criticisms of the einstein field equation - Alpha Institute for Advanced ...

5.3. EINSTEIN AND GENERAL RELATIVITY
curvature until much later, when he tried unsuccessfully with the French
Mathematician Elie Cartan to extend his theory to incorporate light into
his theory to create his fabled unified field theory.
The story of general relativity starts with Euclid and his Euclidian geometry
dealing with flat surfaces, because general relativity allows us to understand the
force of gravity in terms of the geometry. Euclid gave us the means to draw
lines and angles and relate them together by theorems, which explained how
they interacted and depended on one another. Euclid gave us the means to
mathematically construct a box and Einstein was able to make his breakthrough
in general relativity, by considering how an observer inside a box would perceive
the actions of acceleration and gravity. This thought experiment led Einstein
to formulate his equivalence principle, in which he made the important step of
realizing that the acceleration due to gravity has something to do with geometry
- the equivalence principle. The development of general relativity from the
equivalence principle to the famous Einstein-Hilbert Equation field equation of
general relativity required Einstein to become acquainted with developments in
mathematics describing how objects move in time and space.
Vectors came into use at the turn of the eighteenth century and the term
is derived from the Latin verb to carry. The vector points in the specified
direction, with its length giving the magnitude of the force required to ’carry’
in that direction. It was first used by astronomers to describe how the ’radius
vector’, a line drawn from a planet to the focus of an ellipse, ’carries’ the planet
around the centre. Vector usually appeared in the phrase radius vector. The
French term was rayon vecteur as seen in Laplace’s ’Celestial Mechanics, which
was translated by the British Civil List Scientist, astronomer and mathematician
Mary Fairfax-Somerville 1780-1872.
The modern meanings of the terms ’vector’ and ’scalar’ were introduced by
William Rowan Hamilton (1805-1865) of Trinity College Dublin, in his paper
to the Royal Irish Academy in 1844 entitled ’On Quaternions’. Quaternions are
a non-commutative extension of complex numbers which still find use in three
dimensional rotations, but have largely been replaced by vectors. Hamilton also
introduced the term ’tensor’ in 1846.
In 1906, when Einstein started thinking about general relativity, he turned to
his old classmate Marcel Grossmann from his days in Zurich’s ETH University
for advice on how to proceed. Grossmann was a mathematical genius and was
able to acquaint Einstein with the work of Riemann, Christoffel, Ricci and Levi-
Civita on a then new kind of geometry, generally known as Riemann geometry,
in which space and time were merged together in spacetime, and in which the
framework or frame of reference could be dynamic and curve. Bianchi’s work was
also to be of seminal importance, in Einstein’s quest to extend special relativity,
to include the effects of acceleration and gravity.
Professor Luigi Bianchi (1856-1928) was a great Italian mathematician, who
worked in Pisa with Gregorio Ricci-Curbastro (1853-1925) who invented tensor
calculus and Tullio Levi-Civita (1873-1941) who was born in and worked
from Padua. All three mathematicians developed ground breaking mathemati-
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