THE MATHEMATICAL ART FORM
by Julie Peschke, Athabasca University
Photo courtesy of: Einstein's Field Equations of General Relativity Explained
Believe it or not!
This is the mathematical code which encapsulates in a snapshot the entire theory of general relativity. It represents sixteen equations (six of them duplicates) which define the nature of space, light and travel in our universe. If the discipline of mathematics had an art gallery built in its honour, this would be one of its featured works. It is pictorially simple but, in its import, it is enigmatic and deeply complex - as are many of the hieroglyphic forms from the past.
THE COGNITIVE JOURNEY
General Relativity and Gravity (published on August 8, 2007 by stevebd1)
Gravity and Time (published on November 2, 2008 by Marcus Amadeus)
THE HISTORICAL PALETTE of ARTISTS
Einstein’s cognitive journey led to the formulation of this set of equations. They are the mathematical underpinnings of his General Theory of Relativity and are called his field equations. The foundational mathematics supporting these equations was initially developed over a 72- year period from 1828 to 1900 by two German mathematicians, Carl Friedrich Gauss (1877 – 1855) and Bernhard Riemann (1826 – 1866), who laid the groundwork for spatially curved differential geometry models, and two Italian mathematicians, Gregorio Ricci-Curbastro (1853 – 1925), recognized as the inventor of Tensor Calculus, and Tullio Livi-Civita (1873 - 1941).
The mathematical basis for this kind of curved spatial geometry, restricted to only 2-dimensional curved surfaces (for example, the upper half of a sphere of fixed dimensions) in Euclidean 3-space, was first developed by Gauss in his 1828 treatise, entitled Theorema Egregium . He discovered mathematically that such curved surfaces had their own intrinsic geometry which was not dependent on the geometry of the higher dimensional space in which they were embedded – their, so-called, ambient space. That was his remarkable theorem. This concept was extended to higher dimensions (for example, 3-dimensional surfaces in Euclidian 4-space) by one of Gauss’s students, Bernhard Riemann, in his inaugural lecture, entitled On the Hypotheses on which Geometry is Based. This lecture was later published by another mathematician, Richard Dedekind, in 1867, a year after Riemann’s death at 40 years of age and some twelve years before Einstein was born. It was the mathematics of Gaussian and Riemannian Geometry and the subsequent development of tensor calculus which Einstein and his passel of mathematicians used to develop his field equations in the early years of the twentieth century.
THE APPLICATION – the Topography of the Landscape in the State of Hanover, Germany
THE MATHEMATICAL MODEL – a Dynamical System in Continuous Motion
This is a little bit of a leap of the imagination but the intrepid may wish to follow along the journey into outer space. Some background in the study of Calculus and Linear Algebra will help but is not absolutely necessary to understand the model. Just keep track of the dimensions.
Imagine the earth ( which is a solid 3-dimensional mathematical xyz– object) and the outer space surrounding it embedded in a 4-dimensional Euclidean txyz– space where the 0^{th} coordinate t represents time. Let’s call this txyz– space the ambient space of the model , namely the space in which everything happens. The variable t is invisible to the eye but mathematically it has an effect on the earth as a 3-dimensional xyz– object in that space because of the earth’s rotation about its tilted axis and its movement along its orbit over time t. In fact, our notion of time in hours and years is based on the earth’s movements and it is precisely these motions of a solid body which are a significant factor in its own geometry, both internal and external, including the geometry of its surrounding spatial environment.
Now any xyz– object may be Euclidean-flat (the top of a rectangular box) or may be curved in some way (part of a ball). However, curved or not, at every point of the outer surface of such a solid object, there is a unique flat plane, embedded in xyz– space, which touches (or grazes) the surface of that object at that point. Each of these xyz– planes, so as to distinguish them from the object itself, has no curvature and is called the tangent plane to the object at the specified point. Moreover, when the object moves, so do the associated tangent planes at each of it points. Therefore, the earth, as a moving object, and its bundle of tangent planes may be mathematically envisioned as an object in the space-time xyz– continuum.
Movement along a surface may be mathematically represented by vectors (objects with length and direction symbolized by straight arrows) and, because of their straightness, are not embedded in the object itself (unless the object is also flat) but in one of its tangent planes. These vectors/arrows in the tangent plane at a given point define the movement of the object at that point, both how fast it is moving and in which direction. As the point on the surface shifts, the orientation and location of the tangent planes will shift with it and the velocity arrows will follow those shifts.
As an analogy in 2 dimensions, think of a parabola as a 1-dimensional curved object because one can travel only back and forth along it if you want to remain on it. The parabola and its bundle of 1-dimensional tangent lines at each point on the curve are all contained within the 2-dimensional Euclidean xy– plane (its ambient space). The tangent lines will vary in position and orientation (measured by their slopes or their angles of inclination to the horizontal coordinate axis) in the ambient space. These changes of orientation will depend on the geometry (shape and orientation) of the parabola. Therefore, mathematically, each tangent line may be represented as a linear equation in both x and y. Any point moving along the parabola has a direction vector tangent to it which points in the direction of the movement, with its length equalling its speed in that direction. As soon as the point moves, the direction vector changes position and may also change in orientation but always lies in the tangent line to the point at which it is concurrently located.
In cases where the ambient Euclidean xyz–space is 3-dimensional, take the example to be a hemisphere (a 2-dimensional curved surface with longitude and latitude coordinates) and its bundle of 2-dimensional tangent planes at each point on its surface. These 2-dimensional tangent planes will vary in position and direction angles to each of the xyz–spatial coordinate planes depending on the geometry (position and orientation) of the hemisphere in its ambient space.
For illustration and clarification purposes, suppose that the 3-dimensional ambient space has an extrinsic y_{1}y_{2}y_{3}– coordinate system (instead of an xyz– coordinate system because it will become too confusing otherwise). Suppose also that a patch of the 2-dimensional curved surface embedded within that y_{1}y_{2}y_{3}– space has an intrinsic x_{1}x_{2}– coordinate system containing points P and P^{1}. Moreover, depending on the geometry of the surface, at each point on the surface, there will be a unique 2-dimensional tangent plane associated with it. These are referred to as reference frames, which determine the perspective of the observer at each of the points (positions) on the surface. See the diagram below.
In June of 1905, Einstein published a paper called On the Electrodynamics of Moving Bodies, in which he declared his relativity principle that the laws by which the states of physical systems undergo change must be unaffected by a change in the reference frame (the system of coordinates referred to in the uniform translatory motion). Therefore, there had to be a simpler model of mathematical representation for all of these changes in movement which was independent of the coordinate system used. For this very reason, the notion of a tensor, T , which does map vectors onto vectors but can be independent of the underlying coordinate systems, was the mathematical object of choice in the Einsteinian 4-dimensional space-time model. Such tensors are considered to be linear, actually multi-linear (or linear in each of the dimensions), mappings from one flat space to another, defined at each point on the original surface. Their totality over all points on the given surface is sometimes called a linear connection from one tangent plane to another along a surface. See the diagram above.
Now imagine that everything in the figure above is moving continuously in a somewhat restricted rotational fashion about a north-south axis tilted 23 ½ degrees to the vertical, simultaneously pushing forward along a rather large, slightly elliptical, path. This does not take into account the, to us, imperceptible rotational movement of the Milky Way galaxy containing our solar system or the motions within the Local Group cluster of galaxies containing the Milky Way. Keep in mind that there is an invisible time dimension attached to every point directed out of the screen towards you. In stretching the imagination in this way, we have suddenly immersed ourselves into the 4^{th} dimensional space-time model. Here the ambient Euclidean txyz– space is 4-dimensional but the object of interest is the earth and its surrounding space as a 3-dimensional xyz– spatial object curved by its movement in time within that space. At each point on the surface of this object, there is a planar object, also of three dimensions (2 spatial and 1 time) moving in tandem with the object, which is tangential to it but is represented in all four txyz– coordinates because of its movement. These tangent planes will vary in position and direction angles to each of the xyzt– coordinate planes depending on the geometry of the space-time planet. To imagine this 4-dimensional space and its tangent spaces, just add the invisible time coordinate to each spatial object in the diagram above.
In this model, the continuous movement is recorded in small increments mathematically, much like a string of sequential frames in a movie, each frame representing the action in a very small patch of split-second experiences about a given point in space-time. This is why all of the non-constant symbols used in the model are evaluated at single points of the surface in its ambient space.
THE MATHEMATICAL SYMBOLISM
The hieroglyphics in this piece of abstract art consist of both cosmological constants (A,G,c) and mathematical objects called covariant tensors of rank 2 (R_{uv}, g_{uv}, T_{uv}). To understand the mathematics behind the theory, one must have some idea as to what the metric tensor g_{uv}, the Ricci curvature tensor R_{uv}, and the stress momentum energy tensor T_{uv} measure in terms of the physical dynamics of the evolution of earth’s movement through space over time, point by point. Knowledge about derivatives, transformations of vector spaces and their associated matrices will be helpful to understand the symbolism and parts of the recommended videos. However, glossing over the mathematical content and reading only the over-arching import of the mathematical statements are options for the casual reader. For those who are curious about the mathematical details, see The Mathematics of Tensors.
The Metric Tensor g_{uv} : is a covariant tensor of rank 2 on a space of vectors of arbitrary dimension n. It provides a way to measure distance or length on a curved surface. It is a generalization of the Euclidean metric on flat space which is based on Pythagorum’s Theorem regarding the lengths of the sides of a right triangle.
The Scalar Curvature R: sometimes called the Ricci curvature scalar, assigns a number to every point on a surface embedded in flat Euclidean space, which measures the curvature of the surface at that point. The curvature value is found by comparing the volume of a geodesic ball about the point on the surface to the volume of a corresponding ordinary ball of radius 1 in a flat Euclidean space. If there is no deviation in the compared values, the surface has curvature 0 at that point.
The Ricci Curvature Tensor R_{uv}: is used to measure of how the volume of an object (in particular, a small wedge of a geodesic ball) about a point on a curved surface deviates from that of the standard ball in flat Euclidean space. As such, it provides one way of measuring the degree to which the geometry determined by a given metric tensor might differ from the geometry of ordinary Euclidean n-space. As a tensor (in this case, a bilinear map), it is able to track the changes in curvature (including the various contours of a surface) as a point moves across the surface.
The Stress Energy Momentum Tensor T_{uv}: is an attribute of matter, radiation, and non-gravitational force fields. It describes how much energy, flux and momentum a moving body has at each point on it. In Einstein’s mathematical model, it indicates the density of energy and momentum acting on each point in the space-time continuum.
The Cosmological Constant ∧: Modern field theory associates this term with the energy density of a vacuum. It reflects the rate at which the expanding universe is accelerating and is associated with the notion of dark energy. Einstein introduced this constant into his field equations in order extend the forces of gravity to include the notion of ‘push,’ thus leading to acceleration and repulsion from other objects, not just ‘pull’ generating an attraction towards an object.
The Universal Gravitational Constant G: the constant appearing in Newton’s Law of Gravitation. It is related to the gravitational force of attraction between two bodies and is equal to
6.67408313131 x 10^{-11} N m^{2}/kg^{2} , where N is measured in units of newton force. This constant is not the same as the constant gravity g which denotes the acceleration due to gravity.
The Universal Constant c : the speed of light in a vacuum equal to 299,792,458 meters/second. This is the fastest any object can move in our cosmos.
THE SCIENTIFIC EVIDENCE and ITS IMPLICATIONS
Einstein’s Theory of General Relativity is just another example of the predictive powers of abstract mathematics. In this case, his field equations foretold a theory of space, time and gravity that was markedly different from the prevailing Newtonian physics models. Einstein presented his General Theory of Relativity to the scientific community in 1915. However, it was not until four years later, during a solar eclipse on May 29, 1919, that some of the measurements confirming its validity were first scientifically observed. During that eclipse, Sir Arthur Eddington (1882 – 1944), an English astronomer, physicist and mathematician, performed the first experimental test of Albert Einstein's general theory. The measurements taken during that experiment showed that light bends in a gravitational field (one of the predictions of Einstein’s theoretical model).
The findings of this scientific experiment ultimately contributed to the eventual understanding that, while classical Newtonian physics was a good model to describe the movement and attraction of objects in terms of their respective masses, the general theory of relativity better described the complexity of gravitational forces exerted by celestial bodies on the objects around them, particularly in the case of light rays because photons of light have no mass.
A second outcome of Einstein’s theory indicated that all forms of motion are represented by movement in space-time – not by the gravitational equilibrium of the inherent push and pull of the objects in question determined by their respective masses in a magnetic field. This revolutionary theory led to a 4-dimensional space-time coordinate system around the Earth which was curved (see the twisted “spherical-time” coordinates around the earth in the image below). For those who are not aware of the various mathematical coordinate systems of both 2- and 3-dimensional Euclidean space, see a mathematical description of idealized spherical coordinates from which our planetary latitude and longitude coordinates have been derived.
In 2004, almost 85 years after Eddington’s measurements during the solar eclipse, a NASA satellite-based mission, known as Gravity Probe B, measured the 4-dimensional space-time curvature around the Earth. The principal investigator of the probe's mission was Stanford University's Francis Everitt, who said: "The space-time around Earth appears to be distorted just as general relativity predicts."
THE REVOLUTIONARY CONCLUSIONS
“In the General Theory of Relativity, space and time cannot be defined in such a way that differences of the spatial coordinates can be directly measured by the unit measuring-rod, or differences in the time co-ordinate by a standard clock. The method hitherto employed for laying co-ordinates into the space-time continuum in a definite manner thus breaks down ... “
(Einstein, 1916, The Foundation of the General Theory of Relativity)
The book, The Road to Relativity, by H. Gutfreund and J. Renn, contains the original manuscript from which the above quote has been taken.
THE REALISM BEHIND the ABSTRACTION
The Gravity Probe B space mission resulted in a scientific, evidence-based, artistic depiction of that mathematically abstract art form.
Photo courtesy of: http://einstein.stanford.edu/SPACETIME/spacetime4.html
Image copyrighted by James Overduin, Pancho Eekels and Bob Kahn.
The picture above illustrates how space-time is warped and twisted (curvature combinations on the LHS of the field equations) by the mass and spin from the earth’s counterclockwise rotation about its north-south axis and its movement along its orbit around the sun involving the invisible time coordinate (inherent in the RHS of the field equations).
INTERPRETATION of EINSTEIN’S FIELD EQUATIONS
Remember that a mathematical equation is the equivalent of a balancing act between two mathematical quantities.
The mass and movement of a celestial body (RHS) dictates the curvature of the surrounding 4-dimensional space-time surface in which it is embedded (LHS)
and, conversely,
The curvature of the 4-dimensional space-time surface (LHS) determines how a celestial body of a given mass embedded in it will move (RHS).
The equality of these quantities across all ten equations defines the equilibrium of earth’s movement in its orbit over time and the relative smoothness of its rotation about its axis – all of which provides the stability we all need for continued life on this planet as we now know it. Any force or event which upsets this balance significantly will have far-reaching consequences for us and our earthly home.
For more in-depth perspectives on the General Theory of Relativity, see:
SOME VIDEOS
0:00 – 14:45 The general theory illustrated
14:48 – end The mathematics behind the theory, drawing on the discipline of what is now called Differential Geometry. The derivation of the tensors g_{uv}, R_{uv,} T_{uv} is discussed.
A HISTORICAL DRAMA
A BOOK researched and written by an amateur mathematician and physicist, Peter Collier.
✓ A self-study guide to special and general relativity
✓ From simple functions to relativistic cosmology
✓ With numerous fully-solved problems
Updated December 21 2017 by Student & Academic Services
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