Derivation of Lorentz transformations

The derivation of Lorentz transformations can be done in many ways, proceeding from various assumptions.

Lorentz transformations can be obtained abstractly, from group considerations (in this case they are obtained with an indefinite parameter $c$ ${\ displaystyle c}$ $c$ ), as a generalization of the Galilean transformations (which was done by Poincare - see below ). However, for the first time they were obtained as transformations with respect to which the Maxwell equations are covariant (which do not change the form of the laws of electrodynamics and optics when moving to another reference frame). Transformations can be obtained from the assumption of their linearity and the postulate of the same speed of light in all reference frames (which is a simplified formulation of the requirement for covariance of electrodynamics relative to the desired transformations, and the extension of the principle of equality of inertial reference systems - the principle of relativity - to electrodynamics), as is done in special theory of relativity ( STO) (with this parameter $c$ ${\ displaystyle c}$ $c$ in Lorentz transformations it turns out to be definite and coincides with the speed of light).

It should be noted that if we do not limit the class of coordinate transformations to linear ones, then Newton’s first law is satisfied not only for Lorentz transformations, but for a wider class of linear fractional transformations (however, this wider class of transformations - with the exception, of course, of a special case of Lorentz transformations - does not keep the metric constant).

Content

Algebraic Inference

On the basis of several natural assumptions (the main of which is the assumption of the existence of a maximum propagation velocity of interactions), it can be shown that when the ISO changes, the quantity

ds^{2}=c^{2}dt^{2}-dx^{2}-dy^{2}-dz^{2}

{\ displaystyle ds ^ {2} = c ^ {2} dt ^ {2} -dx ^ {2} -dy ^ {2} -dz ^ {2}}

,

called interval . This theorem directly implies the general form of the Lorentz transformations ( see below ). Here we consider only a special case. For clarity, when switching to ISO $K^{'}$ ${\ displaystyle K '}$ moving at speed $v$ ${\ displaystyle v}$ , select in the source system $K$ ${\ displaystyle K}$ axis $X$ ${\ displaystyle X}$ codirectional with $v$ ${\ displaystyle v}$ , and the axis $Y$ ${\ displaystyle Y}$ and $Z$ ${\ displaystyle Z}$ position perpendicular to axis $X$ ${\ displaystyle X}$ . ISO spatial axes $K^{'}$ ${\ displaystyle K '}$ at time $t=0$ ${\ displaystyle t = 0}$ choose co-directed with the ISO axes $K$ ${\ displaystyle K}$ . With such a transformation

y'=y,~~z'=z,~~c^{2}t'^{2}-x'^{2}=c^{2}t^{2}-x^{2}

{\ displaystyle y '= y, ~~ z' = z, ~~ c ^ {2} t '^ {2} -x' ^ {2} = c ^ {2} t ^ {2} -x ^ { 2}}

We will seek linear Lorentz transformations, since for infinitesimal coordinate transformations, the differentials of the new coordinates linearly depend on the differentials of the old coordinates, and due to the homogeneity of space and time, the coefficients cannot depend on the coordinates, only on the relative orientation and speed of the IRF.

The fact that the transverse coordinates cannot change is clear from considerations of the isotropy of space. Indeed, the quantity $y^{'}$ ${\ displaystyle y '}$ cannot change and at the same time not depend on $x$ ${\ displaystyle x}$ (except when spinning around $v$ ${\ displaystyle v}$ , which we exclude from consideration), which is easy to verify by substituting such linear transformations into the expression for the interval. But if it depends on $x$ ${\ displaystyle x}$ , then the point with the coordinate $(0,x,0,0)$ ${\ displaystyle (0, x, 0,0)}$ will have a non-zero coordinate $y^{'}$ ${\ displaystyle y '}$ , which contradicts the symmetry of rotation of the system with respect to $v$ ${\ displaystyle v}$ and isotropy of space. Similarly for $z^{'}$ ${\ displaystyle z '}$ .

The most general form of such transformations:

y'=y,~~z'=z,~~ct'=ct\,\operatorname {ch} \,\alpha -x\,\operatorname {sh} \,\alpha ,~~x'=x\,\operatorname {ch} \,\alpha -ct\,\operatorname {sh} \,\alpha

{\ displaystyle y '= y, ~~ z' = z, ~~ ct '= ct \, \ operatorname {ch} \, \ alpha -x \, \ operatorname {sh} \, \ alpha, ~~ x' = x \, \ operatorname {ch} \, \ alpha -ct \, \ operatorname {sh} \, \ alpha}

Where $\alpha$ ${\ displaystyle \ alpha}$ - some parameter called speed . Inverse transformations have the form

y=y',~~z=z',~~ct=ct'\,\operatorname {ch} \,\alpha +x'\,\operatorname {sh} \,\alpha ,~~x=x'\,\operatorname {ch} \,\alpha +ct'\,\operatorname {sh} \,\alpha

{\ displaystyle y = y ', ~~ z = z', ~~ ct = ct '\, \ operatorname {ch} \, \ alpha + x' \, \ operatorname {sh} \, \ alpha, ~~ x = x '\, \ operatorname {ch} \, \ alpha + ct' \, \ operatorname {sh} \, \ alpha}

It is clear that the point resting in ISO $K$ ${\ displaystyle K}$ will have to move to ISO $K^{'}$ ${\ displaystyle K '}$ with speed $-v$ ${\ displaystyle -v}$ . On the other hand, if the point is at rest, then

dx=dx'\,\operatorname {ch} \,\alpha +c\,dt'\,\operatorname {sh} \,\alpha =0

{\ displaystyle dx = dx '\, \ operatorname {ch} \, \ alpha + c \, dt' \, \ operatorname {sh} \, \ alpha = 0}

{\frac {dx'}{c\,dt'}}=-{\frac {v}{c}}=-\operatorname {th} \,\alpha ~\Rightarrow ~\operatorname {th} \,\alpha ={\frac {v}{c}}

{\ displaystyle {\ frac {dx '} {c \, dt'}} = - {\ frac {v} {c}} = - \ operatorname {th} \, \ alpha ~ \ Rightarrow ~ \ operatorname {th} \, \ alpha = {\ frac {v} {c}}}

Given that when changing the ISO should not change the orientation of the space, we get that

\operatorname {ch} \,\alpha \geqslant 0

{\ displaystyle \ operatorname {ch} \, \ alpha \ geqslant 0}

Therefore, the equation for speed is uniquely solvable:

\operatorname {ch} \,\alpha ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}},~~\operatorname {sh} \,\alpha ={\frac {v}{c{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}

{\ displaystyle \ operatorname {ch} \, \ alpha = {\ frac {1} {\ sqrt {1 - {\ frac {v ^ {2}} {c ^ {2}}}}}}, ~~ \ operatorname {sh} \, \ alpha = {\ frac {v} {c {\ sqrt {1 - {\ frac {v ^ {2}} {c ^ {2}}}}}}}}

and Lorentz transformations have the form

\ x'=\gamma (x-vt)

{\ displaystyle \ x '= \ gamma (x-vt)}

\ t'=\gamma (t-{\frac {v}{c^{2}}}x)

{\ displaystyle \ t '= \ gamma (t - {\ frac {v} {c ^ {2}}} x)}

\gamma =\operatorname {ch} \,\alpha ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}

{\ displaystyle \ gamma = \ operatorname {ch} \, \ alpha = {\ frac {1} {\ sqrt {1 - {\ frac {v ^ {2}} {c ^ {2}}}}}}

Parameter $\gamma$ ${\ displaystyle \ gamma}$ called the Lorentz factor .

Symmetry Group of Maxwell Equations

Visual Conclusion of Lorentz Transforms

We accept the postulates of SRT , which are reduced to the extended principle of relativity, which states that all physical processes proceed exactly the same in all inertial reference frames (the principle of constancy of the speed of light in SRT that clarifies it means the extension of the principle of relativity to electrodynamics, together with the clarifying statement that there is no fundamental physical medium (ether), which would single out one of the reference frames in experience - that is, if there is even ether, then its presence should not violate the principle of relativity in practice). In addition, it is useful to clearly emphasize that the principle of the constancy of the speed of light means the presence of a finite speed (from an experiment equal to the speed of light in vacuum), embedded in the fundamental laws (equations), the same for all inertial reference frames, and in each reference frame the speed of light is the same for any directions of its propagation and does not depend on the speed of the source. The principle of the constancy of the speed of light makes up the second postulate of SRT, which is used below.

Transformation for the transverse axes (p. 1)

Billiard ball near the corner pockets. If performed

y\neq y'

{\ displaystyle y \ neq y '}

, the ball would be blocked in some reference systems and would not be blocked in others

Let there be two infinite planes perpendicular to the y axis. The distance between these planes obviously should not depend on the speed of the planes along themselves, which means that it does not depend on the reference frame, which moves relative to the other along the axis $x$ ${\ displaystyle x}$ . (Indeed, in each such system, the transit time of a ray of light moving along the axis $y$ ${\ displaystyle y}$ from one plane to another, equally according to the postulates of SRT).

You can also imagine how moving along the axis $x$ ${\ displaystyle x}$ the body flies into a fixed hole of the same size. If there is no equality $y=y'$ ${\ displaystyle y = y '}$ , then depending on the reference frame in which the measurement is taken, the body may be larger or smaller than the hole. In reality, the body can go through the hole or not go through in all reference systems.

The same, of course, is true for the axis $z$ ${\ displaystyle z}$ . Therefore, excluding for simplicity, the physically uninteresting case of turning the second coordinate system relative to the first by a constant angle, we obtain:

y=y',z=z'

{\ displaystyle y = y ', z = z'}

Slowing down time (p. 2)

Let us show that any process (for example, the course of the clock) in the moving reference frame is slower than in its own reference frame (relative to which it does not move).

Consider a "light clock" consisting of a point source and a light receiver on the axis $y$ ${\ displaystyle y}$ spaced apart $L$ ${\ displaystyle L}$ and measuring the time interval of the passage of the pulse (flash) of light from the source to the receiver is equal $L/c$ ${\ displaystyle L / c}$ .

If the reference systems are moving relative to each other along the axis $x$ ${\ displaystyle x}$ then the distance $L$ ${\ displaystyle L}$ between two points on the axis $y$ ${\ displaystyle y}$ measured in a system fixed relative to these points is the same as measured in a moving frame of reference, since along the axis $y$ ${\ displaystyle y}$ there is no relative motion of systems. Thus, the units of length will be agreed between the systems. The time units will also be coordinated, since the units of length are coordinated, and the speed of light does not depend on the coordinate system.

Thus, the same light clock can be set in each reference system.

Let us compare the time interval of the pulse passage in the reference system where the light clock is at rest and the time interval of the same hours, measured by identical clocks in a moving reference system.

Let the light clock rest in the frame of reference $K$ ${\ displaystyle K}$ (in the figure on the left), and the reference system $K^{'}$ ${\ displaystyle K '}$ moves right along the axis $x$ ${\ displaystyle x}$ with speed $V$ ${\ displaystyle V}$ . The source at the moment of emission of the pulse is located at the origin of coordinates A of the reference frame $K$ ${\ displaystyle K}$ (red dot in the figure), and the receiver - at point B (blue) on the axis $y$ ${\ displaystyle y}$ . In the reference system $K$ ${\ displaystyle K}$ the emitted light pulse reaches receiver B on the axis $y$ ${\ displaystyle y}$ during $\Delta t=L/c$ ${\ displaystyle \ Delta t = L / c}$ .

In the reference system $K^{'}$ ${\ displaystyle K '}$ a light pulse is emitted from the origin at the moment of its coincidence with the origin of the system $K$ ${\ displaystyle K}$ (point A ) , but gets to receiver B after a while $\Delta t'=L'/c$ ${\ displaystyle \ Delta t '= L' / c}$ measured by clocks moving with the system $K^{'}$ ${\ displaystyle K '}$ . Coordinate $x^{'}$ ${\ displaystyle x '}$ points B - the displacement indicated by the dotted line in the figure to the right is $-V\Delta t'$ ${\ displaystyle -V \ Delta t '}$ , point A indicates the place where the pulse was emitted from, the trajectory of the pulse in $K^{'}$ ${\ displaystyle K '}$ depicted by a green line.

Since the speed of light is the same in any inertial reference frame (does not depend on the speed of the source and direction of radiation), source A at the moment of the pulse can be considered stationary in the reference frame $K^{'}$ ${\ displaystyle K '}$ .

Way $L^{'}$ ${\ displaystyle L '}$ passed by a light pulse from A to B in the reference frame $K^{'}$ ${\ displaystyle K '}$ is equal to the hypotenuse of a right triangle . By the Pythagorean Theorem

{\displaystyle L'}=L^{2}+(V\Delta t')^{2}

{\ displaystyle {\ displaystyle L '} = L ^ {2} + (V \ Delta t') ^ {2}}

,

considering that ${\displaystyle L'}=c\Delta t'$ ${\ displaystyle {\ displaystyle L '} = c \ Delta t'}$ and $L=c\Delta t$ ${\ displaystyle L = c \ Delta t}$ , find the expression for $\Delta t'$ ${\ displaystyle \ Delta t '}$

\Delta t'={\frac {\Delta t}{\sqrt {1-{\frac {V^{2}}{c^{2}}}}}}

{\ displaystyle \ Delta t '= {\ frac {\ Delta t} {\ sqrt {1 - {\ frac {V ^ {2}} {c ^ {2}}}}}}}

.

It follows that $\Delta t'>\Delta t$ ${\ displaystyle \ Delta t '> \ Delta t}$ at $V\neq 0$ ${\ displaystyle V \ neq 0}$ .

So the time interval $\Delta t'$ ${\ displaystyle \ Delta t '}$ any process occurring in the reference system $K$ ${\ displaystyle K}$ measured by hours in a moving reference system $K^{'}$ ${\ displaystyle K '}$ greater than the time interval $\Delta t$ ${\ displaystyle \ Delta t}$ measured by the same clock in its own reference frame $K$ ${\ displaystyle K}$ . The interval increase factor is constant at a constant speed.

Since the reference system $K$ ${\ displaystyle K}$ moves relative to the system $K^{'}$ ${\ displaystyle K '}$ with speed $-V$ ${\ displaystyle -V}$ , then they say that time in a moving frame of reference $K$ ${\ displaystyle K}$ in terms of system $K^{'}$ ${\ displaystyle K '}$ flowing slowly. For example, earthly time of life $\Delta t'$ ${\ displaystyle \ Delta t '}$ short-lived particles that are born at high speeds of other particles, more than their lifetime $\Delta t$ ${\ displaystyle \ Delta t}$ in your own frame of reference.

More clearly, time dilation manifests itself in a slowdown (pace) of the clock moving with the reference system $K$ ${\ displaystyle K}$ . If the source and receiver are equipped with mirrors reflecting a light pulse, then the interval of any duration can be measured by the number of periods between reflections. The oscillation frequency of such a pendulum characterizes the speed of the passage of time. Period $T$ ${\ displaystyle T}$ a repeating process is related to its frequency by equality $\nu ={\frac {1}{T}}$ ${\ displaystyle \ nu = {\ frac {1} {T}}}$ . A greater period corresponds to a lower frequency and inequality $\Delta t'>\Delta t$ ${\ displaystyle \ Delta t '> \ Delta t}$ goes into inequality for frequency $\nu '<\nu$ ${\ displaystyle \ nu '<\ nu}$ where $\nu '$ ${\ displaystyle \ nu '}$ - frequency of the pendulum of the clock moving with the system $K$ ${\ displaystyle K}$ measured by system clock $K^{'}$ ${\ displaystyle K '}$ , $\nu$ ${\ displaystyle \ nu}$ - frequency of the pendulum in its own reference frame $K$ ${\ displaystyle K}$ (regarding which clock is at rest). A moving clock is less frequent than a fixed one.

Since all inertial reference systems are equal, then, by measuring the duration of the pulse in hours moving together with the reference system $K^{'}$ ${\ displaystyle K '}$ clock reference system $K$ ${\ displaystyle K}$ we obtain the inverse inequality $\Delta t'<\Delta t$ ${\ displaystyle \ Delta t '<\ Delta t}$ at $V\neq 0$ ${\ displaystyle V \ neq 0}$ , since in this case $\Delta t'$ ${\ displaystyle \ Delta t '}$ is own time. In terms of reference frame $K$ ${\ displaystyle K}$ moving clock systems $K^{'}$ ${\ displaystyle K '}$ go slower than the system’s own clock $K$ ${\ displaystyle K}$ .

Relativity of simultaneity (item 3)

Change of time on fixed stars in the frame of reference of a moving space ship. When we are near the earth, time changes on Vega and vice versa. As a result, the twin paradox does not arise

In addition to slowing down the time in a moving reference system (slowing down the progress of all hours in a moving laboratory compared to the clock in a stationary laboratory), it turns out that the origin of the time in the moving reference system also does not coincide with that in the stationary one, and the shift of this reference is different at various points - depends on x. A clock in its own frame of reference, showing the same time, shows different lead / lag depending on their location, if you look at them from the frame of reference, relative to which their own frame of reference moves.

In order to understand the very essence of the problem, you will have to think about the question one way or another, and what does it mean that the clock at different points in space (for example, in different cities) runs the same (synchronously), as you can see, or how ( with which procedure) you can synchronize the clock in different places, if initially they were not synchronous.

The already simplest method of synchronization, which consists in the fact that all the clocks are synchronized in one place, and then they are transferred to different points, allows you to make sure that the clocks synchronized in one reference system will look showing different times from another reference system. The fact is that for watches that we transfer to different points along the x axis , their speed relative to another reference frame will be necessarily different, so the time at different points of the x axis will be shifted differently.

This could be carefully considered quantitatively, having obtained the result so sought here. But more simply this can be achieved by considering synchronization with the help of light signals (and the principle of relativity says that any correct synchronization method should give the same result, which, however, can be verified explicitly if desired).

So, consider synchronization using light signals. This process may consist, for example, in the exchange of light signals between two remote chronometers: if the signals are emitted at the same time, then the same time will pass for each clock to receive a signal. But a slightly different (equivalent to this) method is even simpler: it is possible to produce a flash of light precisely in the middle of the segment connecting the chronometers, and state that the light will come to both chronometers at the same time.

In its own reference frame (in which the chronometers are stationary), the picture is symmetrical. However, in any other reference system, both chronometers move (for definiteness we assume that to the right), and then the light from the middle of the segment connecting them at the initial moment will take less time to reach the left chronometer (moving towards the light) than to the right (which light pulse should catch up).

Thus, chronometers running synchronously in their own frame of reference, by the clock of another frame of reference, look asynchronous. The simultaneity of events is relative: events that are simultaneous in one frame of reference are not simultaneous in another.

Simple geometric calculations allow (by depicting the movement of light pulses and chronometers on the xt plane), to obtain an expression for the shift of the origin:

-Vx/c^{2}

{\ displaystyle -Vx / c ^ {2}}

(for simplicity, we considered here only clocks spaced along the x axis, but, of course, everything can be calculated for the general case as well).

Thus, bringing together the results of points 2 and 3, we obtain for time conversion

t'={\frac {t-Vx/c^{2}}{\sqrt {1-V^{2}/c^{2}}}}

{\ displaystyle t '= {\ frac {t-Vx / c ^ {2}} {\ sqrt {1-V ^ {2} / c ^ {2}}}}}

.

This effect can also be proved from the opposite: if it did not exist, or the shift in the reference time would not have been $-Vx/c^{2}$ ${\ displaystyle -Vx / c ^ {2}}$ then the so-called twin paradox would exist.

Lorentz length reduction (Clause 4)

Having considered the movement of the light pulse along the x axis (and not along the y axis, as in step 1), and requiring (based on the postulate of the same speed of light in all inertial reference frames) that the distance between two points should always be equal to the time for which the light goes from one point to another, multiplied by (constant) the speed of light, you can get the factor of reduction of distances along the x axis, and given that the reference offset is $-Vt$ ${\ displaystyle -Vt}$ , you can get the transformation for the x coordinate:

x'={\frac {x-Vt}{\sqrt {1-V^{2}/c^{2}}}}

{\ displaystyle x '= {\ frac {x-Vt} {\ sqrt {1-V ^ {2} / c ^ {2}}}}}

.

It’s even easier now to understand that $x^{'}$ ${\ displaystyle x '}$ expressed in this way, noting that in the plane $x-ct$ ${\ displaystyle x-ct}$ the motion schedule ^[1] of the light pulse should be straight, inclined at 45 ° (due to the fact that the speed of light is always c ), and therefore the scale along the axes $x$ ${\ displaystyle x}$ and $c t$ ${\ displaystyle ct}$ must be the same, and the expressions in the system of units $c=1$ ${\ displaystyle c = 1}$ - symmetrical.

Thus, the Lorentz transformations are quite clearly obtained with collinear spatial axes. Of course, the reverse order of reasoning is also possible: you can first obtain the Lorentz transformations in some more abstract way, for example, using one of the ones mentioned above, and then get all the effects analyzed in the stages of this visual proof as a simple formal consequence of the Lorentz transformations.

Notes

↑ Minkowski called this timeline a world line; however, in this section we will not delve into the connection of the Lorentz transformations with the concept of Minkowski space in its entirety, first of all - so as not to complicate and not interrupt the elementary conclusion, which is more convenient to consider independent of any additional special concepts, restricting ourselves to elementary geometric and algebraic concepts only as much as they are needed. In fact, we are talking specifically about the transformation of coordinates in Minkowski space, and in this section, based on the postulate of the constancy of the speed of light, we just clarify certain properties of this space, as well as Lorentz transformations, as convenient coordinate transformations in it. But once again, for clarity, we emphasize that for the conclusion itself it is not necessary to know anything other than what is explicitly said in the main text of the paragraph.

Links

Lorentz transformations in the book Relativistic world

[1] Minkowski called this timeline a world line; however, in this section we will not delve into the connection of the Lorentz transformations with the concept of Minkowski space in its entirety, first of all - so as not to complicate and not interrupt the elementary conclusion, which is more convenient to consider independent of any additional special concepts, restricting ourselves to elementary geometric and algebraic concepts only as much as they are needed. In fact, we are talking specifically about the transformation of coordinates in Minkowski space, and in this section, based on the postulate of the constancy of the speed of light, we just clarify certain properties of this space, as well as Lorentz transformations, as convenient coordinate transformations in it. But once again, for clarity, we emphasize that for the conclusion itself it is not necessary to know anything other than what is explicitly said in the main text of the paragraph.