Cosmology
The inception of Cosmology was in 1915 when Einstein introduced General Relativity. Special Relativity is limited to inertial frames; GR extends itself to the non-inertial.
The effects of acceleration and gravity are indistinguishable, which is is just an expression of the Equivalence Principle: The equations that describe the laws of physics, appear the same in frames that are falling in a gravitational field, as those observed in an inertial frame in the absence of a gravitational field.
Laws of physics that stay the same, no matter what coordinate system, are said to be covariant. The principle of equivalence is also the Principle of Covariance. The equations of SR are not covariant, they hold only for inertial frames.
Scalars are tensors of rank zero. Vectors are tensors of rank one, and are represented by one-dimensional arrays. We can define tensors in any coordinate system as a number of functions of the coordinates. With tensors, the equations for transformation for their components are linear and homogeneous.
In GR, ds^{2} = g_{μν} dx_{μ} dx_{ν}
The dx_{μ} and dx_{ν} are covariant vectors that are chosen to define a coordinate system. This defines the metric tensor g_{μν} For SR the metric is,
g_{μν} = | g_{00} | g_{10} | g_{20} | g_{30} |
g_{01} | g_{11} | g_{21} | g_{31} | |
g_{02} | g_{11} | g_{12} | g_{13} | |
g_{03} | g_{13} | g_{23} | g_{33} |
= | -1 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | |
0 | 0 | 1 | 0 | |
0 | 0 | 0 | 1 |
Although there are 16 components to the metric tensor, only 10 are independent because it is a symmetric tensor, g_{μν }= g_{νμ }. The metric tensor describes the curvature of spacetime, and replaces the Newtonian concept of a gravitational field.
Particles follow extreme paths (Geodesics) in spacetime. ‘Extreme’ coming from calculus, where we find maxima or minima of functions by setting derivatives to zero: One way to think of GR geodesics is that they are related to the proper time of the worldline. All other observers would have clocks running slower because of time dilation, so the proper time is the maximum value.
The difference in motion between SR particles and GR particles is that an observer in an inertial frame will see the GR geodesics as curving lines. Particles will appear to be accelerated even though they are just travelling along the natural path dictated by the metric. The acceleration of gravity is a result of the curvature of spacetime.
Between any two points in 3-space there are an infinite number of possible paths. In spacetime there is only one possible time-like geodesic between two events. In spacetime we travel on Worldlines between two events rather than on paths between two points.
The Field equations relate the curvature of spacetime to the presence of matter. The Stress-Energy-Momentum tensor for a perfect gas is,
T_{μν} = | ρ | 0 | 0 | 0 |
0 | P | 0 | 0 | |
0 | 0 | P | 0 | |
0 | 0 | 0 | P |
The time-time like component is the proper mass density, composed of the rest mass density plus the density of the mass equivalent to all present energy including electromagnetic fields. The space-space components are the physically measureable Pressure. In a manner similar to the electric and magnetic fields, pressure and density appear differently to different observers but are part of a more fundamental quantity described by the tensor that represents them.
Einstein’s field equations are;
R_{μν} –g_{μν }R/2 = -κT_{μν} +2Λ
Where R is a scalar obtained from the tensor R_{μν} and Λ serves the role of a constant of integration and is called the Cosmological Constant.