Derivations of the Lorentz transformations

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There are many ways to derive the Lorentz transformations using a variety of physical principles, ranging from Maxwell's equations to Einstein's postulates of special relativity, and mathematical tools, spanning from elementary algebra and hyperbolic functions, to linear algebra and group theory.

This article provides a few of the easier ones to follow in the context of special relativity, for the simplest case of a Lorentz boost in standard configuration, i.e. two inertial frames moving relative to each other at constant (uniform) relative velocity less than the speed of light, and using Cartesian coordinates so that the x and x′ axes are collinear.

Lorentz transformation

Main article: Lorentz transformation

In the fundamental branches of modern physics, namely general relativity and its widely applicable subset special relativity, as well as relativistic quantum mechanics and relativistic quantum field theory, the Lorentz transformation is the transformation rule under which all four-vectors and tensors containing physical quantities transform from one frame of reference to another.

The prime examples of such four-vectors are the four-position and four-momentum of a particle, and for fields the electromagnetic tensor and stress–energy tensor. The fact that these objects transform according to the Lorentz transformation is what mathematically defines them as vectors and tensors; see tensor for a definition.

Given the components of the four-vectors or tensors in some frame, the "transformation rule" allows one to determine the altered components of the same four-vectors or tensors in another frame, which could be boosted or accelerated, relative to the original frame. A "boost" should not be conflated with spatial translation, rather it's characterized by the relative velocity between frames. The transformation rule itself depends on the relative motion of the frames. In the simplest case of two inertial frames the relative velocity between enters the transformation rule. For rotating reference frames or general non-inertial reference frames, more parameters are needed, including the relative velocity (magnitude and direction), the rotation axis and angle turned through.

Historical background

Further information: History of Lorentz transformations

The usual treatment (e.g., Albert Einstein's original work) is based on the invariance of the speed of light. However, this is not necessarily the starting point: indeed (as is described, for example, in the second volume of the Course of Theoretical Physics by Landau and Lifshitz), what is really at stake is the locality of interactions: one supposes that the influence that one particle, say, exerts on another can not be transmitted instantaneously. Hence, there exists a theoretical maximal speed of information transmission which must be invariant, and it turns out that this speed coincides with the speed of light in vacuum. Newton had himself called the idea of action at a distance philosophically "absurd", and held that gravity had to be transmitted by some agent according to certain laws.

Michelson and Morley in 1887 designed an experiment, employing an interferometer and a half-silvered mirror, that was accurate enough to detect aether flow. The mirror system reflected the light back into the interferometer. If there were an aether drift, it would produce a phase shift and a change in the interference that would be detected. However, no phase shift was ever found. The negative outcome of the Michelson–Morley experiment left the concept of aether (or its drift) undermined. There was consequent perplexity as to why light evidently behaves like a wave, without any detectable medium through which wave activity might propagate.

In a 1964 paper, Erik Christopher Zeeman showed that the causality-preserving property, a condition that is weaker in a mathematical sense than the invariance of the speed of light, is enough to assure that the coordinate transformations are the Lorentz transformations. Norman Goldstein's paper shows a similar result using inertiality (the preservation of time-like lines) rather than causality.

Physical principles

Einstein based his theory of special relativity on two fundamental postulates. First, all physical laws are the same for all inertial frames of reference, regardless of their relative state of motion; and second, the speed of light in free space is the same in all inertial frames of reference, again, regardless of the relative velocity of each reference frame. The Lorentz transformation is fundamentally a direct consequence of this second postulate.

The second postulate

Assume the second postulate of special relativity stating the constancy of the speed of light, independent of reference frame, and consider a collection of reference systems moving with respect to each other with constant velocity, i.e. inertial systems, each endowed with its own set of Cartesian coordinates labeling the points, i.e. events of spacetime. To express the invariance of the speed of light in mathematical form, fix two events in spacetime, to be recorded in each reference frame. Let the first event be the emission of a light signal, and the second event be it being absorbed.

Pick any reference frame in the collection. In its coordinates, the first event will be assigned coordinates ${\displaystyle x_1,y_1,z_1,c{t}_1}$, and the second ${\displaystyle x_2,y_2,z_2,c{t}_2}$. The spatial distance between emission and absorption is $\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2+(z_2-z_1{)}^2}$, but this is also the distance ${\displaystyle c(t_2-t_1)}$ traveled by the signal. One may therefore set up the equation

$${\displaystyle c^2(t_2-t_1{)}^2-(x_2-x_1{)}^2-(y_2-y_1{)}^2-(z_2-z_1{)}^2=0.}$$

Every other coordinate system will record, in its own coordinates, the same equation. This is the immediate mathematical consequence of the invariance of the speed of light. The quantity on the left is called the spacetime interval. The interval is, for events separated by light signals, the same (zero) in all reference frames, and is therefore called invariant.

Invariance of interval

For the Lorentz transformation to have the physical significance realized by nature, it is crucial that the interval is an invariant measure for any two events, not just for those separated by light signals. To establish this, one considers an infinitesimal interval,

$${\displaystyle d{s}^2=c^{2}d{t}^2-d{x}^2-d{y}^2-d{z}^2,}$$

as recorded in a system ${\displaystyle K}$. Let ${\displaystyle K^{\prime }}$ be another system assigning the interval ${\displaystyle d{s}^{\prime 2}}$ to the same two infinitesimally separated events. Since if ${\displaystyle d{s}^2=0}$, then the interval will also be zero in any other system (second postulate), and since ${\displaystyle d{s}^2}$ and ${\displaystyle d{s}^{\prime 2}}$ are infinitesimals of the same order, they must be proportional to each other, $${\displaystyle d{s}^2=ad{s}^{\prime 2}.}$$

On what may ${\displaystyle a}$ depend? It may not depend on the positions of the two events in spacetime, because that would violate the postulated homogeneity of spacetime. It might depend on the relative velocity ${\displaystyle V^{\prime }}$ between ${\displaystyle K}$ and ${\displaystyle K^{\prime }}$, but only on the speed, not on the direction, because the latter would violate the isotropy of space.

Now bring in systems ${\displaystyle K_1}$ and ${\displaystyle K_2}$, $${\displaystyle d{s}^2=a(V_1)d{s}_1^2,d{s}^2=a(V_2)d{s}_2^2,d{s}_1^2=a(V_{12})d{s}_2^2.}$$ From these it follows, $${\displaystyle \frac{a(V_2)}{a(V_1)}=a(V_{12}).}$$

Now, one observes that on the right-hand side that ${\displaystyle V_{12}}$ depend on both ${\displaystyle V_1}$ and ${\displaystyle V_2}$; as well as on the angle between the vectors ${\displaystyle {\mathbf{\text{V}}}_1}$ and ${\displaystyle {\mathbf{\text{V}}}_2}$. However, one also observes that the left-hand side does not depend on this angle. Thus, the only way for the equation to hold true is if the function ${\displaystyle a(V)}$ is a constant. Further, by the same equation this constant is unity. Thus, $${\displaystyle d{s}^2=d{s}^{\prime 2}}$$ for all systems ${\displaystyle K^{\prime }}$. Since this holds for all infinitesimal intervals, it holds for all intervals.

Most, if not all, derivations of the Lorentz transformations take this for granted. In those derivations, they use the constancy of the speed of light (invariance of light-like separated events) only. This result ensures that the Lorentz transformation is the correct transformation.

Rigorous Statement and Proof of Proportionality of ds² and ds′²

Theorem: Let ${\displaystyle n,p\ge 1}$ be integers, ${\displaystyle d:=n+p}$ and ${\displaystyle V}$ a vector space over ${\displaystyle \mathbb{R}}$ of dimension ${\displaystyle d}$. Let ${\displaystyle h}$ be an indefinite-inner product on ${\displaystyle V}$ with signature type ${\displaystyle (n,p)}$. Suppose ${\displaystyle g}$ is a symmetric bilinear form on ${\displaystyle V}$ such that the null set of the associated quadratic form of ${\displaystyle h}$ is contained in that of ${\displaystyle g}$ (i.e. suppose that for every ${\displaystyle v\in V}$, if ${\displaystyle h(v,v)=0}$ then ${\displaystyle g(v,v)=0}$). Then, there exists a constant ${\displaystyle C\in \mathbb{R}}$ such that ${\displaystyle g=Ch}$. Furthermore, if we assume ${\displaystyle n\ne p}$ and that ${\displaystyle g}$ also has signature type ${\displaystyle (n,p)}$, then we have ${\displaystyle C>0}$.

Standard configuration

The spacetime coordinates of an event, as measured by each observer in their inertial reference frame (in standard configuration) are shown in the speech bubbles.
Top: frame F′ moves at velocity v along the x-axis of frame F.
Bottom: frame F moves at velocity −v along the x′-axis of frame F′.

The invariant interval can be seen as a non-positive definite distance function on spacetime. The set of transformations sought must leave this distance invariant. Due to the reference frame's coordinate system's cartesian nature, one concludes that, as in the Euclidean case, the possible transformations are made up of translations and rotations, where a slightly broader meaning should be allowed for the term rotation.

The interval is quite trivially invariant under translation. For rotations, there are four coordinates. Hence there are six planes of rotation. Three of those are rotations in spatial planes. The interval is invariant under ordinary rotations too.

It remains to find a "rotation" in the three remaining coordinate planes that leaves the interval invariant. Equivalently, to find a way to assign coordinates so that they coincide with the coordinates corresponding to a moving frame.

The general problem is to find a transformation such that $${\displaystyle \begin{array}{rl}& c^2(t_2-t_1{)}^2-(x_2-x_1{)}^2-(y_2-y_1{)}^2-(z_2-z_1{)}^2\\ =& c^2(t_2^{\prime }-t_1^{\prime }{)}^2-(x_2^{\prime }-x_1^{\prime }{)}^2-(y_2^{\prime }-y_1^{\prime }{)}^2-(z_2^{\prime }-z_1^{\prime }{)}^2.\end{array}}$$

To solve the general problem, one may use the knowledge about invariance of the interval of translations and ordinary rotations to assume, without loss of generality, that the frames F and F′ are aligned in such a way that their coordinate axes all meet at t = t′ = 0 and that the x and x′ axes are permanently aligned and system F′ has speed V along the positive x-axis. Call this the standard configuration. It reduces the general problem to finding a transformation such that

$${\displaystyle c^2(t_2-t_1{)}^2-(x_2-x_1{)}^2=c^2(t_2^{\prime }-t_1^{\prime }{)}^2-(x_2^{\prime }-x_1^{\prime }{)}^2.}$$

The standard configuration is used in most examples below. A linear solution of the simpler problem

$${\displaystyle (ct{)}^2-x^2=(c{t}^{\prime }{)}^2-x^{\prime 2}}$$

solves the more general problem since coordinate differences then transform the same way. Linearity is often assumed or argued somehow in the literature when this simpler problem is considered. If the solution to the simpler problem is not linear, then it doesn't solve the original problem because of the cross terms appearing when expanding the squares.

The solutions

As mentioned, the general problem is solved by translations in spacetime. These do not appear as a solution to the simpler problem posed, while the boosts do (and sometimes rotations depending on angle of attack). Even more solutions exist if one only insist on invariance of the interval for lightlike separated events. These are nonlinear conformal ("angle preserving") transformations. One has

Lorentz transformations ⊂ Poincaré transformations ⊂ conformal group transformations.

Some equations of physics are conformal invariant, e.g. the Maxwell's equations in source-free space, but not all. The relevance of the conformal transformations in spacetime is not known at present, but the conformal group in two dimensions is highly relevant in conformal field theory and statistical mechanics. It is thus the Poincaré group that is singled out by the postulates of special relativity. It is the presence of Lorentz boosts (for which velocity addition is different from mere vector addition that would allow for speeds greater than the speed of light) as opposed to ordinary boosts that separates it from the Galilean group of Galilean relativity. Spatial rotations, spatial and temporal inversions and translations are present in both groups and have the same consequences in both theories (conservation laws of momentum, energy, and angular momentum). Not all accepted theories respect symmetry under the inversions.

Using the geometry of spacetime

Landau & Lifshitz solution

These three hyperbolic function formulae (H1–H3) are referenced below:

1. ${\displaystyle {\cosh }^2\mathrm{\Psi }-{\sinh }^2\mathrm{\Psi }=1,}$
2. ${\displaystyle \sinh \mathrm{\Psi }=\frac{\tanh \mathrm{\Psi }}{\sqrt{1-{\tanh }^2\mathrm{\Psi }}},}$
3. ${\displaystyle \cosh \mathrm{\Psi }=\frac{1}{\sqrt{1-{\tanh }^2\mathrm{\Psi }}},}$

The problem posed in standard configuration for a boost in the x-direction, where the primed coordinates refer to the moving system is solved by finding a linear solution to the simpler problem

$${\displaystyle (ct{)}^2-x^2=(c{t}^{\prime }{)}^2-x^{\prime 2}.}$$

The most general solution is, as can be verified by direct substitution using (H1),

${\displaystyle x=x^{\prime }\cosh \mathrm{\Psi }+c{t}^{\prime }\sinh \mathrm{\Psi },ct=x^{\prime }\sinh \mathrm{\Psi }+c{t}^{\prime }\cosh \mathrm{\Psi }.}$

1

To find the role of Ψ in the physical setting, record the progression of the origin of F′, i.e. x′ = 0, x = vt. The equations become (using first x′ = 0),

$${\displaystyle x=c{t}^{\prime }\sinh \mathrm{\Psi },ct=c{t}^{\prime }\cosh \mathrm{\Psi }.}$$

Now divide:

$${\displaystyle \frac{x}{ct}=\tanh \mathrm{\Psi }=\frac{v}{c}\Rightarrow \sinh \mathrm{\Psi }=\frac{\frac{v}{c}}{\sqrt{1-\frac{v^2}{c^2}}},\cosh \mathrm{\Psi }=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}},}$$

where x = vt was used in the first step, (H2) and (H3) in the second, which, when plugged back in (1), gives

$${\displaystyle x=\frac{x^{\prime }+v{t}^{\prime }}{\sqrt{1-\frac{v^2}{c^2}}},t=\frac{t^{\prime }+\frac{v}{c^2}{x}^{\prime }}{\sqrt{1-\frac{v^2}{c^2}}},}$$

or, with the usual abbreviations,

${\displaystyle x=\gamma (x^{\prime }+v{t}^{\prime }),t=\gamma \left(t^{\prime }+\frac{v{x}^{\prime }}{c^2}\right),x^{\prime }=\gamma (x-vt),t^{\prime }=\gamma \left(t-\frac{vx}{c^2}\right).}$

This calculation is repeated with more detail in section hyperbolic rotation.

Hyperbolic rotation

Main article: Hyperbolic rotation

The Lorentz transformations can also be derived by simple application of the special relativity postulates and using hyperbolic identities.

Relativity postulates

Start from the equations of the spherical wave front of a light pulse, centred at the origin:

$${\displaystyle (ct{)}^2-(x^2+y^2+z^2)=(c{t}^{\prime }{)}^2-(x^{\prime 2}+y^{\prime 2}+z^{\prime 2})=0}$$

which take the same form in both frames because of the special relativity postulates. Next, consider relative motion along the x-axes of each frame, in standard configuration above, so that y = y′, z = z′, which simplifies to

$${\displaystyle (ct{)}^2-x^2=(c{t}^{\prime }{)}^2-x^{\prime 2}}$$

Linearity

Now assume that the transformations take the linear form:

$${\displaystyle \begin{array}{rl}{x}^{\prime }& =Ax+Bct\\ c{t}^{\prime }& =Cx+Dct\end{array}}$$

where A, B, C, D are to be found. If they were non-linear, they would not take the same form for all observers, since fictitious forces (hence accelerations) would occur in one frame even if the velocity was constant in another, which is inconsistent with inertial frame transformations.

Substituting into the previous result:

$${\displaystyle (ct{)}^2-x^2=[(Cx{)}^2+(Dct{)}^2+2CDcxt]-[(Ax{)}^2+(Bct{)}^2+2ABcxt]}$$

and comparing coefficients of x², t², xt:

$${\displaystyle \begin{array}{rlr}-1=C^2-A^2& \Rightarrow & A^2-C^2=1\\ c^2=(Dc{)}^2-(Bc{)}^2& \Rightarrow & D^2-B^2=1\\ 2CDc-2ABc=0& \Rightarrow & AB=CD\end{array}}$$

Hyperbolic rotation

The equations suggest the hyperbolic identity ${\displaystyle {\cosh }^2\varphi -{\sinh }^2\varphi =1.}$

Introducing the rapidity parameter ϕ as a hyperbolic angle allows the consistent identifications

$${\displaystyle A=D=\cosh \varphi ,C=B=-\sinh \varphi }$$

where the signs after the square roots are chosen so that x' and t' increase if x and t increase, respectively. The hyperbolic transformations have been solved for:

$${\displaystyle \begin{array}{rl}{x}^{\prime }& =x\cosh \varphi -ct\sinh \varphi \\ c{t}^{\prime }& =-x\sinh \varphi +ct\cosh \varphi \end{array}}$$

If the signs were chosen differently the position and time coordinates would need to be replaced by −x and/or −t so that x and t increase not decrease.

To find how ϕ relates to the relative velocity, from the standard configuration the origin of the primed frame x′ = 0 is measured in the unprimed frame to be x = vt (or the equivalent and opposite way round; the origin of the unprimed frame is x = 0 and in the primed frame it is at x′ = −vt):

$${\displaystyle 0=vt\cosh \varphi -ct\sinh \varphi \Rightarrow \tanh \varphi =\frac{v}{c}=\beta }$$

and hyperbolic identities ${\displaystyle \sinh \mathrm{\Psi }=\frac{\tanh \mathrm{\Psi }}{\sqrt{1-{\tanh }^2\mathrm{\Psi }}},\cosh \mathrm{\Psi }=\frac{1}{\sqrt{1-{\tanh }^2\mathrm{\Psi }}}}$ leads to the relations between β, γ, and ϕ,

$${\displaystyle \cosh \varphi =\gamma ,\sinh \varphi =\beta \gamma .}$$

From physical principles

The problem is usually restricted to two dimensions by using a velocity along the x axis such that the y and z coordinates do not intervene, as described in standard configuration above.

Time dilation and length contraction

The transformation equations can be derived from time dilation and length contraction, which in turn can be derived from first principles. With O and O′ representing the spatial origins of the frames F and F′, and some event M, the relation between the position vectors (which here reduce to oriented segments OM, OO′ and O′M) in both frames is given by:

OM = OO′ + O′M.

Using coordinates (x,t) in F and (x′,t′) in F′ for event M, in frame F the segments are OM = x, OO′ = vt and O′M = x′/γ (since x′ is O′M as measured in F′): $${\displaystyle x=vt+x^{\prime }/\gamma .}$$ Likewise, in frame F′, the segments are OM = x/γ (since x is OM as measured in F), OO′ = vt′ and O′M = x′: $${\displaystyle x/\gamma =v{t}^{\prime }+x^{\prime }.}$$ By rearranging the first equation, we get $${\displaystyle x^{\prime }=\gamma (x-vt),}$$ which is the space part of the Lorentz transformation. The second relation gives $${\displaystyle x=\gamma (x^{\prime }+v{t}^{\prime }),}$$ which is the inverse of the space part. Eliminating x′ between the two space part equations gives

$${\displaystyle t^{\prime }=\gamma t+\frac{\left(1-{\gamma }^2\right)x}{\gamma v}.}$$

that, if ${\displaystyle {\gamma }^2=\frac{1}{1-v^2/c^2}}$, simplifies to:

$${\displaystyle t^{\prime }=\gamma (t-vx/c^2),}$$ which is the time part of the transformation, the inverse of which is found by a similar elimination of x: $${\displaystyle t=\gamma (t^{\prime }+v{x}^{\prime }/c^2).}$$

Spherical wavefronts of light

The following is similar to that of Einstein. As in the Galilean transformation, the Lorentz transformation is linear since the relative velocity of the reference frames is constant as a vector; otherwise, inertial forces would appear. They are called inertial or Galilean reference frames. According to relativity no Galilean reference frame is privileged. Another condition is that the speed of light must be independent of the reference frame, in practice of the velocity of the light source.

Consider two inertial frames of reference O and O′, assuming O to be at rest while O′ is moving with a velocity v with respect to O in the positive x-direction. The origins of O and O′ initially coincide with each other. A light signal is emitted from the common origin and travels as a spherical wave front. Consider a point P on a spherical wavefront at a distance r and r′ from the origins of O and O′ respectively. According to the second postulate of the special theory of relativity the speed of light is the same in both frames, so for the point P: $${\displaystyle \begin{array}{rl}r& =ct\\ r^{\prime }& =c{t}^{\prime }.\end{array}}$$

The equation of a sphere in frame O is given by $${\displaystyle x^2+y^2+z^2=r^2.}$$ For the spherical wavefront that becomes $${\displaystyle x^2+y^2+z^2=(ct{)}^2.}$$ Similarly, the equation of a sphere in frame O′ is given by $${\displaystyle x^{\prime 2}+y^{\prime 2}+z^{\prime 2}=r^{\prime 2},}$$ so the spherical wavefront satisfies $${\displaystyle x^{\prime 2}+y^{\prime 2}+z^{\prime 2}=(c{t}^{\prime }{)}^2.}$$

The origin O′ is moving along x-axis. Therefore, $${\displaystyle \begin{array}{rl}{y}^{\prime }& =y\\ z^{\prime }& =z.\end{array}}$$

x′ must vary linearly with x and t. Therefore, the transformation has the form $${\displaystyle x^{\prime }=\gamma x+\sigma t.}$$ For the origin of O′ x′ and x are given by $${\displaystyle \begin{array}{rl}{x}^{\prime }& =0\\ x& =vt,\end{array}}$$ so, for all t, $${\displaystyle 0=\gamma vt+\sigma t}$$ and thus $${\displaystyle \sigma =-\gamma v.}$$ This simplifies the transformation to $${\displaystyle x^{\prime }=\gamma \left(x-vt\right)}$$ where γ is to be determined. At this point γ is not necessarily a constant, but is required to reduce to 1 for v ≪ c.

The inverse transformation is the same except that the sign of v is reversed: $${\displaystyle x=\gamma \left(x^{\prime }+v{t}^{\prime }\right).}$$

The above two equations give the relation between t and t′ as: $${\displaystyle x=\gamma \left[\gamma \left(x-vt\right)+v{t}^{\prime }\right]}$$ or $${\displaystyle t^{\prime }=\gamma t+\frac{\left(1-{\gamma }^2\right)x}{\gamma v}.}$$

Replacing x′, y′, z′ and t′ in the spherical wavefront equation in the O′ frame, $${\displaystyle x^{\prime 2}+y^{\prime 2}+z^{\prime 2}=(c{t}^{\prime }{)}^2,}$$ with their expressions in terms of x, y, z and t produces: $${\displaystyle {\gamma }^2{\left(x-vt\right)}^2+y^2+z^2=c^2{\left[\gamma t+\frac{\left(1-{\gamma }^2\right)x}{\gamma v}\right]}^2}$$ and therefore, $${\displaystyle {\gamma }^2{x}^2+{\gamma }^2{v}^2{t}^2-2{\gamma }^{2}vtx+y^2+z^2=c^2{\gamma }^2{t}^2+\frac{{\left(1-{\gamma }^2\right)}^2{c}^2{x}^2}{{\gamma }^2{v}^2}+2\frac{\left(1-{\gamma }^2\right)tx{c}^2}{v}}$$ which implies, $${\displaystyle \left[{\gamma }^2-\frac{{\left(1-{\gamma }^2\right)}^2{c}^2}{{\gamma }^2{v}^2}\right]x^2-2{\gamma }^{2}vtx+y^2+z^2=\left(c^2{\gamma }^2-v^2{\gamma }^2\right)t^2+2\frac{\left[1-{\gamma }^2\right]tx{c}^2}{v}}$$ or $${\displaystyle \left[{\gamma }^2-\frac{{\left(1-{\gamma }^2\right)}^2{c}^2}{{\gamma }^2{v}^2}\right]x^2-\left[2{\gamma }^{2}v+2\frac{\left(1-{\gamma }^2\right)c^2}{v}\right]tx+y^2+z^2=\left[c^2{\gamma }^2-v^2{\gamma }^2\right]t^2}$$

Comparing the coefficient of t² in the above equation with the coefficient of t² in the spherical wavefront equation for frame O produces: $${\displaystyle c^2{\gamma }^2-v^2{\gamma }^2=c^2}$$ Equivalent expressions for γ can be obtained by matching the x² coefficients or setting the tx coefficient to zero. Rearranging: $${\displaystyle {\gamma }^2=\frac{1}{1-\frac{v^2}{c^2}}}$$ or, choosing the positive root to ensure that the x and x' axes and the time axes point in the same direction, $${\displaystyle \gamma =\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}}$$ which is called the Lorentz factor. This produces the Lorentz transformation from the above expression. It is given by $${\displaystyle \begin{array}{rl}{x}^{\prime }& =\gamma \left(x-vt\right)\\ t^{\prime }& =\gamma \left(t-\frac{vx}{c^2}\right)\\ y^{\prime }& =y\\ z^{\prime }& =z\end{array}}$$

The Lorentz transformation is not the only transformation leaving invariant the shape of spherical waves, as there is a wider set of spherical wave transformations in the context of conformal geometry, leaving invariant the expression ${\displaystyle \lambda \left(\delta x^2+\delta y^2+\delta z^2-c^2\delta t^2\right)}$. However, scale changing conformal transformations cannot be used to symmetrically describe all laws of nature including mechanics, whereas the Lorentz transformations (the only one implying ${\displaystyle \lambda =1}$) represent a symmetry of all laws of nature and reduce to Galilean transformations at ${\displaystyle v\ll c}$.

Galilean and Einstein's relativity

Galilean reference frames

In classical kinematics, the total displacement x in the R frame is the sum of the relative displacement x′ in frame R′ and of the distance between the two origins x − x′. If v is the relative velocity of R′ relative to R, the transformation is: x = x′ + vt, or x′ = x − vt. This relationship is linear for a constant v, that is when R and R′ are Galilean frames of reference.

In Einstein's relativity, the main difference from Galilean relativity is that space and time coordinates are intertwined, and in different inertial frames t ≠ t′.

Since space is assumed to be homogeneous, the transformation must be linear. The most general linear relationship is obtained with four constant coefficients, A, B, γ, and b: $${\displaystyle x^{\prime }=\gamma x+bt}$$ $${\displaystyle t^{\prime }=Ax+Bt.}$$ The linear transformation becomes the Galilean transformation when γ = B = 1, b = −v and A = 0.

An object at rest in the R′ frame at position x′ = 0 moves with constant velocity v in the R frame. Hence the transformation must yield x′ = 0 if x = vt. Therefore, b = −γv and the first equation is written as $${\displaystyle x^{\prime }=\gamma \left(x-vt\right).}$$

Using the principle of relativity

According to the principle of relativity, there is no privileged Galilean frame of reference: therefore the inverse transformation for the position from frame R′ to frame R should have the same form as the original but with the velocity in the opposite direction, i.o.w. replacing v with -v: $${\displaystyle x=\gamma \left(x^{\prime }-(-v)t^{\prime }\right),}$$ and thus $${\displaystyle x=\gamma \left(x^{\prime }+v{t}^{\prime }\right).}$$

Determining the constants of the first equation

Since the speed of light is the same in all frames of reference, for the case of a light signal, the transformation must guarantee that t = x/c when t′ = x′/c.

Substituting for t and t′ in the preceding equations gives: $${\displaystyle x^{\prime }=\gamma \left(1-v/c\right)x,}$$ $${\displaystyle x=\gamma \left(1+v/c\right)x^{\prime }.}$$ Multiplying these two equations together gives, $${\displaystyle x{x}^{\prime }={\gamma }^2\left(1-v^2/c^2\right)x{x}^{\prime }.}$$ At any time after t = t′ = 0, xx′ is not zero, so dividing both sides of the equation by xx′ results in $${\displaystyle \gamma =\frac{1}{\sqrt{1-\frac{v^2}{c^2}}},}$$ which is called the "Lorentz factor".

When the transformation equations are required to satisfy the light signal equations in the form x = ct and x′ = ct′, by substituting the x and x'-values, the same technique produces the same expression for the Lorentz factor.

Determining the constants of the second equation

The transformation equation for time can be easily obtained by considering the special case of a light signal, again satisfying x = ct and x′ = ct′, by substituting term by term into the earlier obtained equation for the spatial coordinate $${\displaystyle x^{\prime }=\gamma (x-vt),}$$ giving $${\displaystyle c{t}^{\prime }=\gamma \left(ct-\frac{v}{c}x\right),}$$ so that $${\displaystyle t^{\prime }=\gamma \left(t-\frac{v}{c^2}x\right),}$$ which, when identified with $${\displaystyle t^{\prime }=Ax+Bt,}$$ determines the transformation coefficients A and B as $${\displaystyle A=-\gamma v/c^2,}$$ $${\displaystyle B=\gamma .}$$ So A and B are the unique constant coefficients necessary to preserve the constancy of the speed of light in the primed system of coordinates.

Einstein's popular derivation

In his popular book Einstein derived the Lorentz transformation by arguing that there must be two non-zero coupling constants λ and μ such that

$${\displaystyle \{\begin{array}{l}{x}^{\prime }-c{t}^{\prime }=\lambda \left(x-ct\right)\\ x^{\prime }+c{t}^{\prime }=\mu \left(x+ct\right)\end{array}}$$

that correspond to light traveling along the positive and negative x-axis, respectively. For light x = ct if and only if x′ = ct′. Adding and subtracting the two equations and defining

$${\displaystyle \{\begin{array}{l}\gamma =\left(\lambda +\mu \right)/2\\ b=\left(\lambda -\mu \right)/2,\end{array}}$$

gives

$${\displaystyle \{\begin{array}{l}{x}^{\prime }=\gamma x-bct\\ c{t}^{\prime }=\gamma ct-bx.\end{array}}$$

Substituting x′ = 0 corresponding to x = vt and noting that the relative velocity is v = bc/γ, this gives

$${\displaystyle \{\begin{array}{l}{x}^{\prime }=\gamma \left(x-vt\right)\\ t^{\prime }=\gamma \left(t-\frac{v}{c^2}x\right)\end{array}}$$

The constant γ can be evaluated by demanding c²t² − x² = c²t′² − x′² as per standard configuration.

Using group theory

From group postulates

Following is a classical derivation based on group postulates and isotropy of the space.

Coordinate transformations as a group

The coordinate transformations between inertial frames form a group (called the proper Lorentz group) with the group operation being the composition of transformations (performing one transformation after another). Indeed, the four group axioms are satisfied:

1. Closure: the composition of two transformations is a transformation: consider a composition of transformations from the inertial frame K to inertial frame K′, (denoted as K → K′), and then from K′ to inertial frame K′′, [K′ → K′′], there exists a transformation, [K → K′] [K′ → K′′], directly from an inertial frame K to inertial frame K′′.
2. Associativity: the transformations ( [K → K′] [K′ → K′′] ) [K′′ → K′′′] and [K → K′] ( [K′ → K′′] [K′′ → K′′′] ) are identical.
3. Identity element: there is an identity element, a transformation K → K.
4. Inverse element: for any transformation K → K′ there exists an inverse transformation K′ → K.

Transformation matrices consistent with group axioms

Consider two inertial frames, K and K′, the latter moving with velocity v with respect to the former. By rotations and shifts we can choose the x and x′ axes along the relative velocity vector and also that the events (t, x) = (0,0) and (t′, x′) = (0,0) coincide. Since the velocity boost is along the x (and x′) axes nothing happens to the perpendicular coordinates and we can just omit them for brevity. Now since the transformation we are looking after connects two inertial frames, it has to transform a linear motion in (t, x) into a linear motion in (t′, x′) coordinates. Therefore, it must be a linear transformation. The general form of a linear transformation is $${\displaystyle \left[\begin{array}{c}{t}^{\prime }\\ x^{\prime }\end{array}\right]=\left[\begin{array}{cc}\gamma & \delta \\ \beta & \alpha \end{array}\right]\left[\begin{array}{c}t\\ x\end{array}\right],}$$ where α, β, γ and δ are some yet unknown functions of the relative velocity v.

Let us now consider the motion of the origin of the frame K′. In the K′ frame it has coordinates (t′, x′ = 0), while in the K frame it has coordinates (t, x = vt). These two points are connected by the transformation $${\displaystyle \left[\begin{array}{c}{t}^{\prime }\\ 0\end{array}\right]=\left[\begin{array}{cc}\gamma & \delta \\ \beta & \alpha \end{array}\right]\left[\begin{array}{c}t\\ vt\end{array}\right],}$$ from which we get $${\displaystyle \beta =-v\alpha .}$$ Analogously, considering the motion of the origin of the frame K, we get $${\displaystyle \left[\begin{array}{c}{t}^{\prime }\\ -v{t}^{\prime }\end{array}\right]=\left[\begin{array}{cc}\gamma & \delta \\ \beta & \alpha \end{array}\right]\left[\begin{array}{c}t\\ 0\end{array}\right],}$$ from which we get $${\displaystyle \beta =-v\gamma .}$$ Combining these two gives α = γ and the transformation matrix has simplified, $${\displaystyle \left[\begin{array}{c}{t}^{\prime }\\ x^{\prime }\end{array}\right]=\left[\begin{array}{cc}\gamma & \delta \\ -v\gamma & \gamma \end{array}\right]\left[\begin{array}{c}t\\ x\end{array}\right].}$$

Now consider the group postulate inverse element. There are two ways we can go from the K′ coordinate system to the K coordinate system. The first is to apply the inverse of the transform matrix to the K′ coordinates:

$${\displaystyle \left[\begin{array}{c}t\\ x\end{array}\right]=\frac{1}{{\gamma }^2+v\delta \gamma }\left[\begin{array}{cc}\gamma & -\delta \\ v\gamma & \gamma \end{array}\right]\left[\begin{array}{c}{t}^{\prime }\\ x^{\prime }\end{array}\right].}$$

The second is, considering that the K′ coordinate system is moving at a velocity v relative to the K coordinate system, the K coordinate system must be moving at a velocity −v relative to the K′ coordinate system. Replacing v with −v in the transformation matrix gives:

$${\displaystyle \left[\begin{array}{c}t\\ x\end{array}\right]=\left[\begin{array}{cc}\gamma (-v)& \delta (-v)\\ v\gamma (-v)& \gamma (-v)\end{array}\right]\left[\begin{array}{c}{t}^{\prime }\\ x^{\prime }\end{array}\right],}$$

Now the function γ can not depend upon the direction of v because it is apparently the factor which defines the relativistic contraction and time dilation. These two (in an isotropic world of ours) cannot depend upon the direction of v. Thus, γ(−v) = γ(v) and comparing the two matrices, we get $${\displaystyle {\gamma }^2+v\delta \gamma =1.}$$

According to the closure group postulate a composition of two coordinate transformations is also a coordinate transformation, thus the product of two of our matrices should also be a matrix of the same form. Transforming K to K′ and from K′ to K′′ gives the following transformation matrix to go from K to K′′:

$${\displaystyle \begin{array}{rl}\left[\begin{array}{c}{t}^{″}\\ x^{″}\end{array}\right]& =\left[\begin{array}{cc}\gamma (v^{\prime })& \delta (v^{\prime })\\ -v^{\prime }\gamma (v^{\prime })& \gamma (v^{\prime })\end{array}\right]\left[\begin{array}{cc}\gamma (v)& \delta (v)\\ -v\gamma (v)& \gamma (v)\end{array}\right]\left[\begin{array}{c}t\\ x\end{array}\right]\\ & =\left[\begin{array}{cc}\gamma (v^{\prime })\gamma (v)-v\delta (v^{\prime })\gamma (v)& \gamma (v^{\prime })\delta (v)+\delta (v^{\prime })\gamma (v)\\ -(v^{\prime }+v)\gamma (v^{\prime })\gamma (v)& -v^{\prime }\gamma (v^{\prime })\delta (v)+\gamma (v^{\prime })\gamma (v)\end{array}\right]\left[\begin{array}{c}t\\ x\end{array}\right].\end{array}}$$

In the original transform matrix, the main diagonal elements are both equal to γ, hence, for the combined transform matrix above to be of the same form as the original transform matrix, the main diagonal elements must also be equal. Equating these elements and rearranging gives:

$${\displaystyle \begin{array}{rl}\gamma (v^{\prime })\gamma (v)-v\delta (v^{\prime })\gamma (v)& =-v^{\prime }\gamma (v^{\prime })\delta (v)+\gamma (v^{\prime })\gamma (v)\\ v\delta (v^{\prime })\gamma (v)& =v^{\prime }\gamma (v^{\prime })\delta (v)\\ \frac{\delta (v)}{v\gamma (v)}& =\frac{\delta (v^{\prime })}{v^{\prime }\gamma (v^{\prime })}.\end{array}}$$

The denominator will be nonzero for nonzero v, because γ(v) is always nonzero;

$${\displaystyle {\gamma }^2+v\delta \gamma =1.}$$

If v = 0 we have the identity matrix which coincides with putting v = 0 in the matrix we get at the end of this derivation for the other values of v, making the final matrix valid for all nonnegative v.

For the nonzero v, this combination of function must be a universal constant, one and the same for all inertial frames. Define this constant as δ(v)/v γ(v) = κ, where κ has the dimension of 1/v². Solving $${\displaystyle 1={\gamma }^2+v\delta \gamma ={\gamma }^2(1+\kappa v^2)}$$ we finally get $${\displaystyle \gamma =1/\sqrt{1+\kappa v^2}}$$ and thus the transformation matrix, consistent with the group axioms, is given by

$${\displaystyle \left[\begin{array}{c}{t}^{\prime }\\ x^{\prime }\end{array}\right]=\frac{1}{\sqrt{1+\kappa v^2}}\left[\begin{array}{cc}1& \kappa v\\ -v& 1\end{array}\right]\left[\begin{array}{c}t\\ x\end{array}\right].}$$

If κ > 0, then there would be transformations (with κv² ≫ 1) which transform time into a spatial coordinate and vice versa. We exclude this on physical grounds, because time can only run in the positive direction. Thus two types of transformation matrices are consistent with group postulates:

1. with the universal constant κ = 0, and
2. with κ < 0.

Galilean transformations

If κ = 0 then we get the Galilean-Newtonian kinematics with the Galilean transformation, $${\displaystyle \left[\begin{array}{c}{t}^{\prime }\\ x^{\prime }\end{array}\right]=\left[\begin{array}{cc}1& 0\\ -v& 1\end{array}\right]\left[\begin{array}{c}t\\ x\end{array}\right],}$$ where time is absolute, t′ = t, and the relative velocity v of two inertial frames is not limited.

Lorentz transformations

If κ < 0, then we set ${\displaystyle c=1/\sqrt{-\kappa }}$ which becomes the invariant speed, the speed of light in vacuum. This yields κ = −1/c² and thus we get special relativity with Lorentz transformation $${\displaystyle \left[\begin{array}{c}{t}^{\prime }\\ x^{\prime }\end{array}\right]=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\left[\begin{array}{cc}1& \frac{-v}{c^2}\\ -v& 1\end{array}\right]\left[\begin{array}{c}t\\ x\end{array}\right],}$$ where the speed of light is a finite universal constant determining the highest possible relative velocity between inertial frames.

If v ≪ c the Galilean transformation is a good approximation to the Lorentz transformation.

Only experiment can answer the question which of the two possibilities, κ = 0 or κ < 0, is realized in our world. The experiments measuring the speed of light, first performed by a Danish physicist Ole Rømer, show that it is finite, and the Michelson–Morley experiment showed that it is an absolute speed, and thus that κ < 0.

Boost from generators

Using rapidity ϕ to parametrize the Lorentz transformation, the boost in the x direction is

$${\displaystyle \left[\begin{array}{c}c{t}^{\prime }\\ x^{\prime }\\ y^{\prime }\\ z^{\prime }\end{array}\right]=\left[\begin{array}{cccc}\cosh \varphi & -\sinh \varphi & 0& 0\\ -\sinh \varphi & \cosh \varphi & 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{c}ct\\ x\\ y\\ z\end{array}\right],}$$

likewise for a boost in the y-direction

$${\displaystyle \left[\begin{array}{c}c{t}^{\prime }\\ x^{\prime }\\ y^{\prime }\\ z^{\prime }\end{array}\right]=\left[\begin{array}{cccc}\cosh \varphi & 0& -\sinh \varphi & 0\\ 0& 1& 0& 0\\ -\sinh \varphi & 0& \cosh \varphi & 0\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{c}ct\\ x\\ y\\ z\end{array}\right],}$$

and the z-direction

$${\displaystyle \left[\begin{array}{c}c{t}^{\prime }\\ x^{\prime }\\ y^{\prime }\\ z^{\prime }\end{array}\right]=\left[\begin{array}{cccc}\cosh \varphi & 0& 0& -\sinh \varphi \\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ -\sinh \varphi & 0& 0& \cosh \varphi \end{array}\right]\left[\begin{array}{c}ct\\ x\\ y\\ z\end{array}\right].}$$

where e_x, e_y, e_z are the Cartesian basis vectors, a set of mutually perpendicular unit vectors along their indicated directions. If one frame is boosted with velocity v relative to another, it is convenient to introduce a unit vector n = v/v = β/β in the direction of relative motion. The general boost is

$${\displaystyle \left[\begin{array}{c}c{t}^{\prime }\\ x^{\prime }\\ y^{\prime }\\ z^{\prime }\end{array}\right]=\left[\begin{array}{cccc}\cosh \varphi & -n_x\sinh \varphi & -n_y\sinh \varphi & -n_z\sinh \varphi \\ -n_x\sinh \varphi & 1+(\cosh \varphi -1)n_x^2& (\cosh \varphi -1)n_x{n}_y& (\cosh \varphi -1)n_x{n}_z\\ -n_y\sinh \varphi & (\cosh \varphi -1)n_y{n}_x& 1+(\cosh \varphi -1)n_y^2& (\cosh \varphi -1)n_y{n}_z\\ -n_z\sinh \varphi & (\cosh \varphi -1)n_z{n}_x& (\cosh \varphi -1)n_z{n}_y& 1+(\cosh \varphi -1)n_z^2\end{array}\right]\left[\begin{array}{c}ct\\ x\\ y\\ z\end{array}\right].}$$

Notice the matrix depends on the direction of the relative motion as well as the rapidity, in all three numbers (two for direction, one for rapidity).

We can cast each of the boost matrices in another form as follows. First consider the boost in the x direction. The Taylor expansion of the boost matrix about ϕ = 0 is

$${\displaystyle B({\mathbf{e}}_x,\varphi )=\sum _{n=0}^{\mathrm{\infty }}\frac{{\varphi }^n}{n!}{\frac{{\mathrm{\partial }}^{n}B({\mathbf{e}}_x,\varphi )}{\mathrm{\partial }{\varphi }^n}|}_{\varphi =0}}$$

where the derivatives of the matrix with respect to ϕ are given by differentiating each entry of the matrix separately, and the notation |_{ϕ = 0} indicates ϕ is set to zero after the derivatives are evaluated. Expanding to first order gives the infinitesimal transformation

$${\displaystyle B({\mathbf{e}}_x,\varphi )=I+\varphi {\frac{\mathrm{\partial }B}{\mathrm{\partial }\varphi }|}_{\varphi =0}=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]-\varphi \left[\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}$$

which is valid if ϕ is small (hence ϕ² and higher powers are negligible), and can be interpreted as no boost (the first term I is the 4×4 identity matrix), followed by a small boost. The matrix

$${\displaystyle K_x=\left[\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}$$

is the generator of the boost in the x direction, so the infinitesimal boost is

$${\displaystyle B({\mathbf{e}}_x,\varphi )=I-\varphi K_x}$$

Now, ϕ is small, so dividing by a positive integer N gives an even smaller increment of rapidity ϕ/N, and N of these infinitesimal boosts will give the original infinitesimal boost with rapidity ϕ,

$${\displaystyle B({\mathbf{e}}_x,\varphi )={\left(I-\frac{\varphi K_x}{N}\right)}^N}$$

In the limit of an infinite number of infinitely small steps, we obtain the finite boost transformation

$${\displaystyle B({\mathbf{e}}_x,\varphi )=\underset{N\to \mathrm{\infty }}{lim}{\left(I-\frac{\varphi K_x}{N}\right)}^N=e^{-\varphi K_x}}$$

which is the limit definition of the exponential due to Leonhard Euler, and is now true for any ϕ.

Repeating the process for the boosts in the y and z directions obtains the other generators

$${\displaystyle K_y=\left[\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right],K_z=\left[\begin{array}{cccc}0& 0& 0& 1\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 1& 0& 0& 0\end{array}\right]}$$

and the boosts are

$${\displaystyle B({\mathbf{e}}_y,\varphi )=e^{-\varphi K_y},B({\mathbf{e}}_z,\varphi )=e^{-\varphi K_z}.}$$

For any direction, the infinitesimal transformation is (small ϕ and expansion to first order)

$${\displaystyle B(\mathbf{n},\varphi )=I+\varphi {\frac{\mathrm{\partial }B}{\mathrm{\partial }\varphi }|}_{\varphi =0}=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]-\varphi \left[\begin{array}{cccc}0& n_x& n_y& n_z\\ n_x& 0& 0& 0\\ n_y& 0& 0& 0\\ n_z& 0& 0& 0\end{array}\right]}$$

where

$${\displaystyle \left[\begin{array}{cccc}0& n_x& n_y& n_z\\ n_x& 0& 0& 0\\ n_y& 0& 0& 0\\ n_z& 0& 0& 0\end{array}\right]=n_x{K}_x+n_y{K}_y+n_z{K}_z=\mathbf{n}\cdot \mathbf{K}}$$

is the generator of the boost in direction n. It is the full boost generator, a vector of matrices K = (K_x, K_y, K_z), projected into the direction of the boost n. The infinitesimal boost is

$${\displaystyle B(\mathbf{n},\varphi )=I-\varphi (\mathbf{n}\cdot \mathbf{K})}$$

Then in the limit of an infinite number of infinitely small steps, we obtain the finite boost transformation

$${\displaystyle B(\mathbf{n},\varphi )=\underset{N\to \mathrm{\infty }}{lim}{\left(I-\frac{\varphi (\mathbf{n}\cdot \mathbf{K})}{N}\right)}^N=e^{-\varphi (\mathbf{n}\cdot \mathbf{K})}}$$

which is now true for any ϕ. Expanding the matrix exponential of −ϕ(n ⋅ K) in its power series

$${\displaystyle e^{-\varphi \mathbf{n}\cdot \mathbf{K}}=\sum _{n=0}^{\mathrm{\infty }}\frac{1}{n!}(-\varphi \mathbf{n}\cdot \mathbf{K}{)}^n}$$

we now need the powers of the generator. The square is

$${\displaystyle (\mathbf{n}\cdot \mathbf{K}{)}^2=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& n_x^2& n_x{n}_y& n_x{n}_z\\ 0& n_y{n}_x& n_y^2& n_y{n}_z\\ 0& n_z{n}_x& n_z{n}_y& n_z^2\end{array}\right]}$$

but the cube (n ⋅ K)³ returns to (n ⋅ K), and as always the zeroth power is the 4×4 identity, (n ⋅ K)⁰ = I. In general the odd powers n = 1, 3, 5, ... are

$${\displaystyle (\mathbf{n}\cdot \mathbf{K}{)}^n=(\mathbf{n}\cdot \mathbf{K})}$$

while the even powers n = 2, 4, 6, ... are

$${\displaystyle (\mathbf{n}\cdot \mathbf{K}{)}^n=(\mathbf{n}\cdot \mathbf{K}{)}^2}$$

therefore the explicit form of the boost matrix depends only the generator and its square. Splitting the power series into an odd power series and an even power series, using the odd and even powers of the generator, and the Taylor series of sinh ϕ and cosh ϕ about ϕ = 0 obtains a more compact but detailed form of the boost matrix

$${\displaystyle \begin{array}{rl}{e}^{-\varphi \mathbf{n}\cdot \mathbf{K}}& =-\sum _{n=1,3,5\dots }^{\mathrm{\infty }}\frac{1}{n!}{\varphi }^n(\mathbf{n}\cdot \mathbf{K}{)}^n+\sum _{n=0,2,4\dots }^{\mathrm{\infty }}\frac{1}{n!}{\varphi }^n(\mathbf{n}\cdot \mathbf{K}{)}^n\\ & =-\left[\varphi +\frac{{\varphi }^3}{3!}+\frac{{\varphi }^5}{5!}+\cdots \right](\mathbf{n}\cdot \mathbf{K})+I+\left[-1+1+\frac{1}{2!}{\varphi }^2+\frac{1}{4!}{\varphi }^4+\frac{1}{6!}{\varphi }^6+\cdots \right](\mathbf{n}\cdot \mathbf{K}{)}^2\\ & =-\sinh \varphi (\mathbf{n}\cdot \mathbf{K})+I+(-1+\cosh \varphi )(\mathbf{n}\cdot \mathbf{K}{)}^2\end{array}}$$

where 0 = −1 + 1 is introduced for the even power series to complete the Taylor series for cosh ϕ. The boost is similar to Rodrigues' rotation formula,

$${\displaystyle B(\mathbf{n},\varphi )=e^{-\varphi \mathbf{n}\cdot \mathbf{K}}=I-\sinh \varphi (\mathbf{n}\cdot \mathbf{K})+(\cosh \varphi -1)(\mathbf{n}\cdot \mathbf{K}{)}^2.}$$

Negating the rapidity in the exponential gives the inverse transformation matrix,

$${\displaystyle B(\mathbf{n},-\varphi )=e^{\varphi \mathbf{n}\cdot \mathbf{K}}=I+\sinh \varphi (\mathbf{n}\cdot \mathbf{K})+(\cosh \varphi -1)(\mathbf{n}\cdot \mathbf{K}{)}^2.}$$

In quantum mechanics, relativistic quantum mechanics, and quantum field theory, a different convention is used for the boost generators; all of the boost generators are multiplied by a factor of the imaginary unit i = √−1.

From experiments

Main article: Test theories of special relativity

Howard Percy Robertson and others showed that the Lorentz transformation can also be derived empirically. In order to achieve this, it's necessary to write down coordinate transformations that include experimentally testable parameters. For instance, let there be given a single "preferred" inertial frame ${\displaystyle X,Y,Z,T}$ in which the speed of light is constant, isotropic, and independent of the velocity of the source. It is also assumed that Einstein synchronization and synchronization by slow clock transport are equivalent in this frame. Then assume another frame ${\displaystyle x,y,z,t}$ in relative motion, in which clocks and rods have the same internal constitution as in the preferred frame. The following relations, however, are left undefined:

• ${\displaystyle a(v)}$ differences in time measurements,
• ${\displaystyle b(v)}$ differences in measured longitudinal lengths,
• ${\displaystyle d(v)}$ differences in measured transverse lengths,
• ${\displaystyle \epsilon (v)}$ depends on the clock synchronization procedure in the moving frame,

then the transformation formulas (assumed to be linear) between those frames are given by:

$${\displaystyle \begin{array}{rl}t& =a(v)T+\epsilon (v)x\\ x& =b(v)(X-vT)\\ y& =d(v)Y\\ z& =d(v)Z\end{array}}$$

${\displaystyle \epsilon (v)}$ depends on the synchronization convention and is not determined experimentally, it obtains the value ${\displaystyle -v/c^2}$ by using Einstein synchronization in both frames. The ratio between ${\displaystyle b(v)}$ and ${\displaystyle d(v)}$ is determined by the Michelson–Morley experiment, the ratio between ${\displaystyle a(v)}$ and ${\displaystyle b(v)}$ is determined by the Kennedy–Thorndike experiment, and ${\displaystyle a(v)}$ alone is determined by the Ives–Stilwell experiment. In this way, they have been determined with great precision to ${\displaystyle 1/a(v)=b(v)=\gamma }$ and ${\displaystyle d(v)=1}$, which converts the above transformation into the Lorentz transformation.

Science fiction magazine

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Science_fiction_magazine

 

Speculative fiction

A front cover of Imagination, a science fiction magazine in 1956

A science fiction magazine is a publication that offers primarily science fiction, either in a hard-copy periodical format or on the Internet. Science fiction magazines traditionally featured speculative fiction in short story, novelette, novella or (usually serialized) novel form, a format that continues into the present day. Many also contain editorials, book reviews or articles, and some also include stories in the fantasy and horror genres.

History of science fiction magazines

See also: History of US science fiction and fantasy magazines to 1950

Malcolm Edwards and Peter Nicholls write that early magazines were not known as science fiction: "if there were any need to differentiate them, the terms scientific romance or 'different stories' might be used, but until the appearance of a magazine specifically devoted to sf there was no need of a label to describe the category. The first specialized English-language pulps with a leaning towards the fantastic were Thrill Book (1919) and Weird Tales (1923), but the editorial policy of both was aimed much more towards weird-occult fiction than towards sf."

Major American science fiction magazines include Amazing Stories, Astounding Science Fiction, Galaxy Science Fiction, The Magazine of Fantasy & Science Fiction and Isaac Asimov's Science Fiction Magazine. The most influential British science fiction magazine was New Worlds; newer British SF magazines include Interzone and Polluto. Many science fiction magazines have been published in languages other than English, but none has gained worldwide recognition or influence in the world of anglophone science fiction.

There is a growing trend toward important work being published first on the Internet, both for reasons of economics and access. A web-only publication can cost as little as one-tenth of the cost of publishing a print magazine, and as a result, some believe the e-zines are more innovative and take greater risks with material. Moreover, the magazine is internationally accessible, and distribution is not an issue—though obscurity may be. Magazines like Strange Horizons, Ideomancer, InterGalactic Medicine Show, Jim Baen's Universe, and the Australian magazine Andromeda Spaceways Inflight Magazine are examples of successful Internet magazines. (Andromeda provides copies electronically or on paper.)

Web-based magazines tend to favor shorter stories and articles that are easily read on a screen, and many of them pay little or nothing to the authors, thus limiting their universe of contributors. However, multiple web-based magazines are listed as "paying markets" by the SFWA, which means that they pay the "professional" rate of 8c/word or more. These magazines include popular titles such as Strange Horizons, InterGalactic Medicine Show, and Clarkesworld Magazine. The SFWA publishes a list of qualifying magazine and short fiction venues that contains all current web-based qualifying markets.

The World Science Fiction Convention (Worldcon) awarded a Hugo Award each year to the best science fiction magazine, until that award was changed to one for Best Editor in the early 1970s; the Best Semi-Professional Magazine award can go to either a news-oriented magazine or a small press fiction magazine.

Magazines were the only way to publish science fiction until about 1950, when large mainstream publishers began issuing science fiction books. Today, there are relatively few paper-based science fiction magazines, and most printed science fiction appears first in book form. Science fiction magazines began in the United States, but there were several major British magazines and science fiction magazines that have been published around the world, for example in France and Argentina.

The first science fiction magazines

March 1941 cover of the Science Fiction magazine, volume 2, issue 4

The first science fiction magazine, Amazing Stories, was published in a format known as bedsheet, roughly the size of Life but with a square spine. Later, most magazines changed to the pulp magazine format, roughly the size of comic books or National Geographic but again with a square spine. Now, most magazines are published in digest format, roughly the size of Reader's Digest, although a few are in the standard roughly 8.5" x 11" size, and often have stapled spines, rather than glued square spines. Science fiction magazines in this format often feature non-fiction media coverage in addition to the fiction. Knowledge of these formats is an asset when locating magazines in libraries and collections where magazines are usually shelved according to size.

The premiere issue of Amazing Stories (April 1926), edited and published by Hugo Gernsback, displayed a cover by Frank R. Paul illustrating Off on a Comet by Jules Verne. After many minor changes in title and major changes in format, policy and publisher, Amazing Stories ended January 2005 after 607 issues.

Except for the last issue of Stirring Science Stories, the last true bedsheet size sf (and fantasy) magazine was Fantastic Adventures, in 1939, but it quickly changed to the pulp size, and it was later absorbed by its digest-sized stablemate Fantastic in 1953. Before that consolidation, it ran 128 issues.

Much fiction published in these bedsheet magazines, except for classic reprints by writers such as H. G. Wells, Jules Verne and Edgar Allan Poe, is only of antiquarian interest. Some of it was written by teenage science fiction fans, who were paid little or nothing for their efforts. Jack Williamson for example, was 19 when he sold his first story to Amazing Stories. His writing improved greatly over time, and until his death in 2006, he was still a publishing writer at age 98.

Some of the stories in the early issues were by scientists or doctors who knew little or nothing about writing fiction, but who tried their best, for example, David H. Keller. Probably the two best original sf stories ever published in a bedsheet science fiction magazine were "A Martian Odyssey" by Stanley G. Weinbaum and "The Gostak and the Doshes" by Miles Breuer, who influenced Jack Williamson. "The Gostak and the Doshes" is one of the few stories from that era still widely read today. Other stories of interest from the bedsheet magazines include the first Buck Rogers story, Armageddon 2419 A.D, by Philip Francis Nowlan, and The Skylark of Space by coauthors E. E. Smith and Mrs. Lee Hawkins Garby, both in Amazing Stories in 1928.

There have been a few unsuccessful attempts to revive the bedsheet size using better quality paper, notably Science-Fiction Plus edited by Hugo Gernsback (1952–53, eight issues). Astounding on two occasions briefly attempted to revive the bedsheet size, with 16 bedsheet issues in 1942–1943 and 25 bedsheet issues (as Analog, including the first publication of Frank Herbert's Dune) in 1963–1965. The fantasy magazine Unknown, also edited by John W. Campbell, changed its name to Unknown Worlds and published ten bedsheet-size issues before returning to pulp size for its final four issues. Amazing Stories published 36 bedsheet size issues in 1991–1999, and its last three issues were bedsheet size, 2004–2005.

The pulp era

Main article: Science fiction and fantasy pulp magazines

See also: Pulp era of science fiction

Astounding Stories began in January 1930. After several changes in name and format (Astounding Science Fiction, Analog Science Fact & Fiction, Analog) it is still published today (though it ceased to be pulp format in 1943). Its most important editor, John W. Campbell, Jr., is credited with turning science fiction away from adventure stories on alien planets and toward well-written, scientifically literate stories with better characterization than in previous pulp science fiction. Isaac Asimov's Foundation Trilogy and Robert A. Heinlein's Future History in the 1940s, Hal Clement's Mission of Gravity in the 1950s, and Frank Herbert's Dune in the 1960s, and many other science fiction classics all first appeared under Campbell's editorship.

By 1955, the pulp era was over, and some pulp magazines changed to digest size. Printed adventure stories with colorful heroes were relegated to the comic books. This same period saw the end of radio adventure drama (in the United States). Later attempts to revive both pulp fiction and radio adventure have met with very limited success, but both enjoy a nostalgic following who collect the old magazines and radio programs. Many characters, most notably The Shadow, were popular both in pulp magazines and on radio.

Most pulp science fiction consisted of adventure stories transplanted, without much thought, to alien planets. Pulp science fiction is known for clichés such as stereotypical female characters, unrealistic gadgetry, and fantastic monsters of various kinds. However, many classic stories were first published in pulp magazines. For example, in the year 1939, all of the following renowned authors sold their first professional science fiction story to magazines specializing in pulp science fiction: Isaac Asimov, Robert A. Heinlein, Arthur C. Clarke, Alfred Bester, Fritz Leiber, A. E. van Vogt and Theodore Sturgeon. These were among the most important science fiction writers of the pulp era, and all are still read today.

Digest-sized magazines

After the pulp era, digest size magazines dominated the newsstand. The first sf magazine to change to digest size was Astounding, in 1943. Other major digests, which published more literary science fiction, were The Magazine of Fantasy & Science Fiction, Galaxy Science Fiction and If. Under the editorship of Cele Goldsmith, Amazing and Fantastic changed in notable part from pulp style adventure stories to literary science fiction and fantasy. Goldsmith published the first professionally published stories by Roger Zelazny (not counting student fiction in Literary Cavalcade), Keith Laumer, Thomas M. Disch, Sonya Dorman and Ursula K. Le Guin.

There was also no shortage of digests that continued the pulp tradition of hastily written adventure stories set on other planets. Other Worlds and Imaginative Tales had no literary pretensions. The major pulp writers, such as Heinlein, Asimov and Clarke, continued to write for the digests, and a new generation of writers, such as Algis Budrys and Walter M. Miller, Jr., sold their most famous stories to the digests. A Canticle for Leibowitz, written by Walter M. Miller, Jr., was first published in The Magazine of Fantasy & Science Fiction.

Most digest magazines began in the 1950s, in the years between the film Destination Moon, the first major science fiction film in a decade, and the launching of Sputnik, which sparked a new interest in space travel as a real possibility. Most survived only a few issues. By 1960, in the United States, there were only six sf digests on newsstands, in 1970 there were seven, in 1980 there were five, in 1990 only four and in 2000 only three.

Around the world

British science fiction magazines

The first British science fiction magazine was Tales of Wonder, pulp size, 1937–1942, 16 issues, (unless Scoops is taken into account, a tabloid boys' paper that published 20 weekly issues in 1934). It was followed by two magazines, both named Fantasy, one pulp size publishing three issues in 1938–1939, the other digest size, publishing three issues in 1946–1947. The British science fiction magazine, New Worlds, published three pulp size issues in 1946–1947, before changing to digest size. With these exceptions, the pulp phenomenon, like the comic book, was largely a US format. By 2007, the only surviving major British science fiction magazine is Interzone, published in "magazine" format, although small press titles such as PostScripts and Polluto are available.

Transition from print to online science fiction magazines

During recent decades, the circulation of all digest science fiction magazines has steadily decreased. New formats were attempted, most notably the slick-paper stapled magazine format, the paperback format and the webzine. There are also various semi-professional magazines that persist on sales of a few thousand copies but often publish important fiction.

As the circulation of the traditional US science fiction magazines has declined, new magazines have sprung up online from international small-press publishers. An editor on the staff of Science Fiction World, China's longest-running science fiction magazine, claimed in 2009 that, with "a circulation of 300,000 copies per issue", it was "the World's most-read SF periodical", although subsequent news suggests that circulation dropped precipitously after the firing of its chief editor in 2010 and the departure of other editors. The Science Fiction and Fantasy Writers of America lists science fiction periodicals that pay enough to be considered professional markets.

List of current magazines

For a complete list, including defunct magazines, see List of science fiction magazines.

American magazines

• Abyss & Apex Magazine, 2003–present
• Analog Science Fiction and Fact (a.k.a. Astounding Stories, Astounding Science-Fiction and Analog Science Fact & Fiction), 1930–present
• Apex Magazine, 2005–present
• Aphelion the Webzine of Science Fiction and Fantasy, 1997–present
• Ares Magazine (New Edition), 2017–present (Based on defunct magazine Ares)
• Asimov's Science Fiction (a.k.a. Isaac Asimov's Science Fiction Magazine), 1977–present
• Bards and Sages Quarterly, 2009–present
• Bull Spec, 2009–present
• Clarkesworld Magazine, 2006–present
• Compelling Science Fiction, 2016–present
• Daily Science Fiction, 2010–present
• Escape Pod, 2005–present, fiction podcast and online
• FIYAH Literary Magazine, 2016-present
• The Future Fire, 2005–present, US/UK
• Galaxy's Edge Magazine, 2013–present
• GUD Magazine 2006–present, print/pdf
• Hypnos, 2012–present
• Illuminations of the Fantastic (online, 2020–current)
• InterGalactic Medicine Show, 2005–2019
• Leading Edge (a.k.a. The Leading Edge Magazine of Science Fiction and Fantasy), 1981–present
• Lightspeed, 2010–present
• Locus: The Magazine of The Science Fiction & Fantasy Field, 1968–present
• The Magazine of Fantasy & Science Fiction (a.k.a. The Magazine of Fantasy), 1949–present
• Nebula Rift, 2012–present
  
• Not one of us, 1986–present
• Perihelion Science Fiction, 1967–1969, revived 2012–present
• Planet Magazine, 1994–present
• Planetary Stories, 2005–present
• Quantum Muse E-Zine, 1997–present
• Reactor (magazine) (formerly Tor.com), 2008–present
• Shimmer Magazine, 2005–2018
• Space Adventure Magazine, 2011–present
• Space and Time Magazine, 1966–present
• Strange Horizons, 2000–present
• Three-lobed Burning Eye, 1999–present
• Uncanny Magazine, 2014–present
• Unfit Magazine, 2018–present
• Waylines Magazine, 2013–present – US/Japan
• Weird Tales, 1923–1954, revived 1988–present

British magazines

• Arc, 2012–present
• Doctor Who Magazine, 1979–present
• Fever Dreams Magazine, online publication 2012–present
• The Future Fire, 2005–present – US/UK
• Interzone, 1982–present
• SFX, 1995–present
• Starburst, 1977–present

Other magazines

• Albedo One, 1993–present, Ireland
• Andromeda Spaceways Inflight Magazine, 2002–present, Australia
• Aurealis, 1990–present, Australia
• Fantastyka (also known as Nowa Fantastyka), 1982–present, Poland
• Futura, 1992–present, Croatia
• Galaktika, 1972–1995, revived 2004–present, Hungary
• Helice, 2006–present, Spain-Latin America
• Kalpabiswa, 2016–present, India
• Mir Fantastiki, 2003–present, Russia
• Mithila Review, 2016–present, India
• Neo-opsis Science Fiction Magazine, 2003–present, Canada (English)
• NewFoundSpecFic, 2009–present, Canada (English)
• Nova Science Fiction, 1982–1987, revived 2004–present, Sweden
• On Spec, 1989–present, Canada (English)
• Quarber Merkur, Austria
• Portti, 1982–present, Finland
• RBG-Azimuth, 2006–present, Ukraine
• Science Fiction World, 1979–present, China
• Sci Phi Journal, 2014–present, Belgium
• SF Magazine, 1959–present, Japan
• Sirius B, 2011–present, Croatia
• Solaris, 1974–present, Canada (French)
• Tähtivaeltaja, 1982–present, Finland
• Ubiq, 2007–present, Croatia
• Universe Pathways, 2005–present, Greece
• Urania, 1952–present, Italy
• Usva webzine, 2005–present, Finland

History of string theory

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/History_of_string_theory

 

The history of string theory spans several decades of intense research including two superstring revolutions. Through the combined efforts of many researchers, string theory has developed into a broad and varied subject with connections to quantum gravity, particle and condensed matter physics, cosmology, and pure mathematics.

1943–1959: S-matrix theory

String theory represents an outgrowth of S-matrix theory, a research program begun by Werner Heisenberg in 1943 following John Archibald Wheeler's 1937 introduction of the S-matrix. Many prominent theorists picked up and advocated S-matrix theory, starting in the late 1950s and throughout the 1960s. The field became marginalized and discarded in the mid-1970s and disappeared in the 1980s. Physicists neglected it because some of its mathematical methods were alien, and because quantum chromodynamics supplanted it as an experimentally better-qualified approach to the strong interactions.

The theory presented a radical rethinking of the foundations of physical laws. By the 1940s it had become clear that the proton and the neutron were not pointlike particles like the electron. Their magnetic moment differed greatly from that of a pointlike spin-½ charged particle, too much to attribute the difference to a small perturbation. Their interactions were so strong that they scattered like a small sphere, not like a point. Heisenberg proposed that the strongly interacting particles were in fact extended objects, and because there are difficulties of principle with extended relativistic particles, he proposed that the notion of a space-time point broke down at nuclear scales.

Without space and time, it becomes difficult to formulate a physical theory. Heisenberg proposed a solution to this problem: focusing on the observable quantities—those things measurable by experiments. An experiment only sees a microscopic quantity if it can be transferred by a series of events to the classical devices that surround the experimental chamber. The objects that fly to infinity are stable particles, in quantum superpositions of different momentum states.

Heisenberg proposed that even when space and time are unreliable, the notion of momentum state, which is defined far away from the experimental chamber, still works. The physical quantity he proposed as fundamental is the quantum mechanical amplitude for a group of incoming particles to turn into a group of outgoing particles, and he did not admit that there were any steps in between.

The S-matrix is the quantity that describes how a collection of incoming particles turn into outgoing ones. Heisenberg proposed to study the S-matrix directly, without any assumptions about space-time structure. But when transitions from the far-past to the far-future occur in one step with no intermediate steps, it becomes difficult to calculate anything. In quantum field theory, the intermediate steps are the fluctuations of fields or equivalently the fluctuations of virtual particles. In this proposed S-matrix theory, there are no local quantities at all.

Heisenberg proposed to use unitarity to determine the S-matrix. In all conceivable situations, the sum of the squares of the amplitudes must equal 1. This property can determine the amplitude in a quantum field theory order by order in a perturbation series once the basic interactions are given, and in many quantum field theories the amplitudes grow too fast at high energies to make a unitary S-matrix. But without extra assumptions on the high-energy behavior, unitarity is not enough to determine the scattering, and the proposal was ignored for many years.

Heisenberg's proposal was revived in 1956 when Murray Gell-Mann recognized that dispersion relations—like those discovered by Hendrik Kramers and Ralph Kronig in the 1920s (see Kramers–Kronig relations)—allow the formulation of a notion of causality, a notion that events in the future would not influence events in the past, even when the microscopic notion of past and future are not clearly defined. He also recognized that these relations might be useful in computing observables for the case of strong interaction physics. The dispersion relations were analytic properties of the S-matrix, and they imposed more stringent conditions than those that follow from unitarity alone. This development in S-matrix theory stemmed from Murray Gell-Mann and Marvin Leonard Goldberger's (1954) discovery of crossing symmetry, another condition that the S-matrix had to fulfil.

Prominent advocates of the new "dispersion relations" approach included Stanley Mandelstam and Geoffrey Chew, both at UC Berkeley at the time. Mandelstam discovered the double dispersion relations, a new and powerful analytic form, in 1958, and believed that it would provide the key to progress in the intractable strong interactions.

1959–1968: Regge theory and bootstrap models

Main article: Regge theory

By the late 1950s, many strongly interacting particles of ever higher spins had been discovered, and it became clear that they were not all fundamental. While Japanese physicist Shoichi Sakata proposed that the particles could be understood as bound states of just three of them (the proton, the neutron and the Lambda; see Sakata model), Geoffrey Chew believed that none of these particles are fundamental (for details, see Bootstrap model). Sakata's approach was reworked in the 1960s into the quark model by Murray Gell-Mann and George Zweig by making the charges of the hypothetical constituents fractional and rejecting the idea that they were observed particles. At the time, Chew's approach was considered more mainstream because it did not introduce fractional charge values and because it focused on experimentally measurable S-matrix elements, not on hypothetical pointlike constituents.

Chew-Frautschi plot showing the angular momentum J as a function of the square mass of some particles. An example of Regge trajectories.

In 1959, Tullio Regge, a young theorist in Italy, discovered that bound states in quantum mechanics can be organized into families known as Regge trajectories, each family having distinctive angular momenta. This idea was generalized to relativistic quantum mechanics by Stanley Mandelstam, Vladimir Gribov and Marcel Froissart, using a mathematical method (the Sommerfeld–Watson representation) discovered decades earlier by Arnold Sommerfeld and Kenneth M. Watson: the result was dubbed the Froissart–Gribov formula.

In 1961, Geoffrey Chew and Steven Frautschi recognized that mesons had straight line Regge trajectories (in their scheme, spin is plotted against mass squared on a so-called Chew–Frautschi plot), which implied that the scattering of these particles would have very strange behavior—it should fall off exponentially quickly at large angles. With this realization, theorists hoped to construct a theory of composite particles on Regge trajectories, whose scattering amplitudes had the asymptotic form demanded by Regge theory.

In 1967, a notable step forward in the bootstrap approach was the principle of DHS duality introduced by Richard Dolen, David Horn, and Christoph Schmid in 1967, at Caltech (the original term for it was "average duality" or "finite energy sum rule (FESR) duality"). The three researchers noticed that Regge pole exchange (at high energy) and resonance (at low energy) descriptions offer multiple representations/approximations of one and the same physically observable process.

1968–1974: Dual resonance model

The first model in which hadronic particles essentially follow the Regge trajectories was the dual resonance model that was constructed by Gabriele Veneziano in 1968, who noted that the Euler beta function could be used to describe 4-particle scattering amplitude data for such particles. The Veneziano scattering amplitude (or Veneziano model) was quickly generalized to an N-particle amplitude by Ziro Koba and Holger Bech Nielsen (their approach was dubbed the Koba–Nielsen formalism), and to what are now recognized as closed strings by Miguel Virasoro and Joel A. Shapiro (their approach was dubbed the Shapiro–Virasoro model).

In 1969, the Chan–Paton rules (proposed by Jack E. Paton and Hong-Mo Chan) enabled isospin factors to be added to the Veneziano model.

In 1969–70, Yoichiro Nambu, Holger Bech Nielsen, and Leonard Susskind presented a physical interpretation of the Veneziano amplitude by representing nuclear forces as vibrating, one-dimensional strings. However, this string-based description of the strong force made many predictions that directly contradicted experimental findings.

In 1971, Pierre Ramond and, independently, John H. Schwarz and André Neveu attempted to implement fermions into the dual model. This led to the concept of "spinning strings", and pointed the way to a method for removing the problematic tachyon (see RNS formalism).

Dual resonance models for strong interactions were a relatively popular subject of study between 1968 and 1973. The scientific community lost interest in string theory as a theory of strong interactions in 1973 when quantum chromodynamics became the main focus of theoretical research (mainly due to the theoretical appeal of its asymptotic freedom).

1974–1984: Bosonic string theory and superstring theory

In 1974, John H. Schwarz and Joël Scherk, and independently Tamiaki Yoneya, studied the boson-like patterns of string vibration and found that their properties exactly matched those of the graviton, the gravitational force's hypothetical messenger particle. Schwarz and Scherk argued that string theory had failed to catch on because physicists had underestimated its scope. This led to the development of bosonic string theory.

String theory is formulated in terms of the Polyakov action, which describes how strings move through space and time. Like springs, the strings tend to contract to minimize their potential energy, but conservation of energy prevents them from disappearing, and instead they oscillate. By applying the ideas of quantum mechanics to strings it is possible to deduce the different vibrational modes of strings, and that each vibrational state appears to be a different particle. The mass of each particle, and the fashion with which it can interact, are determined by the way the string vibrates—in essence, by the "note" the string "sounds." The scale of notes, each corresponding to a different kind of particle, is termed the "spectrum" of the theory.

Early models included both open strings, which have two distinct endpoints, and closed strings, where the endpoints are joined to make a complete loop. The two types of string behave in slightly different ways, yielding two spectra. Not all modern string theories use both types; some incorporate only the closed variety.

The earliest string model has several problems: it has a critical dimension D = 26, a feature that was originally discovered by Claud Lovelace in 1971; the theory has a fundamental instability, the presence of tachyons (see tachyon condensation); additionally, the spectrum of particles contains only bosons, particles like the photon that obey particular rules of behavior. While bosons are a critical ingredient of the Universe, they are not its only constituents. Investigating how a string theory may include fermions in its spectrum led to the invention of supersymmetry (in the West) in 1971, a mathematical transformation between bosons and fermions. String theories that include fermionic vibrations are now known as superstring theories.

In 1977, the GSO projection (named after Ferdinando Gliozzi, Joël Scherk, and David I. Olive) led to a family of tachyon-free unitary free string theories, the first consistent superstring theories (see below).

1984–1994: First superstring revolution

The first superstring revolution is a period of important discoveries that began in 1984. It was realized that string theory was capable of describing all elementary particles as well as the interactions between them. Hundreds of physicists started to work on string theory as the most promising idea to unify physical theories. The revolution was started by a discovery of anomaly cancellation in type I string theory via the Green–Schwarz mechanism (named after Michael Green and John H. Schwarz) in 1984. The ground-breaking discovery of the heterotic string was made by David Gross, Jeffrey Harvey, Emil Martinec, and Ryan Rohm in 1985. It was also realized by Philip Candelas, Gary Horowitz, Andrew Strominger, and Edward Witten in 1985 that to obtain ${\displaystyle N=1}$ supersymmetry, the six small extra dimensions (the D = 10 critical dimension of superstring theory had been originally discovered by John H. Schwarz in 1972) need to be compactified on a Calabi–Yau manifold. (In string theory, compactification is a generalization of Kaluza–Klein theory, which was first proposed in the 1920s.)

By 1985, five separate superstring theories had been described: type I, type II (IIA and IIB), and heterotic (SO(32) and E₈×E₈).

Discover magazine in the November 1986 issue (vol. 7, #11) featured a cover story written by Gary Taubes, "Everything's Now Tied to Strings", which explained string theory for a popular audience.

In 1987, Eric Bergshoeff [de], Ergin Sezgin [de] and Paul Townsend showed that there are no superstrings in eleven dimensions (the largest number of dimensions consistent with a single graviton in supergravity theories), but supermembranes.

1994–2003: Second superstring revolution

In the early 1990s, Edward Witten and others found strong evidence that the different superstring theories were different limits of an 11-dimensional theory that became known as M-theory (for details, see Introduction to M-theory). These discoveries sparked the second superstring revolution that took place approximately between 1994 and 1995.

The different versions of superstring theory were unified, as long hoped, by new equivalences. These are known as S-duality, T-duality, U-duality, mirror symmetry, and conifold transitions. The different theories of strings were also related to M-theory.

In 1995, Joseph Polchinski discovered that the theory requires the inclusion of higher-dimensional objects, called D-branes: these are the sources of electric and magnetic Ramond–Ramond fields that are required by string duality. D-branes added additional rich mathematical structure to the theory, and opened possibilities for constructing realistic cosmological models in the theory (for details, see Brane cosmology).

In 1997–98, Juan Maldacena conjectured a relationship between type IIB string theory and N = 4 supersymmetric Yang–Mills theory, a gauge theory. This conjecture, called the AdS/CFT correspondence, has generated a great deal of interest in high energy physics. It is a realization of the holographic principle, which has far-reaching implications: the AdS/CFT correspondence has helped elucidate the mysteries of black holes suggested by Stephen Hawking's work and is believed to provide a resolution of the black hole information paradox.

2003–present

In 2003, Michael R. Douglas's discovery of the string theory landscape, which suggests that string theory has a large number of inequivalent false vacua, led to much discussion of what string theory might eventually be expected to predict, and how cosmology can be incorporated into the theory.

A possible mechanism of string theory vacuum stabilization (the KKLT mechanism) was proposed in 2003 by Shamit Kachru, Renata Kallosh, Andrei Linde, and Sandip Trivedi. Much of the present-day research is focused on characterizing the "swampland" of theories incompatible with quantum gravity. The Ryu–Takayanagi conjecture introduced many concepts from quantum information into string theory.