General Relativity - Lesson Installation 1 of 3

One thing Einstein brought to the world, was his amazing conclusion of solving field equations that was describable as a spacetime geometry. The definition of a field equation, is one that describes a physical spacetime field where in the case of Einsteins equations, deal with gravity, which is an all pervading physical field which attracts all other gravitational radiating bodies in the universe.

To begin these lessons, we shall learn about one of Einsteins most famous equations, the Einstein tensor equation. Once we have covered the very basics of his formula, we will then consider the implications of it, through a worded essay.

Einstein’s Tensor Equation

The most important equation in Einstein’s field equations, is the field tensor equation, and is given as:

G_uv=T

Einstein began these equations from a simple equivalence:

G_(αβ)= T_(αβ)

*some times k will not be added, but it is usually more correct to add it

G_(αβ)=kT_(αβ)

And this alone would explain that geometry was being involved in the description of spacetime. It said that particles are fluctuations of a gravitational field, and are themselves fluctuations of gravity. These fluctuations would bend and curve the space around them, not only caused by a generation of mass, but also the acceleration as it moved through the vacuum.

The right hand side of the equation G_(αβ)=T_(αβ) refers to the energy tensor, related to momentum, that naturally causes the space and time surrounding it to curve in on itself. In fact, in many ways, particles are tiny points that are visualized to be fluctuations that have curved in on themselves, so they can be seen as little creases in the very fabric of spacetime.

The provision of equations, also have about them, a conservation, just like the kind of conservation we may use in Kinetic Energy Equations, for example. We (in terms of physics jargon), that the tensor is covariantly conserved as a quantity in space and time, and we denote this conservation as:

▼βT^(αβ)= ▼βG^(αβ)

▼ Is the derivative, also known as the ‘’covariant derivative’’ is the same as the Einstein Tensor,

▼=G_uv

*Notice the lower and higher indices. They can be representable as traces in Einstein’s equations.

The constant k, is given a value of

K=8πG/c^4

These equations lead to a field equation that is most recognizable in physics education,

G_(αβ)=(K=8πG/c^4)T_(αβ)

And in this one equation, if my memory serves me correctly, involves 10 differentiable equations. So there is quite a lot in this equation. In Einstein’s early resolution of the equations, he decided that it was necessary to believe the universe was static, but under closer observations of cosmological evolution, it became apparent that a cosmological constant divided Einstein’s thoughts to either a contracting universe or an expanding universe. It became obvious that it resolved to an expanding universe.

G_(αβ)+Λg^(αβ)=(K=8πG/c^4)T_(αβ)

The cosmological constant ‘’Einstein’s biggest blunder’’ (a phrase he would coin that could be debatable), gave rise to an expanding universe, where an energy density was by a magnitude greater of 10^120, which would mean there was more energy predicted in the universe, than what can be observed – it is considered, understandably, one of the largest discrepancies known in physics today.

The Geometry of Spacetime

And so, what we learn from the Einstein Field equation, is that the right hand side of G_(αβ)=T_(αβ), described the geometry of spacetime. In fact, we learn through relativity, that this geometry is not just simply the three-dimensional presence of matter, but is also related to being distortions in spacetime itself. Because it is related to the momentum, it is also said that these distortions are equivalant to acceleration. So in a nutshell, matter is simply the same as the distortions, which is the same as acceleration; these are also the components of another entity; curvature.

The presence of matter entails the geometry of curvature in spacetime, where it will quite literally distort and derrange the spacetime within its vicinity and curve it within its presence. It is for this reason, that in 1919 Arthur Eddington was able to take a photograph of light being bend around the suns gravitational presence. This happens because light has momentum, and also its own gravity (whilst its gravity is very weak), it is still able to be influenced by gravity itself. The gravitational influence of light in the universe is far too weak (in fact, miniscule) to even have an influence in the evolution of the universal gravitational bodies, such as planets, stars to entire galaxies. Light may have had an effect on the gravitational evolution of bodies perhaps very early on in the universes history.