Contents

Special relativity

Special relativity (SR) (also known as the special theory of relativityorSTR) is the physical theoryofmeasurementininertial frames of reference proposed in 1905 by Albert Einstein (after considerable contributions of Hendrik Lorentz and Henri Poincaré) in the paper "On the Electrodynamics of Moving Bodies".^[1] It generalizes Galileo's principle of relativity – that all uniform motion is relative, and that there is no absolute and well-defined state of rest (noprivileged reference frames) – from mechanics to all the laws of physics, including both the laws of mechanics and of electrodynamics, whatever they may be.^[2] In addition, special relativity incorporates the principle that the speed of light is the same for all inertial observers regardless of the state of motion of the source.^[3]

This theory has a wide range of consequences which have been experimentally verified, ^[4] including counter-intuitive ones such as length contraction, time dilation and relativity of simultaneity, contradicting the classical notion that the duration of the time interval between two events is equal for all observers. (On the other hand, it introduces the space-time interval, which is invariant.) Combined with other laws of physics, the two postulates of special relativity predict the equivalence of matter and energy, as expressed in the mass-energy equivalence formula E = mc², where c is the speed of light in a vacuum.^[5][6] The predictions of special relativity agree well with Newtonian mechanics in their common realm of applicability, specifically in experiments in which all velocities are small compared to the speed of light.

The theory is termed "special" because it applies the principle of relativity only to inertial frames. Einstein developed general relativity to apply the principle generally, that is, to any frame, and that theory includes the effects of gravity. Strictly, special relativity cannot be applied in accelerating frames or in gravitational fields.

Special relativity reveals that c is not just the velocity of a certain phenomenon, namely the propagation of electromagnetic radiation (light)—but rather a fundamental feature of the way space and time are unified as spacetime. A consequence of this is that it is impossible for any particle that has mass to be accelerated to the speed of light.

Postulates

In his autobiographical notes published in November 1949 Einstein described how he had arrived at the two fundamental postulates on which he based the special theory of relativity. After describing in detail the state of both mechanics and electrodynamics at the beginning of the 20th century, he wrote

"Reflections of this type made it clear to me as long ago as shortly after 1900, i.e., shortly after Planck's trailblazing work, that neither mechanics nor electrodynamics could (except in limiting cases) claim exact validity. Gradually I despaired of the possibility of discovering the true laws by means of constructive efforts based on known facts. The longer and the more desperately I tried, the more I came to the conviction that only the discovery of a universal formal principle could lead us to assured results... How, then, could such a universal principle be found?"^[7]

He discerned two fundamental propositions that seemed to be the most assured, regardless of the exact validity of either the (then) known laws of mechanics or electrodynamics. These propositions were:(1) the constancy of the velocity of light, and (2) the independence of physical laws (especially the constancy of the velocity of light) from the choice of inertial system. In his initial presentation of special relativity in 1905 he expressed these postulates as:^[8]

• The Principle of Relativity - The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems of inertial coordinates in uniform translatory motion.
• The Principle of Invariant Light Speed - Light in vacuum propagates with the speed c (a fixed constant) in terms of any system of inertial coordinates, regardless of the state of motion of the light source.

It should be noted that the derivation of special relativity depends not only on these two explicit postulates, but also on several tacit assumptions (which are 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.^[9]

Following Einstein's original presentation of special relativity in 1905, many different sets of postulates have been proposed in various alternative derivations.^[10] However, the most common set of postulates remains those employed by Einstein in his original paper. A more mathematical statement of the Principle of Relativity made later by Einstein, which introduces the concept of simplicity not mentioned above is:^[11]

Special principle of relativity: If a system of coordinates K is chosen so that, in relation to it, physical laws hold good in their simplest form, the same laws hold good in relation to any other system of coordinates K' moving in uniform translation relatively to K.

— Albert Einstein: The foundation of the general theory of relativity, Section A, §1

The two postulates of special relativity imply the applicability to physical laws of the Poincaré group of symmetry transformations, of which the Lorentz transformations are a subset, thereby providing a mathematical framework for special relativity. Many of Einstein's papers present derivations of the Lorentz transformation based upon these two principles.^[12]

Einstein consistently based the derivation of Lorentz invariance (the essential core of special relativity) on just the two basic principles of relativity and light-speed invariance. He wrote:

"The insight fundamental for the special theory of relativity is this: The assumptions relativity and light speed invariance are compatible if relations of a new type ("Lorentz transformation") are postulated for the conversion of coordinates and times of events... The universal principle of the special theory of relativity is contained in the postulate: The laws of physics are invariant with respect to Lorentz transformations (for the transition from one inertial system to any other arbitrarily chosen inertial system). This is a restricting principle for natural laws..."^[7]

Thus many modern treatments of special relativity base it on the single postulate of universal Lorentz covariance, or, equivalently, on the single postulate of Minkowski spacetime.^[13][14]

Mass-energy equivalence

In addition to the papers referenced above—which give derivations of the Lorentz transformation and describe the foundations of special relativity—Einstein also wrote at least four papers giving heuristic arguments for the equivalence (and transmutability) of mass and energy (the famous formula E = m c²).

Mass-energy equivalence does not follow from the two basic postulates of special relativity by themselves.^[15] The first of Einstein's papers on this subject was Does the Inertia of a Body Depend upon its Energy Content? in 1905.^[16] In this first paper and in each of his subsequent three papers on this subject,^[17] Einstein augmented the two fundamental principles by postulating the relations involving momentum and energy of electromagnetic waves implied by Maxwell's equations (the assumption of which, of course, entails among other things the assumption of the constancy of the speed of light). It has been suggested that Einstein's original argument was fallacious.^[18] Other authors suggest that the argument was merely inconclusive by virtue of some implicit assumptions lacking experimental verification at the time. ^[19]

Einstein acknowledged in his 1907 survey paper on special relativity that it was problematic to rely on Maxwell's equations for the heuristic mass-energy argument.^{[20] [21]}

Lack of an absolute reference frame

The principle of relativity, which states that there is no stationary reference frame, dates back to Galileo, and was incorporated into Newtonian Physics. However, in the late 19^th century, the existence of electromagnetic waves led physicists to suggest that the universe was filled with a substance known as "aether", which would act as the medium through which these waves, or vibrations traveled. The aether was thought to constitute an absolute reference frame against which speeds could be measured. In other words, the aether was the only fixed or motionless thing in the universe. Aether supposedly had some wonderful properties: it was sufficiently elastic that it could support electromagnetic waves, and those waves could interact with matter, yet it offered no resistance to bodies passing through it. The results of various experiments, including the Michelson-Morley experiment, indicated that the Earth was always 'stationary' relative to the aether – something that was difficult to explain, since the Earth is in orbit around the Sun. Einstein's elegant solution was to discard the notion of an aether and an absolute state of rest. Special relativity is formulated so as to not assume that any particular frame of reference is special; rather, in relativity, any reference frame moving with uniform motion will observe the same laws of physics. In particular, the speed of light in a vacuum is always measured to be c, even when measured by multiple systems that are moving at different (but constant) velocities.

Consequences

Einstein has said that all of the consequences of special relativity can be derived from examination of the Lorentz transformations.

These transformations, and hence special relativity, lead to different physical predictions than Newtonian mechanics when relative velocities become comparable to the speed of light. The speed of light is so much larger than anything humans encounter that some of the effects predicted by relativity are initially counter-intuitive:

• Time dilation – the time lapse between two events is not invariant from one observer to another, but is dependent on the relative speeds of the observers' reference frames (e.g., the twin paradox which concerns a twin who flies off in a spaceship traveling near the speed of light and returns to discover that his or her twin sibling has aged much more).
• Relativity of simultaneity – two events happening in two different locations that occur simultaneously to one observer, may occur at different times to another observer (lack of absolute simultaneity).
• Lorentz contraction – the dimensions (e.g., length) of an object as measured by one observer may be smaller than the results of measurements of the same object made by another observer (e.g., the ladder paradox involves a long ladder traveling near the speed of light and being contained within a smaller garage).
• Composition of velocities – velocities (and speeds) do not simply 'add', for example if a rocket is moving at ⅔ the speed of light relative to an observer, and the rocket fires a missile at ⅔ of the speed of light relative to the rocket, the missile does not exceed the speed of light relative to the observer. (In this example, the observer would see the missile travel with a speed of 12/13 the speed of light.)
• Inertia and momentum – as an object's speed approaches the speed of light from an observer's point of view, its mass appears to increase thereby making it more and more difficult to accelerate it from within the observer's frame of reference.
• Equivalence of mass and energy, E = mc² – The energy content of an object at rest with mass m equals mc². Conservation of energy implies that in any reaction a decrease of the sum of the masses of particles must be accompanied by an increase in kinetic energies of the particles after the reaction. Similarly, the mass of an object can be increased by taking in kinetic energies.

Reference frames, coordinates and the Lorentz transformation

Full article: Lorentz transformations

Relativity theory depends on "reference frames". A reference frame is an observational perspective in space at rest, or in uniform motion, from which a position can be measured along 3 spatial axes. In addition, a reference frame has the ability to determine measurements of the time of events using a 'clock' (any reference device with uniform periodicity).

An event is an occurrence that can be assigned a single unique time and location in space relative to a reference frame: it is a "point" in space-time. Since the speed of light is constant in relativity in each and every reference frame, pulses of light can be used to unambiguously measure distances and refer back the times that events occurred to the clock, even though light takes time to reach the clock after the event has transpired.

For example, the explosion of a firecracker may be considered to be an "event". We can completely specify an event by its four space-time coordinates: The time of occurrence and its 3-dimensional spatial location define a reference point. Let's call this reference frame S.

In relativity theory we often want to calculate the position of a point from a different reference point.

Suppose we have a second reference frame S', whose spatial axes and clock exactly coincide with that of S at time zero, but it is moving at a constant velocity ${\displaystyle v\,}$ with respect to S along the ${\displaystyle x\,}$-axis.

Since there is no absolute reference frame in relativity theory, a concept of 'moving' doesn't strictly exist, as everything is always moving with respect to some other reference frame. Instead, any two frames that move at the same speed in the same direction are said to be comoving. Therefore S and S' are not comoving.

Let's define the event to have space-time coordinates ${\displaystyle (t,x,y,z)\,}$ in system S and ${\displaystyle (t',x',y',z')\,}$ in S'. Then the Lorentz transformation specifies that these coordinates are related in the following way:

${\displaystyle {\begin{cases}t'=\gamma \left(t-{\frac {vx}{c^{2}}}\right)\\x'=\gamma (x-vt)\\y'=y\\z'=z,\end{cases}}}$

where ${\displaystyle \gamma ={\frac {1}{\sqrt {1-v^{2}/c^{2}}}}}$ is called the Lorentz factor and ${\displaystyle c\,}$ is the speed of light in a vacuum.

The ${\displaystyle y\,}$ and ${\displaystyle z\,}$ coordinates are unaffected, but the ${\displaystyle x\,}$ and ${\displaystyle t\,}$ axes are mixed up by the transformation. In a way this transformation can be understood as a hyperbolic rotation.

A quantity invariant under Lorentz transformations is known as a Lorentz scalar.

Simultaneity

From the first equation of the Lorentz transformation in terms of coordinate differences

${\displaystyle \Delta t'=\gamma \left(\Delta t-{\frac {v\,\Delta x}{c^{2}}}\right)}$

it is clear that two events that are simultaneous in frame S (satisfying ${\displaystyle \Delta t=0\,}$), are not necessarily simultaneous in another inertial frame S' (satisfying ${\displaystyle \Delta t'=0\,}$). Only if these events are colocal in frame S (satisfying ${\displaystyle \Delta x=0\,}$), will they be simultaneous in another frame S'.

Time dilation and length contraction

Writing the Lorentz transformation and its inverse in terms of coordinate differences we get

${\displaystyle {\begin{cases}\Delta t'=\gamma \left(\Delta t-{\frac {v\,\Delta x}{c^{2}}}\right)\\\Delta x'=\gamma (\Delta x-v\,\Delta t)\,\end{cases}}}$

and

${\displaystyle {\begin{cases}\Delta t=\gamma \left(\Delta t'+{\frac {v\,\Delta x'}{c^{2}}}\right)\\\Delta x=\gamma (\Delta x'+v\,\Delta t')\,\end{cases}}}$

Suppose we have a clock at rest in the unprimed system S. Two consecutive ticks of this clock are then characterized by ${\displaystyle \Delta x=0}$. If we want to know the relation between the times between these ticks as measured in both systems, we can use the first equation and find:

${\displaystyle \Delta t'=\gamma \,\Delta t\qquad (\,}$ for events satisfying ${\displaystyle \Delta x=0)\,}$

This shows that the time ${\displaystyle \Delta t'}$ between the two ticks as seen in the 'moving' frame S' is larger than the time ${\displaystyle \Delta t}$ between these ticks as measured in the rest frame of the clock. This phenomenon is called time dilation.

Similarly, suppose we have a measuring rod at rest in the unprimed system. In this system, the length of this rod is written as ${\displaystyle \Delta x}$. If we want to find the length of this rod as measured in the 'moving' system S', we must make sure to measure the distances ${\displaystyle x'}$ to the end points of the rod simultaneously in the primed frame S'. In other words, the measurement is characterized by ${\displaystyle \Delta t'=0}$, which we can combine with the fourth equation to find the relation between the lengths ${\displaystyle \Delta x}$ and ${\displaystyle \Delta x'}$:

${\displaystyle \Delta x'={\frac {\Delta x}{\gamma }}\qquad (\,}$ for events satisfying ${\displaystyle \Delta t'=0)\,}$

This shows that the length ${\displaystyle \Delta x'}$ of the rod as measured in the 'moving' frame S' is shorter than the length ${\displaystyle \Delta x}$ in its own rest frame. This phenomenon is called length contractionorLorentz contraction.

These effects are not merely appearances; they are explicitly related to our way of measuring time intervals between events which occur at the same place in a given coordinate system (called "co-local" events). These time intervals will be different in another coordinate system moving with respect to the first, unless the events are also simultaneous. Similarly, these effects also relate to our measured distances between separated but simultaneous events in a given coordinate system of choice. If these events are not co-local, but are separated by distance (space), they will not occur at the same spacial distance from each other when seen from another moving coordinate system.

See also the twin paradox.

Causality and prohibition of motion faster than light

In diagram 2 the interval AB is 'time-like'; i.e., there is a frame of reference in which event A and event B occur at the same location in space, separated only by occurring at different times. If A precedes B in that frame, then A precedes B in all frames. It is hypothetically possible for matter (or information) to travel from A to B, so there can be a causal relationship (with A the cause and B the effect).

The interval AC in the diagram is 'space-like'; i.e., there is a frame of reference in which event A and event C occur simultaneously, separated only in space. However there are also frames in which A precedes C (as shown) and frames in which C precedes A. If it were possible for a cause-and-effect relationship to exist between events A and C, then paradoxes of causality would result. For example, if A was the cause, and C the effect, then there would be frames of reference in which the effect preceded the cause. Although this in itself won't give rise to a paradox, one can show^[22][23] that faster than light signals can be sent back into one's own past. A causal paradox can then be constructed by sending the signal if and only if no signal was received previously.

Therefore, one of the consequences of special relativity is that (assuming causality is to be preserved), no information or material object can travel faster than light. On the other hand, the logical situation is not as clear in the case of general relativity, so it is an open question whether there is some fundamental principle that preserves causality (and therefore prevents motion faster than light) in general relativity.

Even without considerations of causality, there are other strong reasons why faster-than-light travel is forbidden by special relativity. For example, if a constant force is applied to an object for a limitless amount of time, then integrating F = dp/dt gives a momentum that grows without bound, but this is simply because ${\displaystyle p=m\gamma v}$ approaches infinity as v approaches c. To an observer who is not accelerating, it appears as though the object's inertia is increasing, so as to produce a smaller acceleration in response to the same force. This behavior is in fact observed in particle accelerators.

See also the Tachyonic Antitelephone.

Composition of velocities

If the observer in S sees an object moving along the x axis at velocity w, then the observer in the S' system, a frame of reference moving at velocity v in the x direction with respect to S, will see the object moving with velocity w' where

${\displaystyle w'={\frac {w-v}{1-wv/c^{2}}}.}$

This equation can be derived from the space and time transformations above. Notice that if the object were moving at the speed of light in the S system (i.e. ${\displaystyle w=c}$), then it would also be moving at the speed of light in the S' system. Also, if both w and v are small with respect to the speed of light, we will recover the intuitive Galilean transformation of velocities: ${\displaystyle w'\approx w-v}$.

Relativistic mechanics

In addition to modifying notions of space and time, special relativity forces one to reconsider the concepts of mass, momentum, and energy, all of which are important constructs in Newtonian mechanics. Special relativity shows, in fact, that these concepts are all different aspects of the same physical quantity in much the same way that it shows space and time to be interrelated.

There are a couple of (equivalent) ways to define momentum and energy in SR. One method uses conservation laws. If these laws are to remain valid in SR they must be true in every possible reference frame. However, if one does some simple thought experiments using the Newtonian definitions of momentum and energy, one sees that these quantities are not conserved in SR. One can rescue the idea of conservation by making some small modifications to the definitions to account for relativistic velocities. It is these new definitions which are taken as the correct ones for momentum and energy in SR.

The energy and momentum of an object with invariant mass m (also called rest mass in the case of a single particle), moving with velocity v with respect to a given frame of reference, are given by

${\displaystyle {\begin{cases}E&=\gamma mc^{2}\\\mathbf {p} &=\gamma m\mathbf {v} \end{cases}}}$

respectively, where γ (the Lorentz factor) is given by

${\displaystyle \gamma ={\frac {1}{\sqrt {1-(v/c)^{2}}}}.}$

The quantity γm is often called the relativistic mass of the object in the given frame of reference, ^[24] although recently this concept is falling in disuse, and Lev B. Okun suggested that "this terminology [...] has no rational justification today", and should no longer be taught. ^[25] Other physicists, including Wolfgang Rindler and T. R. Sandin, have argued that relativistic mass is a useful concept and there is little reason to stop using it. ^[26] See Mass in special relativity for more information on this debate. Some authors use the symbol m to refer to relativistic mass, and the symbol m₀ to refer to rest mass. ^[27]

The energy and momentum of an object with invariant mass m are related by the formulas

${\displaystyle E^{2}-($pc)^{2}=(mc^{2})^{2}\,}

${\displaystyle \mathbf {p} c^{2}=E\mathbf {v} \,.}$

The first is referred to as the relativistic energy-momentum equation. While the energy E and the momentum p depend on the frame of reference in which they are measured, the quantity E² − (pc)² is invariant, being equal to the squared invariant mass of the object (up to the multiplicative constant c⁴).

It should be noted that the invariant mass of a system

${\displaystyle m_{\text{tot}}={\frac {\sqrt {E_{\text{tot}}^{2}-(p_{\text{tot}}c^{2})^{2}}}{c^{2}}}}$

isgreater than the sum of the rest masses of the particles it is composed of (unless they are all stationary with respect to the center of mass of the system, and hence to each other). The latter quantity is not even always conserved in closed systems.

For massless particles, m is zero and γ is undefined (infinity) because they always travel at the speed of light; thus, the above formulae for E and p cannot be used. However, the relativistic energy-momentum equation still holds, and, by substituting m with 0, the relation E = pc is obtained. A particle which has no rest mass can nevertheless contribute to the total invariant mass of a system.

Looking at the above formulas for energy one sees that, when an object is at rest (v = 0, γ = 1), there is a non-zero energy remaining:

${\displaystyle E_{\text{rest}}=mc^{2}.}$

This energy is referred to as rest energy. Being a constant, rest energy isn't included in the kinetic energy, which is defined as E − E_rest = (γ − 1)mc².

This article may be confusing or unclear to readers. Please help clarify the article. There might be a discussion about this on the talk page. (Learn how and when to remove this message)

Taking this formula at face value, we see that in relativity, mass is simply another form of energy. In 1927 Einstein remarked about special relativity:

Under this theory mass is not an unalterable magnitude, but a magnitude dependent on (and, indeed, identical with) the amount of energy.^[28]

This formula becomes important when one measures the masses of different atomic nuclei. By looking at the difference in masses, one can predict which nuclei have extra stored energy that can be released by nuclear reactions, providing important information which was useful in the development of nuclear energy and, consequently, the nuclear bomb. The implications of this formula on 20th-century life have made it one of the most famous equations in all of science.

Force

In special relativity, Newton's second law does not hold in its form F = ma, but it does if it is expressed as

${\displaystyle \mathbf {F} ={\frac {d\mathbf {p} }{dt}}}$

where p is the momentum as defined above. Thus, the force is given by

${\displaystyle \mathbf {F} =m{\frac {d(\gamma \mathbf {v} )}{dt}}=m\left({\frac {d\gamma }{dt}}\mathbf {v} +\gamma {\frac {d\mathbf {v} }{dt}}\right).}$

Carrying out the derivatives gives

${\displaystyle \mathbf {F} ={\frac {m\gamma ^{3}\mathbf {v} }{c^{2}}}\left(\mathbf {v} \cdot \mathbf {a} \right)+m\gamma \mathbf {a} .}$

If the acceleration is separated into the part parallel to the velocity and the part perpendicular to it, one gets

${\displaystyle \mathbf {F} ={\frac {m\gamma ^{3}v^{2}}{c^{2}}}\mathbf {a} _{\parallel }+m\gamma (\mathbf {a} _{\parallel }+\mathbf {a} _{\perp })\,}$

${\displaystyle =m\gamma ^{3}\left({\frac {v^{2}}{c^{2}}}+1-{\frac {v^{2}}{c^{2}}}\right)\mathbf {a} _{\parallel }+m\gamma \mathbf {a} _{\perp }=m\gamma ^{3}\mathbf {a} _{\parallel }+m\gamma \mathbf {a} _{\perp }\,.}$

Consequently in some old texts, γm³ is referred to as the longitudinal mass, and γm is referred to as the transverse mass, which is the same as the relativistic mass.

For the four-force, see below.

Classical limit

For velocities much smaller than those of light, γ can be expanded into a Taylor series, obtaining:

${\displaystyle {\begin{cases}E-mc^{2}&={\frac {1}{2}}mv^{2}+{\frac {3}{8}}m{\frac {v^{4}}{c^{2}}}+\cdots \\\mathbf {p} &=m\mathbf {v} +{\frac {1}{2}}m{\frac {v^{2}\mathbf {v} }{c^{2}}}+\cdots .\end{cases}}}$

Neglecting the terms with c² in the denominator, these formulas agree with the standard definitions of Newtonian kinetic energy and momentum. This is as it should be, for special relativity must agree with Newtonian mechanics at low velocities.

The geometry of space-time

SR uses a 'flat' 4-dimensional Minkowski space, which is an example of a space-time. This space, however, is very similar to the standard 3 dimensional Euclidean space, and fortunately by that fact, very easy to work with.

The differential of distance (ds) in cartesian 3D space is defined as:

${\displaystyle ds^{2}=dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}}$

where ${\displaystyle (dx_{1},dx_{2},dx_{3})}$ are the differentials of the three spatial dimensions. In the geometry of special relativity, a fourth dimension is added, derived from time, so that the equation for the differential of distance becomes:

${\displaystyle ds^{2}=dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}-c^{2}dt^{2}}$

If we wished to make the time coordinate look like the space coordinates, we could treat time as imaginary: x₄ = ict . In this case the above equation becomes symmetric:

${\displaystyle ds^{2}=dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}+dx_{4}^{2}}$

This suggests what is in fact a profound theoretical insight as it shows that special relativity is simply a rotational symmetry of our space-time, very similar to rotational symmetry of Euclidean space. Just as Euclidean space uses a Euclidean metric, so space-time uses a Minkowski metric. Basically, SR can be stated in terms of the invariance of space-time interval (between any two events) as seen from any inertial reference frame. All equations and effects of special relativity can be derived from this rotational symmetry (the Poincaré group) of Minkowski space-time. According to Misner (1971 §2.3), ultimately the deeper understanding of both special and general relativity will come from the study of the Minkowski metric (described below) rather than a "disguised" Euclidean metric using ict as the time coordinate.

If we reduce the spatial dimensions to 2, so that we can represent the physics in a 3-D space

${\displaystyle ds^{2}=dx_{1}^{2}+dx_{2}^{2}-c^{2}dt^{2}}$

We see that the null geodesics lie along a dual-cone:

defined by the equation

${\displaystyle ds^{2}=0=dx_{1}^{2}+dx_{2}^{2}-c^{2}dt^{2}}$

or

${\displaystyle dx_{1}^{2}+dx_{2}^{2}=c^{2}dt^{2}}$

Which is the equation of a circle with r=c×dt. If we extend this to three spatial dimensions, the null geodesics are the 4-dimensional cone:

${\displaystyle ds^{2}=0=dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}-c^{2}dt^{2}}$

${\displaystyle dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}=c^{2}dt^{2}}$

This null dual-cone represents the "line of sight" of a point in space. That is, when we look at the stars and say "The light from that star which I am receiving is X years old", we are looking down this line of sight: a null geodesic. We are looking at an event ${\displaystyle d={\sqrt {x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}}}$ meters away and d/c seconds in the past. For this reason the null dual cone is also known as the 'light cone'. (The point in the lower left of the picture below represents the star, the origin represents the observer, and the line represents the null geodesic "line of sight".)

The cone in the -t region is the information that the point is 'receiving', while the cone in the +t section is the information that the point is 'sending'.

The geometry of Minkowski space can be depicted using Minkowski diagrams, which are also useful in understanding many of the thought-experiments in special relativity.

Physics in spacetime

Here, we see how to write the equations of special relativity in a manifestly Lorentz covariant form. The position of an event in spacetime is given by a contravariant four vector whose components are:

${\displaystyle x^{\nu }=\left(ct,x,y,z\right)}$

Where ${\displaystyle x^{1}=x}$ and ${\displaystyle x^{2}=y}$ and ${\displaystyle x^{3}=z}$ as usual. We define ${\displaystyle x^{0}=ct}$ so that the time coordinate has the same dimension of distance as the other spatial dimensions; in accordance with the general principle that space and time are treated equally, so far as possible.^[29][30][31] Superscripts are contravariant indices in this section rather than exponents except when they indicate a square. Subscripts are covariant indices which also range from zero to three as with the spacetime gradient of a field φ:

${\displaystyle \partial _{0}\phi ={\frac {1}{c}}{\frac {\partial \phi }{\partial t}},\quad \partial _{1}\phi ={\frac {\partial \phi }{\partial x}},\quad \partial _{2}\phi ={\frac {\partial \phi }{\partial y}},\quad \partial _{3}\phi ={\frac {\partial \phi }{\partial z}}.}$

Metric and transformations of coordinates

Having recognised the four-dimensional nature of spacetime, we are driven to employ the Minkowski metric, η, given in components (valid in any inertial reference frame) as:

${\displaystyle \eta _{\alpha \beta }={\begin{pmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}}}$

which is equal to its reciprocal, ${\displaystyle \eta ^{\alpha \beta }}$, in those frames.

Then we recognize that coordinate transformations between inertial reference frames are given by the Lorentz transformation tensor Λ. For the special case of motion along the x-axis, we have:

${\displaystyle \Lambda ^{\mu '}{}_{\nu }={\begin{pmatrix}\gamma &-\beta \gamma &0&0\\-\beta \gamma &\gamma &0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}}}$

which is simply the matrix of a boost (like a rotation) between the x and ct coordinates. Where μ' indicates the row and ν indicates the column. Also, β and γ are defined as:

${\displaystyle \beta ={\frac {v}{c}},\ \gamma ={\frac {1}{\sqrt {1-\beta ^{2}}}}.}$

More generally, a transformation from one inertial frame (ignoring translations for simplicity) to another must satisfy:

${\displaystyle \eta _{\alpha \beta }=\eta _{\mu '\nu '}\Lambda ^{\mu '}{}_{\alpha }\Lambda ^{\nu '}{}_{\beta }\!}$

where there is an implied summation of ${\displaystyle \mu '\!}$ and ${\displaystyle \nu '\!}$ from 0 to 3 on the right-hand side in accordance with the Einstein summation convention. The Poincaré group is the most general group of transformations which preserves the Minkowski metric and this is the physical symmetry underlying special relativity.

All proper physical quantities are given by tensors. So to transform from one frame to another, we use the well-known tensor transformation law

${\displaystyle T_{\left[j_{1}',j_{2}',\dots ,j_{q}'\right]}^{\left[i_{1}',i_{2}',\dots ,i_{p}'\right]}=\Lambda ^{i_{1}'}{}_{i_{1}}\Lambda ^{i_{2}'}{}_{i_{2}}\cdots \Lambda ^{i_{p}'}{}_{i_{p}}\Lambda _{j_{1}'}{}^{j_{1}}\Lambda _{j_{2}'}{}^{j_{2}}\cdots \Lambda _{j_{q}'}{}^{j_{q}}T_{\left[j_{1},j_{2},\dots ,j_{q}\right]}^{\left[i_{1},i_{2},\dots ,i_{p}\right]}}$