Special relativity

[p 1] Maxwell's equations of electromagnetism appeared to be incompatible with Newtonian mechanics, and the Michelson–Morley experiment failed to detect the Earth's motion against the hypothesized luminiferous aether.

Special relativity corrects the hitherto laws of mechanics to handle situations involving all motions and especially those at a speed close to that of light (known as relativistic velocities).

[3][4] Even so, the Newtonian model is still valid as a simple and accurate approximation at low velocities (relative to the speed of light), for example, everyday motions on Earth.

As long as the universe can be modeled as a pseudo-Riemannian manifold, a Lorentz-invariant frame that abides by special relativity can be defined for a sufficiently small neighborhood of each point in this curved spacetime.

In his initial presentation of special relativity in 1905 he expressed these postulates as:[p 1] The constancy of the speed of light was motivated by Maxwell's theory of electromagnetism[13] and the lack of evidence for the luminiferous ether.

The derivation of special relativity depends not only on these two explicit postulates, but also on several tacit assumptions (made in almost all theories of physics), including the isotropy and homogeneity of space and the independence of measuring rods and clocks from their past history.

The term reference frame as used here is an observational perspective in space that is not undergoing any change in motion (acceleration), from which a position can be measured along 3 spatial axes (so, at rest or constant velocity).

Although it is not as easy to perform exact computations using them as directly invoking the Lorentz transformations, their main power is their ability to provide an intuitive grasp of the results of a relativistic scenario.

are related to the unprimed coordinates through the Lorentz transformations and could be approximately measured from the graph (assuming that it has been plotted accurately enough), but the real merit of a Minkowski diagram is its granting us a geometric view of the scenario.

⁠, there are three cases to note:[21][27]: 25–39 The interweaving of space and time revokes the implicitly assumed concepts of absolute simultaneity and synchronization across non-comoving frames.

Among his numerous contributions to the foundations of special relativity were independent work on the mass–energy relationship, a thorough examination of the twin paradox, and investigations into rotating coordinate systems.

His name is frequently attached to a hypothetical construct called a "light-clock" (originally developed by Lewis and Tolman in 1909[32]), which he used to perform a novel derivation of the Lorentz transformation.

The concept of time dilation is frequently taught using a light-clock that is traveling in uniform inertial motion perpendicular to a line connecting the two mirrors.

Although observer A is traveling rapidly along a train, from her point of view the emission and receipt of the pulse occur at the same place, and she measures the interval using a single clock located at the precise position of these two events.

[39] The reciprocity of time dilation between two observers in separate inertial frames leads to the so-called twin paradox, articulated in its present form by Langevin in 1911.

This result appears puzzling because both the traveler and an Earthbound observer would see the other as moving, and so, because of the reciprocity of time dilation, one might initially expect that each should have found the other to have aged less.

These time intervals (which can be, and are, actually measured experimentally by relevant observers) are different in another coordinate system moving with respect to the first, unless the events, in addition to being co-local, are also simultaneous.

Interpreted in such a fashion, they are commonly referred to as the relativistic velocity addition (or composition) formulas, valid for the three axes of S and S′ being aligned with each other (although not necessarily in standard configuration).

The classical calculation of the displacement takes two forms and makes different predictions depending on whether the receiver, the source, or both are in motion with respect to the medium.

Nevertheless, relativistic Doppler shift for the longitudinal case, with source and receiver moving directly towards or away from each other, can be derived as if it were the classical phenomenon, but modified by the addition of a time dilation term, and that is the treatment described here.

where An identical expression for relativistic Doppler shift is obtained when performing the analysis in the reference frame of the receiver with a moving source.

Although, as discussed above, subsequent scholarship has established that his arguments fell short of a broadly definitive proof, the conclusions that he reached in this paper have stood the test of time.

Besides the vigorous debate that continues until this day as to the formal correctness of his original derivation, the recognition of special relativity as being what Einstein called a "principle theory" has led to a shift away from reliance on electromagnetic phenomena to purely dynamic methods of proof.

The γ factor can be written as Transformations describing relative motion with uniform velocity and without rotation of the space coordinate axes are called boosts.

For instance, demonstrating relativistic invariance of Maxwell's equations in their usual form is not trivial, while it is merely a routine calculation, really no more than an observation, using the field strength tensor formulation.

But at macroscopic scales and in the absence of strong gravitational fields, special relativity is experimentally tested to extremely high degree of accuracy (10−20)[84] and thus accepted by the physics community.

This is true more generally: certain pairs of "timelike" and "spacelike" quantities naturally combine on equal footing under the same Lorentz transformation.

In Newtonian mechanics, quantities that have magnitude and direction are mathematically described as 3d vectors in Euclidean space, and in general they are parametrized by time.

The notation makes it clear the equations are manifestly covariant under the Poincaré group, thus bypassing the tedious calculations to check this fact.

Given the four-dimensional nature of spacetime the Minkowski metric η has components (valid with suitably chosen coordinates), which can be arranged in a 4 × 4 matrix:

Albert Einstein around 1905, the year his " Annus Mirabilis papers " were published. These included Zur Elektrodynamik bewegter Körper , the paper founding special relativity.
Figure 2–1. The primed system is in motion relative to the unprimed system with constant velocity v only along the x -axis, from the perspective of an observer stationary in the unprimed system. By the principle of relativity , an observer stationary in the primed system will view a likewise construction except that the velocity they record will be − v . The changing of the speed of propagation of interaction from infinite in non-relativistic mechanics to a finite value will require a modification of the transformation equations mapping events in one frame to another.
Figure 4–1. The three events (A, B, C) are simultaneous in the reference frame of some observer O . In a reference frame moving at v = 0.3 c , as measured by O , the events occur in the order C, B, A. In a reference frame moving at v = −0.5 c with respect to O , the events occur in the order A, B, C. The white lines, the lines of simultaneity , move from the past to the future in the respective frames (green coordinate axes), highlighting events residing on them. They are the locus of all events occurring at the same time in the respective frame. The gray area is the light cone with respect to the origin of all considered frames.
Figure 4–2. Hypothetical infinite array of synchronized clocks associated with an observer's reference frame
Figure 4–3. Thought experiment using a light-clock to explain time dilation
Figure 4-4. Doppler analysis of twin paradox
Figure 4–6. Light cone
Figure 5–1. Highly simplified diagram of Fizeau's 1851 experiment.
Figure 5–2. Illustration of stellar aberration
Figure 5–3. Transverse Doppler effect for two scenarios: (a) receiver moving in a circle around the source; (b) source moving in a circle around the receiver.
Figure 5–4. Comparison of the measured length contraction of a cube versus its visual appearance.
Figure 5–5. Comparison of the measured length contraction of a globe versus its visual appearance, as viewed from a distance of three diameters of the globe from the eye to the red cross.
Figure 5–6. Galaxy M87 sends out a black-hole-powered jet of electrons and other sub-atomic particles traveling at nearly the speed of light.
Figure 6–2. Newtonian analysis of the elastic collision of a moving particle with an equal mass stationary particle
Figure 6–3. Relativistic elastic collision between a moving particle incident upon an equal mass stationary particle
Figure 7–2. Plot of the three basic Hyperbolic functions : hyperbolic sine ( sinh ), hyperbolic cosine ( cosh ) and hyperbolic tangent ( tanh ). Sinh is red, cosh is blue and tanh is green.
Figure 7–4. Dewan–Beran–Bell spaceship paradox
Figure 7–5. The curved lines represent the world lines of two observers A and B who accelerate in the same direction with the same constant magnitude acceleration. At A' and B', the observers stop accelerating. The dashed lines are lines of simultaneity for either observer before acceleration begins and after acceleration stops.
Figure 7–6. Accelerated relativistic observer with horizon. Another well-drawn illustration of the same topic may be viewed here .
Figure 10–1. Orthogonality and rotation of coordinate systems compared between left: Euclidean space through circular angle φ , right: in Minkowski spacetime through hyperbolic angle φ (red lines labelled c denote the worldlines of a light signal, a vector is orthogonal to itself if it lies on this line). [ 87 ]
Figure 10–2. Three-dimensional dual-cone.
Figure 10–3. Concentric spheres, illustrating in 3-space the null geodesics of a 4-dimensional cone in spacetime.