Introduction to General Relativity

Basic Idea

With the general theory of relativity, in which Einstein managed to reconcile relativity and gravitation, he had to discard the classical physics worldview, which saw space as a stage on which the events of the world unfold. Instead, space-time is a dynamic entity, which is distorted by any matter that is contained in it, and which in turn tells that matter how to move and evolve. This interaction between spacetime and matter is described by Einstein's geometric, relativistic theory of gravity. As always in the following pages I will try to give a simplified vision, to highlight the fundamental aspects of this theory.

The Problem of Gravity 

Special Relativity provides that there can be no signs that travel faster than the speed of light (300.000 km/s). But until then the force of gravity, that of the Newton's Law of Universal Gravitation, was conceived as instantaneous. If we assume, for example, that the Sun moves a little, the Earth will feel a different strength instantly? According to Newton yes. But according to the theory of special relativity, this is not possible! So how does gravity work?

Inertial Mass & Gravitational Mass

The law that describes the dynamics of moving bodies is the second law of dynamics: F=ma 

Where m is the so-called inertial mass. The law that describes the gravitational force is the law of Universal gravitation of Newton: F=mM/r²

Where m is the so-called gravitational mass. These two masses come from two different ideas, so they are not necessarily equal. Some (inaccurate) experiments said that they could be very similar. But Einstein proved theoretically for the first time, that there will never be a way to distinguish between the effects of a uniform gravitational field or constant acceleration.

Space-time

So, he managed to write a law of relativity for ALL systems reference. The starting point is the new idea of the space-time. According to General Relativity it isn't flat, but curved. All the bodies move along the lines of the universe of a curved space, no need to invoke the force of gravity of Newton. Special Relativity continues to be valid, but it's extended to accelerated systems (non-inertial). Be careful when you see this kind of image of space-time. It is only a visual representation, not a physical one. 

Gravity & Metric

Moreover, according to Einstein the space in the presence of masses is bent by the masses. So, the force of gravity doesn't exist, it's just convenient to use it. The physical reality of space-time is described by an equation that binds the presence of matter/energy to the metric, that is the geometry of curved spacetime. The metric is a function of four independent variables (3 spatial and 1 temporal): (x,y,z,t). There are several metrics that describe different possible situations and it's here that the theory becomes extremely complex. But in any case, matter tells space-time how to «bend», and in turn moves in curved spacetime.

Gravity Equation 

Rᵤᵥ - ½R.gᵤᵥ + Λ. gᵤᵥ = (8πG/c⁴)Tᵤᵥ

This is the famous Einstein's equation of General Relativity. Briefly, the member to the left of equality measures the curvature and geometry of space-time in 'x', while the right one measures the density and flow of matter and energy in 'x'. This equation contains a numerical term called: cosmological constant (Λ) . Einstein introduces a negative value of it, to indicate a static universe. Well this is probably the most serious mistake he made in his life (but we absolve him for this), because later observations showed that the universe is expanding (Λ = positive).

Final Consequences

1) Effects on space-time: lengths and times depend on the presence of masses nearby so, as in special relativity, they are no longer fixed quantities. This cause, for example, an effect that is called Gravitational redshift, which acts on the wavelength of a light ray. 

2) Effects on massless particles: a light ray, like an object with mass, must suffer the action of the gravitational field, and possibly to be bent, because it's simply moving into the fabric of curved space-time.

 3) Gravitational waves: they are a direct result of accelerated masses, that generate ripples in the metric of spacetime that propagate at the speed of light.

A Daily Effect of GR : The GPS

GPS satellites send an electromagnetic wave towards our receiver, the beam comes reflected and by the delay between issuance and reception we can measure the distance. Combining the data of three satellites we can locate a point on Earth's surface. But the clocks on the satellites are subjected to the 'force' of gravity, which is different from that on Earth. So we measure a different time from what they would find on Earth. If we didn't consider the effects of general relativity (and even those of special relativity) there would be a few kilometres error in the position of GPS.