General Introduction to terms of General Relativity
The General Relativity is one of the successful theories in our time. From description of planets orbiting around a Star, to prediction of black holes and further dark energy and dark matter, it has a special role to play. It has been experimentally verified time and again. But the fundamentals of the mathematics involved are not as easy as the Newtonian gravity equations. At first look, a fresher in the topic will find the equation beautiful and elegant , however as he progresses with the terms in the equation , it is sure that his mind will be boggled.
The formulation of this theory took Einstein almost 10 -15 years of hard work and some rigorous mathematics to reach the final equation. However this was not the end, the proper solution of the equation requires further tampering, setting of parameters and calculations on different case basis. The equations of General Relativity often termed as Einstein’s Field equations represent 10 nonlinear partial differential equation in four different variables. So before trying to make sense with the equation itself, we have to trace the constituents of the equations independently.
The following article will include the brief description of the terms of the Einstein’s Field Equations of General Relativity.
Ricci Curvature Tensor (Rμν )
The Ricci curvature Tensor defines the difference in volume of a geodesic ball in a Riemann Manifold to that in an Euclidian Manifold. It determines a part of the space time curvature in which matter will either converge or diverge in time. In Einstein’s field equations it also represents the amount of matter content in the universe. It is a second order, symmetric tensor related to the curvature of the Riemann manifold.
The Riemann manifold is the space defined by the Riemann tensor as the inner product on the tangent space. The Riemann tensor makes it possible in a Riemann manifold to calculate the volume, arc length, gradient etc of the functions or divergence vectors acting on it. It thus determines the curvature of a surface as the path integrals of the distance along the surface and it is not necessary to regard the position of the surface in a 3 dimensional space. The non-zero contraction form of the Riemann Curvature Tensor also yield the Ricci Curvature Tensor.
Euclidian space is a two or more dimensional space represented by the Euclidean geometry. The Euclidian geometry consists of a vector plane represented by the cartesian coordinates in a set of Real numbers. The basic notion is a representation of a point in the space as (x,y,z) as vector spaces in a 3 dimensional plane.
The scalar Curvature (R) represents the LaGrange density for the Einstein Hilbert Space. To simplify it represents the small volume of the geodesic object that affects the Riemann manifolds. The representation of a particle in an Euclidean space or in Hilbert space generalised by some kind of metric is the scalar tensor or the scalar curvature.
Lagrange density is the formalism of representation of independent variables in spacetime in terms of events in space time.(x,y,z,t)
The metric tensor ( gμν ) is a term that basically tells the distance between two objects in a geometry. It can be taken as a tool that can be used to characterize geometrical properties of space in arithmetic way within a coordinate system within the Riemann manifolds.
The stress energy tensor (Tμν ) in the field equations of general relativity describe the flux of energy density and momentum in a spacetime. It is the source of Gravitation in General Relativity. It is a symmetric tensor of the second rank and can be represented by a 4x4 matrix.
The other terms used in the field equations are the Universal Gravitational Constant, The Cosmological Constant and the term for speed of light.
The gravitational constant G and term indicating speed of light © are familiar terms while the cosmological constant (Λ) represent the mass energy distribution of the empty space or vacuum. It was used by Einstein to balance the contraction of the space time due to the gravity. The value of the cosmological constant also determines possibility of the expansion or contraction of the spacetime.
These mathematical terms act such that the stress energy tensor directs the metric tensor to follow scalar curvature in a Riemann manifold or the metric representation of the scalar curvature in a Riemann Manifold tells the stress energy tensor to move.
In simple terms mass tells space to curve and infact the curved space tells mass to move.
The solution of the equations of the general relativity are the solutions for the metric tensor of the field. These metrices determine the movement of objects in the spacetime and the curvature of the space time. Since the equations are not linear, it is very difficult to find the exact solutions of the equations mathematically. One way to solve the equations is to set hypothetical assumptions and solve equations on the basis of those.
All the curvature terms and the mass energy terms are so arranged that the left side of the equation represents the curvature terms and the right hand side represents the mass and gravitation terms. It is not a mathematical formulation but instead the nature of the mathematics itself that define the equivalence of the terms themselves.
So, here we have defined the terms that are included in the field equations of the general relativity. When Einstein derived his equations of general relativity he was also unsure of the solutions of the equations. However , for the experimental verification of the perihelion of Mercury and calculation of the bending of light from stars in solar eclipse, he had applied linearization to the equations assuming weak gravitational field and space time as Minkowski space.
As how complex these equations of relativity may be , still after a century, these equations are proving to be more important than Einstein ever thought they would be.
