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Einstein’s Relativity

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Art drawing about the theory of relativity(Art drawing about the theory of relativity)

The theory of relativity includes both the theory of special relativity as that of general relativity formulated by Albert Einstein in the early twentieth century, who sought to resolve the incompatibility between Newtonian mechanics and electromagnetism.

The theory of special relativity, published in 1905, deals with the physics of the movement of bodies in the absence of gravitational forces, in which Maxwell’s equations of electromagnetism were made compatible with a reformulation of the laws of motion.

The theory of general relativity, published in 1915, is a theory of gravity that replaces Newtonian gravity, although it coincides numerically with gravity for weak gravitational fields and “small” velocities. The general theory is reduced to the special theory in the absence of gravitational fields.

On March 7, 2010, the Israeli Academy of Sciences publicly exhibited Einstein’s original manuscripts (written in 1905). The document, which contains 46 pages of texts and mathematical formulas written by hand, was donated by Einstein to the Hebrew University of Jerusalem in 1925 on the occasion of its inauguration.

The basic assumption of the theory of relativity is that the location of physical events, both in time and space, are relative to the state of movement of the observer : thus, the length of an object in motion or the moment in which something happens, unlike what happens in Newtonian mechanics, are not absolute invariants, and different observers in relative movement with each other will differ with respect to them (lengths and time intervals, in relativity, are relative and not absolute).

Formalism of the theory of relativity

Representation of the world line of a particle
Source https://en.wikipedia.org/wiki/File:Brane-wlwswv.png

(Representation of the world line of a particle. As it is not possible to reproduce a four-dimensional space-time, in the figure only the projection on two spatial dimensions and one temporal is represented. )

Particles

In the theory of relativity a point particle is represented by a pair (γ(τ),m), where γ(τ) it is a differentiable curve, called the world’s line of the particle, and m is a scalar that represents the mass at rest. The vector tangent to this curve is a temporary vector called quadripole, the product of this vector by the mass at rest of the particle is the quadrimomentum. This quadrimomentum is a vector of four components, three of these components are called spatial and represent the relativistic analogue of the linear momentum of classical mechanics, the other component called temporal component represents the relativistic generalization of kinetic energy. Furthermore, given an arbitrary curve in spacetime, the so-called relativistic interval , which is obtained from the metric tensor, can be defined along it . The relativistic interval measured along the path of a particle is proportional to the time interval itself or time interval perceived by that particle.

Fields

When considering fields or continuous distributions of mass, some kind of generalization is needed for the notion of particle. A physical field has momentum and energy distributed in spacetime, the concept of quadrimomentum is generalized by the so-called energy-impulse tensor that represents the distribution in spacetime of both energy and linear momentum. In turn, a field depending on its nature can be represented by a scalar, a vector or a tensor. For example, the electromagnetic field is represented by a second order fully antisymmetric or 2-way tensor. If the variation of a field or a distribution of matter is known, in space and time then there are procedures to build your energy-impulse tensor.

Physical magnitudes

In relativity, these physical magnitudes are represented by 4-dimensional vectors or by mathematical objects called tensors, which generalize the vectors, defined over a space of four dimensions. Mathematically, these 4-vectors and 4-tensors are defined elements of the vector space tangent to spacetime (and tensors are defined and constructed from the tangent or cotangent bundle of the manifold that represents spacetime).

Correspondence between E3 and M4:

  • Euclidean three-dimensional space >>> Minkowski spacetime
  • Point >>> Event
  • Length >>> Interval
  • Speed >>> Quad speed
  • Momentum >>> Quadrimomentum

In addition to quadrivectors, quadritensors are defined (ordinary tensors defined on the tangent bundle of spacetime conceived as a Lorentzian manifold). The curvature of spacetime is represented by a 4-tensor (fourth-order tensor), while the energy and momentum of a continuous medium or electromagnetic field are represented by 2-tensors (symmetric energy-impulse tensor, antisymmetric electromagnetic field). The quadrivectors are in fact 1-tensors, in this terminology. In this context it is said that a magnitude is a relativistic invariant if it has the same value for all observers, obviously all the relativistic invariants are scalar (0-tensors), frequently formed by the contraction of tensor magnitudes.

The relativistic interval

The relativistic interval can be defined in any spacetime, be this plane as in special relativity, or curved as in general relativity. However, for simplicity, we will initially discuss the concept of interval for the case of a flat spacetime. Minkowski’s flat spacetime metric tensor is designated with the letter ηij.

(Reproduction of a cone of light, in which two spatial dimensions and one temporal are represented (axis of ordinates). The observer is situated at the origin, while the absolute future and past are represented by the lower and upper parts of the temporal axis. The plane corresponding to t = 0 is called the simultaneity plane or the present hypersurface (also called the “Minkowski diagram”). The events within the cones are linked to the observer by time intervals. Those who are outside, by spatial intervals. )

The intervals can be classified into three categories: space intervals, temporary intervals and null intervals. The null intervals are those that correspond to particles that move at the speed of light, such as photons (the distance traveled by the photon equals its speed (c) multiplied by time).

The null intervals can be represented in the form of a cone of light, popularized by the famous book by Stephen Hawking, History of Time. Be an observer located at the origin, the absolute future (the events that will be perceived by the individual) is displayed on the top of the axis of ordinates, the absolute past (the events that have already been perceived by the individual) in the part lower, and the present perceived by the observer in point 0. The events that are outside the cone of light do not affect us, and therefore they are said to be located in areas of spacetime that have no causal relationship with ours.

Imagine, for a moment, that in the Andromeda galaxy, located 2.5 million light-years away from us, a cosmic cataclysm happened 100,000 years ago. Since, first: the light of Andromeda takes two million years to reach us and second: nothing can travel at a speed higher than that of photons, it is clear that we have no way of knowing what happened in said Galaxy only 100 000 years ago. It is said therefore that the interval between said hypothetical cosmic catastrophe and us, observers of the present, is a spatial interval, and therefore can not affect the individuals who live in the Earth at present: that is, there is no causal relationship between that event and us.

(Image of the Andromeda galaxy, taken by the Spitzer telescope, as it waAndromeda galaxys 2.5 million years ago (being located at 2.5 million light years). The events that occurred 1,000,000 years ago will be observed from Earth in one and a half million years. It is therefore said that between such events and us there is a spatial interval.)

Analysis

The only problem with this hypothesis is that when entering a black hole, spacetime is annulled, and as we already know, something that contains some volume or mass, must have at least a space to be located, time in that case does not matter, but space plays a very important role in the location of volumes, so this is very unlikely, but not impossible for technology.

We can choose another historical episode that is even more illustrative: that of the star of Bethlehem, as interpreted by Johannes Kepler. This German astronomer considered that this star was identified with a supernova that took place in the year 5 BC, whose light was observed by contemporary Chinese astronomers, and which was preceded in previous years by several planetary conjunctions in the Pisces constellation. That supernova probably exploded thousands of years ago, but its light did not reach the earth until the year 5 BC. Hence, the interval between that event and the observations of the Egyptian and megalithic astronomers (which took place several centuries before Christ) is a spatial interval, because the radiation of the supernova could never reach them. On the contrary, the explosion of the supernova on the one hand, and the observations made by the three magicians in Babylon and by Chinese astronomers in the year 5 BC, on the other hand, are linked together by a temporary interval , since the light could reach these observers.

The own time and the interval are related by the following equivalence: cdr = ds, that is, the interval is equal to the local time multiplied by the speed of light. One of the characteristics of both the local time and the interval is its invariance for the coordinate transformations. Whatever our point of reference, whatever our speed, the interval between a certain event and us remains invariant.

This invariance is expressed through the so-called hyperbolic geometry : The interval equation has the structure of a hyperbola in four dimensions, whose independent term coincides with the value of the square of the interval, which, as was just said in the previous paragraph, is constant. The asymptotes of the hyperbola would come to coincide with the cone of light.

Quadrivelocity, acceleration and quadrimomentum

In the Minkowski space, the kinematic properties of the particles is mainly represented by three factors: The quadrivelocity (or tetravelocity), the quadriacceleration and quadrimomentum (or tetramomentum).

The quadrivelocity is a quadrivector tangent to the world line of the particle, related to the coordinate speed of a body measured by an observer at any rest. The Newtonian measure of speed is not useful in relativity theory, because the Newtonian velocities measured by different observers are not easily related because they are not covariant magnitudes. Thus in relativity a modification is introduced in the expressions that account for the speed, introducing a relativistic invariant. This invariant is the proper time of the particle that is easily related to the coordinate time of different observers.

The coordinate velocity of a body with mass depends whimsically on the reference system that we choose, while the quadrivelocity itself is a quantity that is transformed according to the principle of covariance and always has an constant value equivalent to the interval divided by the own time, or, what is the same, at the speed of light c. For massless particles, such as photons, the above procedure can not be applied, and quadrivelocity can be defined simply as a vector tangent to the trajectory followed by them.

The four-acceleration can be defined as the time derivative of the quadrivelocity. Its magnitude is equal to zero in the inertial systems, whose world lines are geodesic, straight in the plain space-time of Minkowski. On the contrary, the curved world lines correspond to particles with acceleration different from zero, to non-inertial systems.

Along with the principles of invariance of interval and quadripolarity, the law of conservation of the quadrimomentum plays a fundamental role. The Newtonian definition of momentum is applicable here as the mass (in this case preserved) multiplied by the speed (in this case, the quadrivelocity). The amount of momentum conserved is defined as the square root of the norm of the quadrimomentum vector. The conserved momentum, as well as the interval and its own quadrivelocity, remains invariant to the transformations of coordinates , although here we also have to distinguish between bodies with mass and photons. In the former, the magnitude of the quadriomentum is equal to the mass multiplied by the speed of light. On the contrary, the conserved quadrimomentum of the photons is equal to the magnitude of its three-dimensional momentum.

As both the speed of light and the quadrimomentum are conserved magnitudes, so is their product, resulting the conserved energy for bodies E = mc2, the famous formula of Einstein, and in photons E = pc.

The appearance of special relativity put an end to the secular dispute that the mechanistic and energetic schools maintained within classical mechanics. The first argued, following Descartes and Huygens, that the magnitude conserved in all movement was constituted by the total momentum of the system, while the others -which were based on the studies of Leibniz- considered that the conserved magnitude was the sum of two quantities: live force , equivalent to half the mass, multiplied by the speed squared, mv2/2, which we would now call “kinetic energy”, and the dead force, equivalent to the height multiplied by the constant g (hg), which would correspond to the “potential energy”. It was the German physicist Hermann von Helmholtz who first gave the Leibnizian forces the generic denomination of energy and who formulated the law of conservation of energy , which is not restricted to mechanics, which also extends to other physical disciplines such as thermodynamics.

Newtonian mechanics gave the reason to both postulates, affirming that both momentum and energy are conserved magnitudes in all movement subjected to conservative forces. However, special relativity went a step further, since from the work of Einstein and Minkowski, momentum and energy ceased to be considered as independent entities and they were considered as two aspects, two facets of a single conserved magnitude: the quadrimomentum.

The stress-energy tensor (Tab)
The stress-energy tensor
Source https://en.wikipedia.org/wiki/File:StressEnergyTensor.PNG

(The stress-energy tensor. )

Three are the fundamental equations that in Newtonian physics describe the phenomenon of universal gravitation : the first, affirms that the gravitational force between two bodies is proportional to the product of their masses and inversely proportional to the square of their distance (1); the second, that the gravitational potential at a certain point is equal to the mass multiplied by the constant G and divided by the distance r (2); and the third, finally, is the so-called Poisson equation (3), which indicates that the Laplacian of the gravitational potential is equal to  a 4πGρ, where ρ is the density of mass in a certain spherical region.

However, these equations are not compatible with special relativity for two reasons:

  • First, the mass is not an absolute magnitude, but its measurement results in different results depending on the relative speed of the observer. Hence, the density of mass cannot serve as a parameter of gravitational interaction between two bodies.
  • Secondly, if the concept of space is relative, so is the notion of density. It is evident that the contraction of the space produced by the increase of the speed of an observer, prevents the existence of densities that remain invariable to the Lorentz transformations.
The electromagnetic tensor (Fab)

The equations deduced by the Scottish physicist James Clerk Maxwell showed that electricity and magnetism are not more than two manifestations of the same physical phenomenon: the electromagnetic field. Now, to describe the properties of this field, physicists of the late nineteenth century had to use two different vectors, the corresponding electric and magnetic fields.

It was the arrival of special relativity that allowed describing the properties of electromagnetism with a single geometric object, the quadripotential vector, whose temporal component corresponded to the electrical potential, while its spatial components were the same as those of the magnetic potential.

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