Converting the thought experiment into mathematical equations
Einstein thought out all the basic principles derived from his thought experiment in his head and had ultimate confidence in their correctness. Only then did he attempt to depict them in mathematical terms. This was by no means an easy task. In fact it took over a decade until his theory was ultimately propounded in 1915, and in the end he was almost beaten to the punch by the renowned German mathematician David Hilbert (1862 - 1943)[1].
Einstein’s theory of special relativity involved observers at rest and those in constant velocity motion, and had nothing to say about the effects of gravity or observers in a complex state of accelerated motion. These elements form the cornerstone of his theory of general relativity and are described in the Einstein Field Equations (EFE), a set of 10 equations first published in 1915 which equate spacetime curvature (as expressed by the Einstein tensor (footnote [5]), named after him) with the energy and momentum in spacetime, as expressed by the stress-energy tensor.
Matter tells spacetime how to curve
General relativity describes how spacetime is curved under the influence of matter and energy. Newton's classical theory of gravity as a universal force depended on mass alone. The solutions of the Einstein field equations are metrics of spacetime. Hence they are often simply referred to as metrics. The metrics describe the structure of the spacetime including the inertial motion of objects in the spacetime. As the field equations are non-linear, they cannot always be completely solved without making approximations. The EFE and the geodesic equation - which dictates how freely-falling matter moves through space-time - form the core of the mathematical formulation of general relativity[2] which can be reduced to its basic formulae in various ways.
It was Hermann Minkowski (1864-1909), Einstein’s erstwhile teacher, who first conceived a mathematical approach to space and time which placed them on an equal footing. His system described objects as having not only x, y and z axes of position, but also a t corresponding to their position on a fourth axis, time. An object that stayed in the same place would be represented by a straight line, with only its position on the t axis changing, but if you plot the path of a body moving through both space and time, their combined paths will appear to curve.
Einstein at first regarded this idea as “superfluous learnedness”. He knew that ordinary Euclidean geometry based on lines drawn on flat surfaces does not apply when, say, a circle is drawn on a sphere. On a curved surface, the rules of Euclidean geometry are changed: parallel lines can meet, and the sum of the angles in a triangle can be more or less than 180 degrees, depending on how space is curved. For spheres with the same radius, a circle drawn on a sphere has a shorter circumference than one drawn on a flat sheet of paper. Aircraft utilise this principle every day in their journeys across the globe, the shortest distance between two cities being the curved route. Thus non-stop flights from the United States to Europe fly over parts of Greenland. On a flat map the plane's flight path looks curved, but on a globe, that path is the shortest one.
Einstein’s theory postulated a four-dimensional spacetime with curvature that varies from place to place, and depended upon charting a body’s spatial location (space curvature in the presence of mass) in time simultaneously. Ordinary Euclidean geometry was useless to describe it. Karl Frederick Gauss, Janos Bolyai and Nikolai Lobachevsky had successively sewed the seeds of non-Euclidean geometry in the early 1800s, but their geometries had constant curvature. Then, in the mid 1800s, Bernard Riemann made it possible to vary the curvature[3]. His geometry was therefore differential. Without the considerable contribution made by these eminent scientists and that of Minkowski, Einstein would never have been able to write down the equations for this theory, so he fell back on Minkowski’s idea of the use of “tensors” translating the quantities from one coordinate system to another.[4].
The whole of general relativity is couched in the language of tensors - mathematical objects that are “ranked” according to their indices, a common measure of their complexity[5]. A tensor of rank 0 is a constant, of rank 1 a vector[6], and of rank 2 a set of complex matrices with multiple components that can be used to translate one set of co-ordinates into another. Rank 2 tensors allow space and time to be brought together, redefining them in the process[7].
The EFE are commonly depicted as [8]:where the right side represents energy and density, while the left side treats the curvature of space time that results from it. On the left side, the Reimann tensor represents the measured distance between two points in curved space. On the right side is the Ricci tensor determines space time curvature. The indices μ and v stand for variables that change over the four co-ordinates representing the dimensions of space – x, y and z – and time – t - at each point[9]. “With Einstein’s tools in hand, a phycisist can determine how the presence of mass, energy density, time and pressure influenced the geometry of space-time. He or she can not only determine the path of objects in a gravitational field, but can also apply them consistently to the universe as a whole. This had not been the case with Newtonian theory”[10].
A vastly simplified version of the EFE appears as:
Gμν = 8πTμν or even more simply G = T [11]
Lawrence Krauss[12] depicts the situation, including the cosmological constant and the Einstein tensor, more fully thus:
Left hand side Right hand side
Curvature of spacetime Energy and momentum of spacetime
Gμν = 8πTμν
Gμν - Λgμν[13] = 8πTμν
The left hand side of the equation refers to a set of terms that characterise the geometry of spacetime, that is its curvature at a given point. The right hand side represents a set of terms that describes the distribution of the energy and momentum of spacetime. The EFE are therefore a set of equations which dictate how the distribution of matter and energy determines the curvature of spacetime.
However, if we move the cosmological constant portion of the equation to the right hand side, we come up with:
Gμν = 8πTμν + Λgμν
Giving rise to the question, is the right hand side of the equation here, “the energy of nothing”, in other words, vacuum energy or dark energy? As to which, see /the-cosmological-constant-and-the-expansion-of-space.html
[1] The story of their rivalry is narrated by Walter Isaacson in “How Einstein invented reality”, Scientific American Special Issue – 100 years of general relativity, September 2015, 28.
[2] David Zaslavsky and Luboš Motl in http://physics.stackexchange.com/questions/3009/how-exactly-does-curved-space-time-describe-the-force-of-gravity
[3] On the role of Gauss, Bolyai and Lobachevsky in the formulation of non-Euclidean geometry, see Dana Mackenzie, The Universe in Zero Words – The Story of Mathematics, Elwin Street Productions, London, 2012, Chapter 15, “The geometry of Whales and Ants – Non-Euclidean geometry”, 123-127; on Riemann, ibid, 127, 131-133; on Minkowski, ibid, 153. On the issues in this paragraph (the workings of non-Euclidean geometry in general relativity) , see also John Farrell, The Day Without Yesterday - Lemaître, Einstein and the Birth of Modern Cosmology, Thunder's Mouth Press, New York, 2005, 7.
[4] Crease, op cit, 191-3.
[5] Simply, tensors defined mathematically are arrays of numbers, or functions, that transform according to certain rules under a change of coordinates. In physics, tensors characterize the properties of a physical system: Warren Davis, http://www.physlink.com/education/askexperts/ae168.cfm
[6] Vectors are mathematical quantities that are fully described by both a magnitude and a direction – the so-called arrow of spacetime: Mackenzie, op cit, 169. By contrast, scalars are quantities that are fully described by a magnitude (or numerical value) alone.
[7] Robert P. Crease 192.
[8] See, for example John Farrell, The Day Without Yesterday, op cit above, 76; also the site http://backreaction.blogspot.com.au/2007/11/cosmological-constant.html
[9] Farrell, op cit.
[10] Ibid.
[11] More fully, including indices and constants Gμν + Λgμν = 8πTμν, where Gμν is the Einstein tensor, Λ is the cosmological constant, and G is the gravitational constant which derives from Newton's law of universal gravitation. Einstein's full calculations are far more labyrinthine.
[12] Lawrence Krauss, A Universe from nothing: Why there is something rather than nothing, Scientific Thinker’s Channel, reproduced 30 March 2012 as a YouTube Video: http://thesciencenetwork.org/ (even Miley Cyrus gets a mention). Otherwise reproduced as https://www.youtube.com/watch?v=VdUYw59ztyw Bear in mind that Krauss`s “nothing” is in fact a vacuum state full of energy and obeying the laws of quantum mechanics, which is something different from the philosophers’ nothing in the sense of the complete absence of being.
[13] The Greek symbol Λ (Lambda) represents the cosmological constant.