Special theory of relativity

The Special theory of Relativity (shortly, Special or Restricted Relativity, RE), first published by Albert Einstein in 1905, describes the physics of motion in the absence of gravitational fields.

Before it, most physicists believed that Isaac Newton’s classical mechanics, based on the so-called relativity of Galileo (origin of the mathematical equations known as Galileo transformations), described the concepts of speed and force for all observers ( or reference systems). 

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However, Hendrik Lorentz and others had found that Maxwell’s equations, which govern electromagnetism, did not behave according to Newton’s laws when the reference system changes (for example, when the same physical problem is considered from the point of view of two observers moving with respect to each other).

It is the notion of transformation of the laws of physics with respect to observers that gives the theory its name, which is adjusted with the qualification of special or restricted because it is limited 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 these fields.

Motivation of the theory

Newton’s laws consider that time and space are the same for different observers of the same physical phenomenon. Before formulating 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 person to person moving relatively early on at near speeds to those of light. 

Thus, one can observe the absence of a magnetic field while the other observes said field in the same physical space.

Lorentz suggested an ether theory in which objects and observers would travel through a stationary ether, undergoing 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 combined by applying the Lorentz transformation, which would come to replace the Galilean transformation 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 theory of the ether was criticized even by Lorentz himself due to this nature.

When Lorentz suggested his 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 invariants”. 

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It was Max Planck who later suggested the term “relativity” to highlight the notion of transformation of laws of physics between observers moving relatively to each other.

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

Invariance of the speed of light

To support RE, 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 postulates a logical reasoning principle, in the form of the anthropic principle.

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

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

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

Time and space are no longer invariant when changing the reference system, becoming dependent on the relative speeds of the observers’ 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 the same place in one system, they can occur in different places in another.

The time intervals between events depend on the frame of reference in which they are measured (for example, the famous paradox of the twins). 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 highly disputed, as it threw out 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. It was previously believed that the universe traveled through a substance known as ether (identifiable as absolute space) in relation to which speeds could be measured. 

The results of various experiments, however, resulting in the famous Michelson-Morley experiment, proposed that either the Earth had been stationary (which is absurd), or the notion of an absolute reference system was incorrect and should be discarded. Einstein concluded with the special theory of relativity that any movement is relative, there being no universal concept of “stationary”.

Equivalence of mass and energy

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

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 speed v relative to the observer, the body’s total energy is:

E = gamma cdot m cdot c ^ 2

where

gamma = {1 over {sqrt {1- {v ^ 2 over c ^ 2}}}}

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

gamma = left ({1- {v ^ 2 over c ^ 2}} right) ^ {- {1 over 2}} = 1- left (- {1 over 2} right) cdot {v ^ 2 over c ^ 2 } + 0 cdot {v ^ 4 over c ^ 4} + ... approx 1 + {1 over 2} cdot {v ^ 2 over c ^ 2}

so,

 gamma cdot m cdot c ^ 2 approx m cdot c ^ 2 + {1 over 2} cdot m cdot v ^ 2

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

Furthermore, at the speed of light, the energy will be infinite, which prevents particles that have mass at rest from reaching the speed of light.

The most practical implication of the theory is that it places an upper limit on the laws of classical mechanics and gravity proposed by Isaac Newton when speeds approach those of light. Nothing that can transport 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 massless particles, such as photons, can reach that speed (and indeed must be translated in any reference system at that speed) which is approximately 300,000 kilometers per second (3 · 10 8 ms -1 ).

The name tachyon has been used to name hypothetical particles that could move faster than the speed of light. Currently, 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 space-time without changing speed, the theory calls this line a time interval. Since an observer following said line could not feel movement (it would be at rest), but only travel in time according to its reference system. Similarly, a spatial interval means a straight line in space-time along which neither light nor any slower signal could travel. 

Events throughout a spatial interval cannot influence each other by transmitting light or matter, and may appear simultaneous to an observer in a suitable frame of reference. To observers in different reference systems, event A may appear to precede 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, unlike confirmed general relativity , but with experiences that might not exclude some alternative theory of gravitation. However, there is still a group of people opposed to SR in various fields, several alternatives having been proposed, such as the so-called Ether Theories.

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The theory

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

ds 2 = dx 1 2 + dx 2 2 + dx 3 2

where dx 1 , dx 2 , dx 3 are differentials 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:

ds 2 = dx 1 2 + dx 2  + dx 3 2 -c 2 dt 2

Dual cone.

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

ds 2 = dx 1 2 + dx 2 2 -c 2 dt 2

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

defined by the equation

ds 2 = 0 = dx 1 2 + dx 2 2 -c 2 dt 2

, or

dx 1 2 + dx 2 2 = c 2 dt 2

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

Geodesic.

ds 2 = 0 = dx 1 2 + dx 2 2 + dx 3 2 -c 2 dt 2

dx 1 2 + dx 2 2 + dx 3 2 = c 2 dt 2

This double cone of zero distances represents the “horizon of vision” of a point in space i.e., when we look at the stars and say “The star I am receiving light from is X years old.”, We are traveling through that line of sight: a geodesic of zero distance. We are seeing an event at d =? {x 1 2 + x 2 2 + x 3 2 } meters, and d / c seconds in the past. 

For this reason the double cone is also known as the light cone. (The bottom point on the left of the bottom diagram represents the star, the origin represents the observer, and the line represents the null geodesic, the “horizon of vision” or light cone .)

Vision horizon.

Geometrically, all the “points” along the cone of light give information (represent) the same point in space-time (because the distance between them is 0). This can be thought of as a ‘point of neutralization’ of forces. (“The connection occurs when two movements, each of which exclusive of the other, come together in a moment.” – quote from James Morrison) 

It is where the events in space-time intersect, where space interacts with itself. It is like a point sees the rest of the universe and is seen. The cone in the -t region includes the information that the point receives, while the + t regionThe cone includes the information that the point sends. In this way, what we can envision is a space of horizons of vision 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 points in uniform motion of relative inertial systems . This means that the geometry of the universe remains the same regardless of the speed ( ? X /? T ) ( inertial ) of the observer. Thus we fall back on 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, the geometry does not remain constant when acceleration ( ? X 2 /? T 2 ) is involved, which implies an application of force ( F = ma ), and consequently a change in energy, which leads to general relativity, in which the intrinsic curvature of space-time is directly proportional to the energy density at that point.

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