On Gravity: Remembrance of the Association Between a Guru …
63
where κ depends on G the fundamental constant of gravity, first used by Newton. Even
though this is a simple looking equation, in the inside it is highly complicated. The
exact form of the Einstein tensor is G ab = R ab − (1/2)g ab R, where R ab is a second
rank tensor called Ricci tensor which represents the curvature of space–time and R is
the scalar corresponding to it. The object g ab is known as the metric components of
the space–time which is the basic quantity carrying the geometry of the given space–
time. The immediate question that may arise is “if R ab is representing the curvature
of the space–time, then what is the reason for using another second rank tensor G ab
for representing the same curvature of the space-time in writing down the Einstein
equation?”. The reason is the following. If one uses R ab alone in equation (1) instead
of G ab , then the equation will posses the problem of unequal divergence on both sides
of the equation. That is the divergence of R ab is not equal to zero while divergence of
T ab is zero. It was Einstein’s intelligence which ultimately frame the left-hand side of
the equation with G ab , a second rank tensor, representing curvature and at the same
time posses zero divergence. In the absence of matter, the equation of gravity will
reduces to R ab = 0 under linear limit. The immediate success of Einstein’s equation
is that it explains the perihelion shift of planet Mercury accurately and predicts the
effect of gravity on light that the light will bend under gravity. The bending of light
was later observationally verified by a historic event by a team led by Eddington.
Even though the success of reformulating gravity as the curvature of space–time
is remarkable, mathematically the situation becomes more complicated than that
of Newton’s law. The difficulty is that the quantity R ab contains second-order partial derivatives of g ab and also the products of metric components, which makes
the equation highly non-linear. So finding solutions to Einstein’s equation becomes
a Himalayan task. Schwarzschild comes with the first solution for compact mass
distribution, where he took advantage of the simple spherically symmetric distribution of matter, which simplified the process of deriving the solution a lot. The other
prominent situation in which the equation possess a well-defined solution is for an
isotropic and homogeneous distribution of matter, and is the case for our universe,
which was first obtained by Friedmann and later by Lamaitre, Robertson and walker.
Unless there exists convenient symmetries, it becomes quite impossible to find an
analytical solution for Einstein’s field equation.
3.1 Searching for Quantum Version of Gravity
Gravity is a peculiar force compared to other fundamental forces in the sense that it is
the only one which can affect the geometry of the space–time. In essence a theory of
gravity is a theory of the space–time geometry itself. In the later part of the twentieth
century it was showed that all systems are basically quantum mechanical in nature.
Hence theories of all fundamental laws of nature must be described in terms of the
principles of quantum theory. Towards the end of the last century, the classical laws
of electrodynamics got modified in to a description based on quantum principles, and
thus we have quantum electrodynamics. Therefore, one expects that gravity should
63
where κ depends on G the fundamental constant of gravity, first used by Newton. Even
though this is a simple looking equation, in the inside it is highly complicated. The
exact form of the Einstein tensor is G ab = R ab − (1/2)g ab R, where R ab is a second
rank tensor called Ricci tensor which represents the curvature of space–time and R is
the scalar corresponding to it. The object g ab is known as the metric components of
the space–time which is the basic quantity carrying the geometry of the given space–
time. The immediate question that may arise is “if R ab is representing the curvature
of the space–time, then what is the reason for using another second rank tensor G ab
for representing the same curvature of the space-time in writing down the Einstein
equation?”. The reason is the following. If one uses R ab alone in equation (1) instead
of G ab , then the equation will posses the problem of unequal divergence on both sides
of the equation. That is the divergence of R ab is not equal to zero while divergence of
T ab is zero. It was Einstein’s intelligence which ultimately frame the left-hand side of
the equation with G ab , a second rank tensor, representing curvature and at the same
time posses zero divergence. In the absence of matter, the equation of gravity will
reduces to R ab = 0 under linear limit. The immediate success of Einstein’s equation
is that it explains the perihelion shift of planet Mercury accurately and predicts the
effect of gravity on light that the light will bend under gravity. The bending of light
was later observationally verified by a historic event by a team led by Eddington.
Even though the success of reformulating gravity as the curvature of space–time
is remarkable, mathematically the situation becomes more complicated than that
of Newton’s law. The difficulty is that the quantity R ab contains second-order partial derivatives of g ab and also the products of metric components, which makes
the equation highly non-linear. So finding solutions to Einstein’s equation becomes
a Himalayan task. Schwarzschild comes with the first solution for compact mass
distribution, where he took advantage of the simple spherically symmetric distribution of matter, which simplified the process of deriving the solution a lot. The other
prominent situation in which the equation possess a well-defined solution is for an
isotropic and homogeneous distribution of matter, and is the case for our universe,
which was first obtained by Friedmann and later by Lamaitre, Robertson and walker.
Unless there exists convenient symmetries, it becomes quite impossible to find an
analytical solution for Einstein’s field equation.
3.1 Searching for Quantum Version of Gravity
Gravity is a peculiar force compared to other fundamental forces in the sense that it is
the only one which can affect the geometry of the space–time. In essence a theory of
gravity is a theory of the space–time geometry itself. In the later part of the twentieth
century it was showed that all systems are basically quantum mechanical in nature.
Hence theories of all fundamental laws of nature must be described in terms of the
principles of quantum theory. Towards the end of the last century, the classical laws
of electrodynamics got modified in to a description based on quantum principles, and
thus we have quantum electrodynamics. Therefore, one expects that gravity should