Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
. Thus a body can be brought in a Born rigid way from rest into any
translational
Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...
motion, but it cannot be brought in a Born rigid way from rest into rotational motion.
Stresses and Born rigidity
It was shown by Herglotz (1911), that a relativistic
theory of elasticity
Solid mechanics, also known as mechanics of solids, is the branch of continuum mechanics that studies the behavior of solid materials, especially their motion and deformation under the action of forces, temperature changes, phase changes, and ot ...
can be based on the assumption, that stresses arise when the condition of Born rigidity is broken.
An example of breaking Born rigidity is the
Ehrenfest paradox
The Ehrenfest paradox concerns the rotation of a "rigid" disc in the theory of relativity.
In its original 1909 formulation as presented by Paul Ehrenfest in relation to the concept of Born rigidity within special relativity, it discusses an ideal ...
: Even though the state of
uniform circular motion
In physics, circular motion is a movement of an object along the circumference of a circle or rotation along a circular path. It can be uniform, with constant angular rate of rotation and constant speed, or non-uniform with a changing rate of rot ...
of a body is among the allowed Born rigid motions of
class B, a body cannot be brought from any other state of motion into uniform circular motion without breaking the condition of Born rigidity during the phase in which the body undergoes various accelerations. But if this phase is over and the
centripetal acceleration
In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by the ...
becomes constant, the body can be uniformly rotating in agreement with Born rigidity. Likewise, if it is now in uniform circular motion, this state cannot be changed without again breaking the Born rigidity of the body.
Another example is
Bell's spaceship paradox
Bell's spaceship paradox is a thought experiment in special relativity. It was designed by E. Dewan and M. Beran in 1959 and became more widely known when J. S. Bell included a modified version.J. S. Bell: ''How to teach special relativity'', Prog ...
: If the endpoints of a body are accelerated with constant proper accelerations in rectilinear direction, then the leading endpoint must have a lower proper acceleration in order to leave the proper length constant so that Born rigidity is satisfied. It will also exhibit an increasing Lorentz contraction in an external inertial frame, that is, in the external frame the endpoints of the body are not accelerating simultaneously. However, if a different acceleration profile is chosen by which the endpoints of the body are simultaneously accelerated with same proper acceleration as seen in the external inertial frame, its Born rigidity will be broken, because constant length in the external frame implies increasing proper length in a comoving frame due to relativity of simultaneity. In this case, a fragile thread spanned between two rockets will experience stresses (which are called Herglotz–Dewan–Beran stresses
) and will consequently break.
Born rigid motions
A classification of allowed, in particular rotational, Born rigid motions in flat
Minkowski spacetime
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 inert ...
was given by Herglotz,
which was also studied by
Friedrich Kottler Friedrich Kottler (December 10, 1886 – May 11, 1965) was an Austrian theoretical physicist. He was a Privatdozent before he got a professorship in 1923 at the University of Vienna.
Life
In 1938, after the Anschluss, he lost his profess ...
(1912, 1914),
[Kottler (1912); Kottler (1914a)] Georges Lemaître
Georges Henri Joseph Édouard Lemaître ( ; ; 17 July 1894 – 20 June 1966) was a Belgian Catholic priest, theoretical physicist, mathematician, astronomer, and professor of physics at the Catholic University of Louvain. He was the first to th ...
(1924),
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. ...
(1940), George Salzmann &
Abraham H. Taub
Abraham Haskel Taub (; February 1, 1911 – August 9, 1999) was a distinguished American mathematician and physicist, well known for his important contributions to the early development of general relativity, as well as differential geometry and ...
(1954).
[Salzmann & Taub (1954)] Herglotz pointed out that a continuum is moving as a rigid body when the world lines of its points are
equidistant curve
In hyperbolic geometry, a hypercycle, hypercircle or equidistant curve is a curve whose points have the same orthogonal distance from a given straight line (its axis).
Given a straight line and a point not on , one can construct a hypercycle ...
s in
. The resulting worldliness can be split into two classes:
Class A: Irrotational motions
Herglotz defined this class in terms of equidistant curves which are the orthogonal trajectories of a family of
hyperplanes
In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...
, which also can be seen as solutions of a
Riccati equation In mathematics, a Riccati equation in the narrowest sense is any first-order ordinary differential equation that is quadratic in the unknown function. In other words, it is an equation of the form
: y'(x) = q_0(x) + q_1(x) \, y(x) + q_2(x) \, y^2(x ...
(this was called "plane motion" by Salzmann & Taub
or "irrotational rigid motion" by Boyer
). He concluded, that the motion of such a body is completely determined by the motion of one of its points.
The general metric for these irrotational motions has been given by Herglotz, whose work was summarized with simplified notation by Lemaître (1924). Also the
Fermi metric in the form given by
Christian Møller
Christian Møller (22 December 1904 in Hundslev, Als (island), Als14 January 1980 in Ordrup) was a Danish people, Danish chemist and physicist who made fundamental contributions to the theory of relativity, theory of gravitation and quantum chemi ...
(1952) for rigid frames with arbitrary motion of the origin was identified as the "most general metric for irrotational rigid motion in special relativity". In general, it was shown that irrotational Born motion corresponds to those Fermi congruences of which any worldline can be used as baseline (homogeneous Fermi congruence).
Already Born (1909) pointed out that a rigid body in translational motion has a maximal spatial extension depending on its acceleration, given by the relation