Science

# Special Relativity, General Relativity and Lorentz Transformation

## General theory of relativity

The General Theory of Relativity is the accepted name for the gravitational theory published by Albert Einstein in 1915. According to the general theory of relativity, the force of gravity is a manifestation of the local geometry of spacetime. Although modern theory is due to Einstein, its origins lie in the axioms of Euclidean geometry and the many attempts to prove, throughout the centuries, Euclid’s fifth postulate, which says that parallel lines always remain equidistant, and that culminated in the finding by Bolyai and Gauss that this axiom is not necessarily true. The general mathematics of non-Euclidean geometry were developed by Riemann, a disciple of Gauss;

Gauss showed that there is no reason why the geometry of space should be Euclidean, which means that if a physicist puts a mark, and a cartographer remains at a certain distance and its length is measured by triangulation based on Euclidean geometry, then no The same answer is guaranteed to be given if the physicist carries the mark with him and measures his length directly. Of course, the difference between the two measures could not be measured in practice, but there are equivalent measures that must detect the non-Euclidean geometry of space-time directly, for example the Pound-Rebka experiment (1959) detected the change in the wavelength of light from a cobalt source emerging by 22.5 meters against gravity at a local Jefferson Physics Laboratory at Harvard,

The fundamental idea in relativity is that we cannot talk about the physical amounts of velocity or acceleration without first defining their reference system. And said reference system is defined by particular choice. In that case, every movement is defined and quantified relatively to another subject. In the special theory of relativity it is assumed that reference systems can be extended indefinitely in all directions in spacetime. But in the general theory it is recognized that only the definition of approximate systems is possible locally and for a finite time for finite regions of space (similar to how we can draw flat maps of regions of the earth’s surface but cannot extend them to cover the surface of the entire earth without distortion). In general relativity, Newton’s laws are assumed only in relation to local reference systems. In particular, free particles travel by drawing straight lines in local inertial systems (Lorentz). When those lines extend, they do not appear as straight lines, being geodesic calls. Then, Newton’s first law is replaced by the law of the geodesic movement.

We distinguish inertial reference systems, in which bodies maintain a uniform movement without the action of or on other bodies, from non-inertial reference systems in which bodies that move freely suffer acceleration derived from the reference system itself. In non-inertial reference systems, force derived from the reference system is perceived, not by the direct influence of another matter. We feel “gravitational” forces when we go in a car and turn in a curve as the physical basis of our reference system. Similarly, the Coriolis effect and centrifugal force act when we define reference systems based on rotating matter (such as the Earth or a spinning child). The principle of equivalence in general relativity establishes that there are no local experiments that are able to distinguish a non-rotational fall in a gravitational field from uniform motion in the absence of a gravitational field. That is, there is no gravity in a free fall reference system. From this perspective, the gravity observed on the Earth’s surface is the force observed in a reference system defined by matter on the surface that is not free (is bound) but is activated downward by Earth’s matter, and is analogous to the “gravitational” force felt in a car giving a curve. there is no gravity in a free fall reference system. From this perspective, the gravity observed on the Earth’s surface is the force observed in a reference system defined by matter on the surface that is not free (is bound) but is activated downward by Earth’s matter, and is analogous to the “gravitational” force felt in a car giving a curve. there is no gravity in a free fall reference system. From this perspective, the gravity observed on the Earth’s surface is the force observed in a reference system defined by matter on the surface that is not free (is bound) but is activated downward by Earth’s matter, and is analogous to the “gravitational” force felt in a car giving a curve.

Mathematically, Einstein modeled space-time by a pseudo-Riemanian variety, and its field equations establish that the curvature of the variety at one point is directly related to its energy tensor at that point; said tensor is a measure of the density of matter and energy. The curvature tells the matter how to move, and in a reciprocal way the matter tells the space how to bend. The possible field equation is not unique, with the possibility of other models without contradicting the observation. The general relativity is distinguished from other theories of gravity by the simplicity of coupling between matter and curvature, although its unification with quantum mechanics and the replacement of the field equation with an adequate quantum law has not yet been resolved. Few physicists doubt that such a theory,

The Einstein field equation contains a parameter called “cosmological constant”? which was originally introduced by Einstein to allow a static universe. This effort was unsuccessful for two reasons: the instability of the universe resulting from such theoretical efforts, and the observations made by Hubble a decade later confirm that our universe is in fact not static but expanding. So ? It was abandoned, but quite recently, astronomical techniques found that a nonzero value for? It is necessary to be able to explain some observations.

The field equations are read as follows:

where R i k is the Ricci curvature tensor, R is the Ricci curvature scalar,g i k is the metric tensor, it is the cosmological constant,T i k is the energy tensor, p is pi,c is the speed of light in a vacuum and G is the universal gravitational constant, similar to what happens in Newtonian gravity.g i kIt describes the variety metric and is a 4×4 symmetric tensor, so it has 10 independent components. Given the freedom of choice of the four spatio-temporal coordinates, the independent equations are reduced to six.

## Special theory of relativity

The Theory (Special or Restricted) of Relativity (in short, special or restricted relativity, RE), first published by Albert Einstein in 1905, describes the physics of movement in the absence of gravitational fields. Before it, most physicists thought that Isaac Newton’s classical mechanics, based on the so-called Galileo relativity (origin of the mathematical equations known as Galileo transformations) described the concepts of speed and force for all observers ( or reference systems). However, Hendrik Lorentz and others had verified that Maxwell’s equations, which govern electromagnetism, did not behave according to Newton’s laws when the reference system changes (for example, when considering the same physical problem from the point of view of two observers who move with respect to each other). The notion of transformation of the laws of physics with respect to observers is the one that gives name to the theory, which adjusts with the qualification of special or restricted by sticking to cases of systems in which gravitational fields are not taken into account. An extension of this theory is the general theory of relativity, also published by Einstein in 1916 and including such fields.

Motivation of the theory

Newton’s laws consider that time and space are the same for different observers of the same physical phenomenon. Prior to the formulation of the special theory of relativity, Hendrik Lorentz and others had already discovered that electromagnetism differed from Newtonian physics in that observations of a phenomenon could differ from one person to another that was moving relatively to the first at close speeds. to those of the light. Thus, one can observe the absence of a magnetic field while the other observes that field in the same physical space.

Lorentz suggested an ether theory in which objects and observers would travel through a stationary ether, suffering a physical shortening (Lorentz contraction hypothesis) and a change in the passage of time (time dilation). This provided a partial reconciliation between Newtonian physics and electromagnetism, which were conjugated by applying the Lorentz transformation, which would replace the transformation of Galileo in force in the Newtonian system. When the speeds involved are much less than c (the speed of light), the resulting laws are in practice the same as in Newton’s theory, and the transformations are reduced to those of Galileo. In any case, the ether theory was criticized even by Lorentz himself because of its ad hoc nature.

When Lorentz suggested its transformation as an accurate mathematical description of the results of the experiments, Einstein derived these equations from two fundamental hypotheses: the constancy of the speed of light, c, and the need for the laws of physics to be equal ( invariants in different inertial systems, that is, for different observers.From this idea came the original title of the theory,? Theory of the invariants ?. It was Max Planck who subsequently suggested the term “relativity” to highlight the notion of transformation of the laws of physics among observers moving relatively together.

Special relativity studies the behavior of objects and observers who remain at rest or move with uniform motion (ie, constant relative velocity). In this case, it is said that the observer is in an inertial reference system. The comparison of spaces and times between inertial observers can be performed using Lorentz transformations. The special theory of relativity could also predict the behavior of accelerated bodies when such acceleration does not involve gravitational forces, in which case general relativity is necessary

Invariance of the speed of light

To support the ER, Einstein postulated that the speed of light in a vacuum is the same for all inertial observers. Likewise, he stressed that all physical theory must be described by laws that have a similar mathematical form in any inertial reference system. The first postulate is in accordance with Maxwell’s equations of electromagnetism, and the second uses a logical reasoning principle, in the form of the anthropic principle.

Einstein showed that from these principles Lorentz’s equations are deduced, and, when applied under these concepts, the resulting mechanics have several interesting properties:

When the speeds of the objects considered are much lower than the speed of light, the resulting laws are those described by Newton.

Likewise, electromagnetism is no longer a set of laws that require a different transformation from that applied in mechanics.

Time and space cease to be invariant when changing the reference system, becoming dependent on the relative speeds of the observer reference systems: Two events that occur simultaneously in different places for a reference system, can occur in different times in another reference system (simultaneity is relative). Similarly, if they occur in one place in one system, they can occur in different places in another.

The time intervals between events depend on the reference system in which they are measured (for example, the famous paradox of the twins. The distances between events, too.

The first two properties were very attractive, since any new theory must explain the existing observations, and these indicated that Newton’s laws were very precise. The third conclusion was initially much discussed, since it threw down many well-known and apparently obvious concepts, such as the concept of simultaneity.

Absence of an absolute reference system

Another consequence is the rejection of the notion of a single and absolute reference system. Previously it was believed that the universe traveled through a substance known as ether (identifiable as absolute space) in relation to which velocities could be measured. However, the results of several experiments, which culminated in the famous Michelson-Morley experiment, suggested that either the Earth was always stationary (which is absurd), or the notion of an absolute reference system was wrong and should If discarded Einstein concluded with the special theory of relativity that any movement is relative, there being no universal concept of “stationary.”

## Mass and energy equivalence

famous formula on a German postal stamp

But perhaps much more important was the demonstration that energy and mass, previously considered differentiated measurable properties, were equivalent, and related through what is undoubtedly the most famous equation of the theory:

E = m · c 2

where E is the energy, m is the mass and c is the speed of light in a vacuum. If the body is moving at the velocity v relative to the observer, the total energy of the body is:

where

The term ? It is frequent in relativity. It is derived from the Lorentz transformation equations. When v is much smaller than c can the following approximation of? (obtained by Taylor’s serial development):

so,

which is precisely equal to resting energy, mc 2, plus Newtonian kinetic energy, ½mv 2. This is an example of how the two theories coincide when the speeds are small.

In addition, at the speed of light, the energy will be infinite, which prevents particles that have resting mass from reaching the speed of light.

The most practical implication of the theory is that it puts an upper limit on the laws of classical mechanics and gravity proposed by Isaac Newton when the velocities approach those of light. Nothing that can carry mass or information can move faster than that speed. When an object approaches the speed of light (in any system) the amount of energy required to continue increasing its speed increases rapidly and asymptotically towards infinity, making it impossible to reach the speed of light. Only particles without mass, such as photons, can reach that speed (and in fact they must be transferred in any reference system at that speed) that is approximately 300,000 kilometers per second (3·108 ms -1).

The name tachyon has been used to name hypothetical particles that could move faster than the speed of light. At present, no experimental evidence of its existence has yet been found.

Special relativity also shows that the concept of simultaneity is relative to the observer: If matter can travel along a line (trajectory) in spacetime without changing speed, the theory calls this time interval line , since an observer following this line could not feel movement (would be at rest), but only travel in time according to their reference system. Similarly, a spatial intervalIt means a straight line in space-time along which neither light nor another slower signal could travel. Events over a spatial interval cannot influence each other by transmitting light or matter, and may appear as simultaneous to an observer in a suitable reference system. For observers in different reference systems, event A may appear prior to B or vice versa. This does not happen when we consider events separated by time intervals.

Special Relativity is universally accepted by the physical community today, contrary to the general relativity that is confirmed, but with experiences that might not exclude some alternative theory of gravitation. However, there is still a group of people opposed to the RE in several fields, several alternatives having been proposed, such as the so-called Ether Theories.

## The theory

The RE uses tensors or quadrivectors to define a non-Euclidean space. This space, however, is similar in many ways and easy to work with. The distance differential (ds) in a Euclidean space is defined as:

where dx 1, dx 2, dx 3 are differential of the three spatial dimensions. In the geometry of special relativity, a fourth dimension, time, has been added, but it is treated as an imaginary quantity with units of c, leaving the equation for distance, differentially, as:

Dual Cone

If we reduce the spatial dimensions to 2, we can make a physical representation in a three-dimensional space,

ds 2= dx 12+ dx 22-c 2 dt 2

We can see that geodesics with zero measurement form a dual cone:

defined by the equation

ds 2=0= dx 12+ dx 22-c 2 dt 2

or

dx 12+ dx 22= c 2 dt 2

The previous equation is that of circle with r = c * dt. If we extend the above to the three spatial dimensions, the zero geodesics are concentric spheres, with radius = distance = c *(+ or -) time.

Geodetic

This double cone of null distances represents the “horizon of vision” of a point in space. That is, when we look at the stars and say “The star from which I am receiving light is X years old.”, We are traveling through that line of sight: a geodetic of zero distance. We are seeing an event at d =?X 12+ x 22+ x 32 meters, and d / c seconds in the past. For this reason the double cone is also known as the cone of light.(The lower point on the left of the lower diagram represents the star, the origin represents the observer and the line represents the zero geodetic, the “horizon of vision” or cone of light)

Horizon of vision

Geometrically, all the “points” along the cone of light give information (represent) the same point in spacetime (because the distance between them is 0). This can be thought of as a ‘neutralization point’ of forces.(“The connection occurs when two movements, each of which excluding the other, come together in a moment.”- quote from James Morrison) It is where events in space-time intersect, where space interacts with itself. It’s like a point sees the rest of the universe and is seen. The cone in the -t region includes the information the point receives, while the + t regionof the cone includes the information that the point sends. In this way, what we can see is a space of vision horizons and fall back on the concept of cellular automaton, applying it in a continuous space-time sequence.

Space of vision horizons.

Inertial systems.

This also counts for uniform moving points of relative translation inertial systems . Which means that the geometry of the universe remains the same regardless of the speed(? X /? T) ( inertial ) of the observer. Thus we fall into Newton’s first law of motion:“An object in motion tends to remain in motion; an object at rest tends to remain at rest

Law of conservation of kinetic energy

However, geometry does not remain constant when acceleration is involved (? X 2/? T 2), which implies an application of force (F = ma), and consequently a change in energy, which leads us to relativity In general , in which the intrinsic curvature of spacetime is directly proportional to the energy density at that point.

## Modifications of special relativity

At the beginning of the 21st century a certain number of modified versions of special relativity have been postulated.

## Postulate tests of special relativity

• Michelson-Morley experiment (ether drag)
• Hamar experiment (blockage of ether flow)
• Trouton-Noble experiment (torque in a capacitor)
• Kennedy-Thorndike experiment (time contraction)
• Experiment on emission forms

## Lorentz Transformation

The Lorentz transformations, within the theory of special relativity, are a set of relationships that account for how the measurements of a physical magnitude obtained by two different observers are related. These relationships established the mathematical basis of Einstein’s theory of special relativity, since Lorentz’s transformations specify the type of spacetime geometry required by Einstein’s theory.

Historically, Lorentz’s transformations were introduced by Hendrik Antoon Lorentz (1853-1928), who had introduced them phenomenally to solve certain inconsistencies between electromagnetism and classical mechanics. Lorentz had discovered in 1900 that Maxwell’s equations were invariant under this set of transformations, now called Lorentz transformations. Like the other physicists, before the development of the theory of relativity, he assumed that the invariant speed for the transmission of electromagnetic waves referred to the transmission through a privileged reference system, a fact that is known by the name of hypothesis of the ether. But nevertheless,

Lorentz’s transformations were published in 1904 but his initial mathematical formalism was incorrect. The French mathematician Poincaré developed the set of equations in the consistent manner in which they are known today. The works of Minkowski and Poincaré showed that Lorentz’s relations could be interpreted as the transformation formulas for rotation in four-dimensional space-time, which had been introduced by Minkowski.

The Lorentz transformations The Lorentz transformations relate the measurements of a physical magnitude made by two different inertial observers, being the relativistic equivalent of the Galileo transformation used in physics until then.

The Lorentz transformation allows the value of constant light speed to be preserved for all inertial observers.

Lorentz transformations of the coordinates One of the consequences that – unlike what happens in classical mechanics – in relativistic mechanics there is no absolute time, is that both the time interval between two events, and the effective distances measured by different Observers in different states of motion are different. This implies that the coordinates of time and space measured by two inertial observers differ from each other. However, due to the objectivity of physical reality, the measurements of both observers are related by fixed rules: Lorentz transformations for coordinates.

To examine the concrete form that these transformations of the coordinates take, two inertial reference systems or inertial observers are considered: and and it is assumed that each of them represents the same event S or point of space-time (representable for a moment of time and three spatial coordinates) by two different coordinate systems:

Since the two sets of four coordinates represent the same point of spacetime, they must be relatable in some way. Lorentz transformations say that if the system is in uniform motion at velocity along the X axis of the system and at the initial moment () the origin of the coordinates of both systems coincide, then the coordinates attributed by the two observers are related by The following expressions:

Or equivalently for the inverse relationships of the previous ones:

The previous Lorentz transformation takes that form on the assumption that the coordinate origin of both reference systems is the same for t=0; If this restriction is eliminated, the concrete form of the equations becomes complicated. If, in addition, the restriction that the relative velocity between the two systems is given along the X axis and that the axes of both coordinate systems are parallel is eliminated, the Lorentz transformation expressions are further complicated, denominating the expression general transformation of Poincaré.

### Lorentz transformations for momentum and energy

The covariance requirement of the theory of relativity requires that any vector magnitude of Newtonian mechanics be represented in relativistic mechanics by a quadrisect or quadritensor in relativity theory. Thus, the linear momentum needs to be extended to a quadrivector called quadrivector energy-momentum or quadrimoment , which is given by four components, a temporal component (energy) and three spatial components (linear moments in each coordinate direction):

When examining the quadrimoments measured by two inertial observers, it is found that both measure different components of the moment according to their velocity relative to the observed particle (something that also happens in Newtonian mechanics). If the quadrimoment measured by two inertial observers and with Cartesian coordinate systems of parallel axes and in relative motion along the X axis are denoted , as those considered in the previous section, the quadrimoments measured by both observers are related by a transformation of Lorentz given by:

And the inverse transformation is similarly given by:

Or equivalently in matrix form the two previous sets of equations are represented as:

Where the Lorentz factor and the relative velocity with respect to light have been introduced again for short.

Until now, only inertial systems in relative motion with respect to the X axis have been considered, but equally parallel axis systems with respect to the Y and Z axes could have been considered and, in that case, the coordinate transformation matrices would be given by similar matrices to those considered in the previous sections of the form:

The above transformations are sometimes called boosts, space-time rotations or sometimes Lorentz transformations themselves. The product of any number of transformations of the above type also constitutes a Lorentz transformation. All these products make up a subgroup of Lorentz’s own group. In general, Lorentz’s own group consists of:

• Spatio-temporal rotations or boosts, which can be written as the product of a finite number of boosts of type [*].
• Spatial rotations, consisting of an axis rotation. This type of transformation is also part of the Galileo group.

The Lorentz group defined in this way is a related Lie group. If improper transformations such as temporal inversions and spatial reflections are added to these own transformations , the entire Lorentz group is formed, consisting of four related components each homeomorphic to the Lorentz group itself. Once the Lorentz group is defined we can write the most general linear transformations possible between measurements taken by inertial observers whose coordinate axes coincide at the initial moment:

Where in addition to the boost given by the coordinate transformation according to the relative separation speed, the two rotations in terms of Euler’s angles have been included:

• The matrix R (?1,?2,?3) aligns the first coordinate system so that the transformed X axis becomes parallel to the separation speed of the two systems.
• The matrix R (?1?2,?3) is the inverse rotation of which the X axis of the second observer would align with the separation speed.

In a more compact form we can write the last transformation in tensor form using the Einstein summation agreement as:

\ rho ^

\ sigma [\ Lambda_ { (X)}]

\ sigma ^

\ rho {[R (\ varphi)]}

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\ beta ^

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General tensor form of Lorentz transformations

Suppose now that instead of measuring vector magnitudes, two observers measure the components of some other tensorial magnitude, suppose that the observers and measure in their coordinate systems the same tensor magnitude but each their own coordinate system reaching:

The postulate that there is an objective reality independent of the observers and that their measurements can be compared by means of appropriate covariance transformations leads to the fact that if these observers are inertial, their measures will be related by the following relationships:

Where the matrices? They are defined, like the previous section by the product of two spatial rotations and a simple temporal rotation (boost).

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