special relativity in a mathematical nutshell

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Special relativity

Un article de Quantic.

Special relativity is an extension of rational mechanics where velocity is limited to the speed of light c, a

constant in the vacuum of matter and radiation.

Sommaire

1 Speed limit

2 Derivation of the Lorentz transformation

2.1 Galilean reference frames

2.2 Light invariance principle

2.2.1 Relativity principle

2.3 Expression of the Lorentz transformation

3 Velocity addition law4 Minkowski metric

5 Time dilation

6 Length contraction

7 Acceleration transformation

8 Relativistic Newton's Second Law of Motion

9 Kinetic energy

10 Total relativistic energy E=mc!

11 Notes

Speed limit

In classical mechanics, the absolute velocity is the sum of the velocity of the moving reference frame and thevelocity relative to the moving reference frame. In special relativity one has to take into account the speedlimit, e.g. the light speed. For colinear velocities, we get, as we shall show below :

where vx

is the"absolue" et v'x

the relative velocity. According to the relativity theory, all the velocities are

relative; that's why the absolute velocity is replaced by the velocity in R, the observer's frame. The relativevelocity v'

xis the velocity of the frame R' moving relatively to the observer in frame R. The frame R' moves at

velocity v relatively to R. This formula gives a speed limit as may be seen by replacing v'x

by c to get vx=c. For

an infinite light speed one gets the galilean addition of velocities.

Derivation of the Lorentz transformation

Galilean reference frames

In classical kinematics, the total displacement x in reference frame R is the sum of the relative displacement xin R and of the displacement vt of R relative to R at a velocity v : x = x+vt or, equivalently, x=x-vt. Thisrelation is linear when the velocity v is constant, that is when the frames R and R' are galilean. Time t is thesame in R and R, which is no more valid in special relativity, where t " t. The more general relationship, withfour constants #, $, % and v is :

The Lorentz transformation becomes the Galilean one for $ = % = 1 et # = 0.

Light invariance principle

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The velocity of light is independent of the velocity of the source, as was shown by Michelson. We thus need tohave x = ct if x = ct. Replacing x and x' in these two equations, we have

Replacing t' from the second equation, the first one becomes

After simplification by t and dividing by c$, one obtains :

Relativity principle

This derivation does not use the speed of light and allows therefore to separate it from the principle ofrelativity. The inverse transformation of

is :

In accord with the principle of relativity, the expressions of x and t should write :

They should be identical to the original expressions except for the sign of the velocity :

We should then have the following identities, verified independently of x and t :

This gives the following equalities :

Expression of the Lorentz transformation

Using the above relationship

we get :

and, finally:


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We have now all the four coe&cients needed for the Lorentz transformation which writes in two dimensions :

The inverse Lorentz transformation writes, using the Lorentz factor % :

These four equations are used according to the needs.

The true base of special relativity is the Lorentz transformation generalizing that of Galieo at velocities nearthat of light. The Lorentz transformation expresses the transformation of space and time, both depending onthe relative velocity between the observer's and relative frames R and R'. Another demonstration may be found

in Einstein's book Albert Einstein, Relativity: The Special and General Theory (http://web.mit.edu/birge/Public/books/Einstein-Relativity.pdf) . The Lorentz transformation is, in two dimensions:

In the following, in order to simplify writing, the Lorentz factor will be used:

The inverse Lorentz transformation is:

We have then four equations to be used as needed.

Velocity addition law

The Lorentz transformation remains valid in di'erential form for a constant velocity :

From these two formulas we get the formula at the top of this page:

Minkowski metric

The euclidean space is caracterised by the validity of the Pythagoras theorem which may be written as atwo-dimensional metric:

Wih y=ict, one obtains the Minkowski metric representing the pseudo-euclidean space of special relativity:


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where v is the velocity of the frame R' relative to R. The di'erential form of the Lorentz transformation writes:

Replacing x and t as a function of x' and t' with the Lorentz transformation , one obtains the same Minkowskimetric except for the primes:

Let us develop and simplify:

The Minkowski metric is conserved in the Lorentz transformation.

Time dilation

Let us consider a clock in its rest frame R' moving at a velocity v relative to a frame R where is an observer.The clock rate is (t at rest, in its proper frame R' and (t viewed from R. Since the clock is at rest in R', itsposition is constant in R', say x'=0. To apply the Lorentz transformation, we have to choose the right equationamong the four of the direct and reciprocal Lorentz transformation. We choose the one containing (t, (t andx':

The time interval between two beats appears larger on a moving clock than on a clock at rest. One says thattime is dilated or that the clock is running slow. The time of the moving clock does not flow any more when

the clock moves at light speed, but only for the distant observer, at rest. A high speed particle of limitedlifetime like a meson coming from outer space, wil have an apparently much larger lifetime when viewed fromthe Earth but its proper lifetime remains unchanged.

Let us consider now that an observer places himself in the moving frame R' and looks at a clock placed in therest frame R. We shall have the same formula, but with t and t' reversed. Indeed the movement is relative;there is no absolute movement but a symmetry between both galilean frames.

Length contraction

Now consider a ruler at rest in a frame R' moving at a constant velocity v relative to a frame R where is anobserver. The length at rest of this ruler is (x' for an observer in R'. The ruler appears to have a length (x forthe observer in R. In order to measure the length of the ruler, the latter has to take an instantaneous picture

of the two rulers, for example at time t=0, with (t=0. He obtains then their lengths (x and (x. He will usethe equation of the Lorentz transformation where (x, (x and (t appear:

This formula has the same form as for the time, except that the primes are on the left side. For this reason,lengths contract instead of dilating for the time. Then one writes usually:

Acceleration transformation

In classical kinematics, accelerations do not depend on the velocity of the galilean frame since the velocity of


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the frame beeing constant, its derivative, the acceleration, is zero. In special relativity, due to both timedilation and length contraction, the change of galilean frame changes acceleration.

Let be dvx/dt and dv'

x/dt the accelerations of a particle of abscissas x and x' in the frames R of the observer

and R', moving. Since the acceleration is the second derivative of space relative to time and if the frames R andR' are approximately galilean, the Lorentz factor % is a constant to the fourth order, as pointed out by Einstein(the slowly accelerated electron), we have:

Let us bind the accelerated particle to its frame R'. We then have v=vx; the frames are thus no more galilean.

At low speeds, we are in the newtonian domain where %)1; the accelerations are practically equal in R et R'.For a velocity near the speed of light, the variation dv/dt of the velocity is small and, then, the acceleration, asviewed from R, is low. In both cases the frames are approximately galilean. It is only at intermediate velocitiesthat % may be large with a variation of the velocity not negligible. This approximation seems to be validaccording to the rares experimental data.

We may write v'=vx'

:

Using an identity due, as it seems, to Lorentz, we have

one obtains then:

Relativistic Newton's Second Law of Motion

Let us multiply both sides of the acceleration transformation equation by the constant rest mass m0:

In the frame R' where the velcity of the particle is low (in fact zero), one may apply Newton's Second Law ofMotion. The right side represents the force F' in the frame R'. If one admits that the force does not depend oneframe since it is applied to the particle, we have F=F' and, then

where mr

is the relativistic mass, appearing to the distant observer, varying in fonction of the velocity:

Kinetic energy

In a frame moving at velocity v relative to the observer, contrarily to the Galilean transformation, the Lorentztransformation gives an acceleration depending on the relative speeds of the referentials, even galileans (welimit ourselves to the case where velocity and acceleration are colinear). In order to produce the acceleration a= dv/dt, it is necessary to apply a force, defined by the relativistic Newton's Second Law of Motion which is aderivative relative to time of the momentum m

rv. The variation dT of the kinetic energy being equal to the

work of the applied force F for a displacement dx, we have:

Let us use an identity similar to that of Lorentz above:


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The variation of the kinetic energy becomes dT=m0d%. Integrating this equation, one obtains:

The kinetic energy should be zero when the velocity v is zero, e.g. when %=1. The integration constant is thus:

The kinetic energy is:

equal to the di'erence between rest mass m0

and relativistic mass mr

multiplied by the universal factor .

These two masses have indices in order to avoid any confusion.

Total relativistic energy E=mc!

The total relativistic energy E = mc!

must not be confused with the total classical mechanical energy. The sumof the kinetic energy and the potential energy remains a constant value without being an absolute value.

Drivers know that the distance they may travel is proportional to their volume or mass of gas with a coe&cientK depending on its heat content. It may be assumed that there is a maximum value of energy of any typecontained in a given mass. The maximum energy avalable in a given mass is obtained when all the mass isconverted into energy (radiative,thermal, mechanical, electrical). A higher energy content is impossible,because there is no matter anymore. The problem is to find the universal coe&cient K. Let us apply relativity.The maximum energy available is then E

r= Km

rin the observer's frame and E

0=Km

0in its proper frame of the

object of mass m0. The di'erence between these two energies is

is due uniquely to the velocity, the relativistic mass depending only on the rest mass and the relative velocitybetween the object and the observer. The application of the Lorentz transformation, of newton's law and ofthe definition of energy has shown in the preceding paragraph that the relativistic kinetic energy is:

Identifying these last equations, one finds

The total relativistic energy is then:

where m is the mass, static or dynamic. We have derived the most celebrated equation of the twentiethcentury with the Lorentz transformation and Newton's laws.

Notes

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Catgorie: Physics

Dernire modification de cette page le 23 juillet 2009 10:18.


2/08/09 6:10

special relativity in a mathematical nutshell

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