17. Relativistic Mechanics

17.1: Introduction to Relativistic Mechanics¶

Newtonian mechanics incorporates the Newtonian concept of the complete separation of space and time. This theory reigned supreme from inception, in 1687, until November 1905 when Einstein pioneered the Special Theory of Relativity. Relativistic mechanics undermines the Newtonian concepts of absoluteness of time that is inherent to Newton’s formulation, as well as when recast in the Lagrangian and Hamiltonian formulations of classical mechanics. Relativistic mechanics has had a profound impact on twentieth-century physics and the philosophy of science. Classical mechanics is an approximation of relativistic mechanics that is valid for velocities much less than the velocity of light in vacuum. The term “relativity” refers to the fact that physical measurements are always made relative to some chosen reference frame. Naively one may think that the transformation between different reference frames is trivial and contains little underlying physics. However, Einstein showed that the results of measurements depend on the choice of coordinate system, which revolutionized our concept of space and time.

Einstein’s work on relativistic mechanics comprised two major advances. The first advance is the 1905 Special Theory of Relativitywhich refers to nonaccelerating frames of reference. The second major advance was the 1916 General Theory of Relativity which considers accelerating frames of reference and their relation to gravity. The Special Theory is a limiting case of the General Theory of Relativity. The mathematically complex General Theory of Relativity is required for describing accelerating frames, gravity, plus related topics like Black Holes, or extremely accurate time measurements inherent to the Global Positioning System. The present discussion will focus primarily on the mathematically simple Special Theory of Relativity since it encompasses most of the physics encountered in atomic, nuclear and high energy physics. This chapter uses the basic concepts of the Special Theory of Relativity to investigate the implications of extending Newtonian, Lagrangian and Hamiltonian formulations of classical mechanics into the relativistic domain. The Lorentz-invariant extended Hamiltonian and Lagrangian formalisms are introduced since they are applicable to the Special Theory of Relativity. The General Theory of Relativity incorporates the gravitational force as a geodesic phenomena in a four-dimensional Reimannian structure based on space, time, and matter. A superficial outline is given to the fundamental concepts and evidence that underlie the General Theory of Relativity.

17.2: Galilean Invariance¶

As discussed in chapter 2.3, an inertial frame is one in which Newton’s Laws of motion apply. Inertial frames are non-accelerating frames so that pseudo forces are not induced. All reference frames moving at constant velocity relative to an inertial reference, are inertial frames. Newton’s Laws of nature are the same in all inertial frames of reference and therefore there is no way of determining absolute motion because no inertial frame is preferred over any other. This is called Galilean-Newtonian invariance. Galilean invariance assumes that the concepts of space and time are completely separable. Time is assumed to be an absolute quantity that is invariant to transformations between coordinate systems in relative motion. Also the element of length is the same in different Galilean frames of reference.

Consider two coordinate systems shown in Figure 17.2.1, where the primed frame is moving along the $x$ axis of the fixed unprimed frame. A Galilean transformation implies that the following relations apply;

Note that at any instant $t$, the infinitessimal units of length in the two systems are identical since

These are the mathematical expression of the Newtonian idea of space and time. An immediate consequence of the Galilean transformation is that the velocity of light must differ in different inertial reference frames.

At the end of the 19$^{th}$ century physicists thought they had discovered a way of identifying an absolute inertial frame of reference, that is, it must be the frame of the medium that transmits light in vacuum. Maxwell’s laws of electromagnetism predict that electromagnetic radiation in vacuum travels at $c = \frac{1}{\sqrt{\mu_o \varepsilon_o}} = 2.998 \times 10^8$ $m/s$. Maxwell did not address in what frame of reference that this speed applied. In the nineteenth century all wave phenomena were transmitted by some medium, such as waves on a string, water waves, sound waves in air. Physicists thus envisioned that light was transmitted by some unobserved medium which they called the ether. This ether had mystical properties, it existed everywhere, even in outer space, and yet had no other observed consequences. The ether obviously should be the absolute frame of reference.

In the 1880’s, Michelson and Morley performed an experiment in Cleveland to try to detect this ether. They transmitted light back and forth along two perpendicular paths in an interferometer, shown in Figure 17.2.2, and assumed that the earth’s motion about the sun led to movement through the ether.

The time taken to travel a return trip takes longer in a moving medium, if the medium moves in the direction of the motion, compared to travel in a stationary medium. For example, you lose more time moving against a headwind than you gain travelling back with the wind. The time difference $\Delta t$, for a round trip to a distance $L$, between travelling in the direction of motion in the ether, versus travelling the same distance perpendicular to the movement in the ether, is given by $\Delta t \approx \frac{L}{c} \left(\frac{v}{c}\right)^2$ where $v$ is the relative velocity of the ether and $c$ is the velocity of light.

Interference fringes between perpendicular light beams in an optical interferometer provides an extremely sensitive measure of this time difference. Michelson and Morley observed no measurable time difference at any time during the year, that is, the relative motion of the earth within the ether is less than $1/6$ the velocity of the earth around the sun. Their conclusion was either, that the ether was dragged along with the earth, or the velocity of light was dependent on the velocity of the source, but these did not jibe with other observations. Their disappointment at the failure of this experiment to detect evidence for an absolute inertial frame is important and confounded physicists for two decades until Einstein’s Special Theory of Relativity explained the result.

17.3: Special Theory of Relativity¶

Einstein Postulates¶

In November 1905, at the age of 26, Einstein published a seminal paper entitled ”On the electrodynamics of moving bodies”. He considered the relation between space and time in inertial frames of reference that are in relative motion. In this paper he made the following postulates.

1. The laws of nature are the same in all inertial frames of reference.

2. The velocity of light in vacuum is the same in all inertial frames of reference.

Note that Einstein’s first postulate, coupled with Maxwell’s equations, leads to the statement that the velocity of light in vacuum is a universal constant. Thus the second postulate is unnecessary since it is an obvious consequence of the first postulate plus Maxwell’s equations which are basic laws of physics. This second postulate explained the null result of the Michelson-Morley experiment. However, it was not this experimental result that led Einstein to the theory of special relativity; he deduced the Special Theory of Relativity from consideration of Maxwell’s equations of electromagnetism. Although Einstein’s postulates appear reasonable, they lead to the following surprising implications.

Lorentz transformation¶

Galilean invariance leads to violation of the Einstein postulate that the velocity of light is a universal constant in all frames of reference. It is necessary to assume a new transformation law that renders physical laws relativistically invariant. Maxwell’s equations are relativistically invariant, which led to some electromagnetic phenomena that could not be explained using Galilean invariance. In 1904 Lorentz proposed a new transformation to replace the Galilean transformation in order to explain such electromagnetic phenomena. Einstein’s genius was that he derived the transformation, that had been proposed by Lorentz, directly from the postulates of the Special Theory of Relativity. The Lorentz transformation satisfies Einstein’s theory of relativity, and has been confirmed to be correct by many experiments.

For the geometry shown in Figure 17.2.1, the Lorentz transformations are:

where the Lorentz $\gamma$ factor

The inverse transformations are

The Lorentz $\gamma$ factor, defined above, is the key feature differentiating the Lorentz transformations from the Galilean transformation. Note that $\gamma \geq 1$; also $\gamma \rightarrow 1.0$ as $v \rightarrow 0$ and increases to infinity as $\frac{v}{c} \rightarrow 1$ as illustrated in Figure 17.3.1. A useful fact that will be used later is that for $\frac{v}{c} << 1$;

Note that for $v << c$ then $\gamma = 1$ and the Lorentz transformation is identical to the Galilean transformation.

Time Dilation¶

Consider that a clock is fixed at $x^{\prime}_o$ in a moving frame and measures the time interval between two events in the moving frame, i.e. $\Delta t^{\prime}_p = t^{\prime}_1 − t^{\prime}_2$. According to the Lorentz transformation, the times in the fixed frame are given by:

Thus the time interval is given by:

The time between events in the rest frame of the clock, $\Delta \tau \equiv \Delta t^{\prime}_p$ is called the proper time which always is the shortest time measured for a given event and is represented by the symbol $\tau$. That is

Note that the time interval for any other frame of reference, moving with respect to the clock frame, will show larger time intervals because $\gamma \geq 1.0$ which implies that the fixed frame perceives that the moving clock is slow by the factor $\gamma$.

The plausibility of this time dilation can be understood by looking at the simple geometry of the space ship example shown in Figure 17.3.2. Pretend that the clock in the proper frame of the space ship is based on the time for the light to travel to and from the mirror in the space ship. In this proper frame the light has the shortest distance to travel, and the proper transit time is

In the fixed frame, $b$, the component of velocity in the direction of the mirror is $\sqrt{c^2 − v^2}$ using the Pythagorus theorem, assuming that the light cannot travel faster the $c$. Thus the transit time towards and back from the mirror must be

which is the predicted time dilation.

There are many experimental verifications of time dilation in physics. For example, a stationary muon has a mean lifetime of $\tau_p = 2$ $\mu s$, whereas the lifetime of a fast moving muon, produced in the upper atmosphere by high-energy cosmic rays, was observed in 1941 to be longer and given by $\gamma\tau_p$ as described in example 17.3.1. In 1972 Hafely and Keating used four accurate cesium atomic clocks to confirm time dilation. Two clocks were flown on regularly scheduled airlines travelling around the World, one westward and the other eastward. The other two clocks were used for reference. The westward moving clock was slow by $(273 \pm 7)$ $n s$ compared to the predicted value of $(275 \pm 10)$ $n s$. The Global Positioning System of 24 geosynchronous satellites is used for locating positions to within a few meters. It has an accuracy of a few nanoseconds which requires allowance for time dilation and is a daily tribute to the correctness of Einstein’s Theory of Relativity.

Length Contraction¶

The Lorentz transformation leads to a contraction of the apparent length of an object in a moving frame as seen from a fixed frame. The length of a ruler in its own frame of reference is called the proper length. Consider an accurately measured rod of known proper length $L_p = x^{\prime}_2 − x^{\prime}_1$ that is, at rest in the moving primed frame. The locations of both ends of this rod are measured at a given time in the stationary frame, $t_1 = t_2$, by taking a photograph of the moving rod. The corresponding locations in the moving frame are:

Since $t_2 = t_1$, the measured lengths in the two frames are related by:

That is, the lengths are related by:

Note that the moving rod appears shorter in the direction of motion. As $v \rightarrow c$ the apparent length shrinks to zero in the direction of motion while the dimensions perpendicular to the direction of motion are unchanged. This is called the Lorentz contraction. If you could ride your bicycle at close to the speed of light, you would observe that stationary cars, buildings, people, all would appear to be squeezed thin along the direction that you are travelling. Also objects that are further away down any side street would be distorted in the direction of travel. A photograph taken by a stationary observer would show the moving bicycle to be Lorentz contracted along the direction of travel and the stationary objects would be normal.

Simultaneity¶

The Lorentz transformations imply a new philosophy of space and time. A surprising consequence is that the concept of simultaneity is frame dependent in contrast to the prediction of Newtonian mechanics.

Consider that two events occur in frame $S$ at $(x_1, t_1)$ and $(x_2, t_2)$. In frame $S^{\prime}$ these two events occur at $(x^{\prime}_1, t^{\prime}_1)$ and $(x^{\prime}_2, t^{\prime}_2)$. From the Lorentz transformation the time difference is

If an event is simultaneous in frame $S$, that is $(t_2 − t_1)=0$ then

Thus the event is not simultaneous in frame $S^{\prime}$ if $(x_2 − x_1) = L_p \neq 0$. That is, an event that is simultaneous in one frame is not simultaneous in the other frame if the events are spatially separated. The equivalent statement is that for two clocks, spatially separated by a distance $L_p$, which are synchronized in their rest frame, then in a moving frame they are not simultaneous.

Einstein discussed the problem of lightning striking both ends of a railway carriage that is moving at a velocity $v$. Assume that the lightning strikes both the front and rear of the carriage simultaneously, according to a stationary observer. A woman riding in the center of the train will se the lightning flash arrive from the front of the carriage before the wavefront from the rear of the carriage arrives since the carriage is moving towards the approaching wavefront and away from the wavefront from the rear of the train. If the length of the carriage is $L$, then the time difference between the light flash from front and rear of the carriage will be $\Delta t = \gamma L_p \frac{v}{c^2}$. As a consequence she observes that the two signals are not simultaneous. Thus a photograph of a rapidly moving body will appear to have a shorter distance. The relativistic snake discussed in chapter 17, exercise 1 is a similar example of the role of simultaneity in relativistic mechanics.

17.4: Relativistic Kinematics¶

Velocity Transformations¶

Consider the two parallel coordinate frames with the primed frame moving at a velocity $v$ along the $x^{\prime}_1$ axis as shown in Figure 17.2.1. Velocities of an object measured in both frames are defined to be

Using the Lorentz transformations $(17.3.1)$, $(17.3.3)$ between the two frames moving with relative velocity $v$ along the $x_1$ axis, gives that the velocity along the $x^{\prime}_1$ axis is

Similarly we get the velocities along the perpendicular $x^{\prime}_2$ and $x^{\prime}_3$ axes to be

When $\dfrac{u_1v}{ c^2} \rightarrow 0$ these velocity transformations become the usual Galilean relations for velocity addition. Do not confuse $\mathbf{u}$ and $\mathbf{u}^{\prime}$ with $\mathbf{v}$; that is, $\mathbf{u}$ and $\mathbf{u}^{\prime}$ are the velocities of some object measured in the unprimed and primed frames of reference respectively, whereas $\mathbf{v}$ is the relative velocity of the origin of one frame with respect to the origin of the other frame.

Momentum¶

Using the classical definition of momentum, that is $\mathbf{p} = m\mathbf{u}$, the linear momentum is not conserved using the above relativistic velocity transformations if the mass $m$ is a scalar quantity. This problem originates from the fact that both $\mathbf{x}$ and $t$ have non-trivial transformations and thus $\mathbf{u} = \dfrac{d\mathbf{x}}{dt}$ is frame dependent.

Linear momentum conservation can be retained by redefining momentum in a form that is identical in all frames of reference, that is by referring to the proper time $\tau$ as measured in the rest frame of the moving object. Therefore we define relativistic linear momentum as

But we know the time dilation relation

Note that the $\gamma_u$ in this relation refers to the velocity $u$ between the moving object and the frame; this is quite different from the $\gamma = \dfrac{ 1}{\sqrt{ (1− \dfrac{v^2}{ c^2} )}}$ which refers to the transformation between the two frames of reference. Thus the new relativistic definition of momentum is

The relativistic definition of linear momentum is the same as the classical definition with the rest mass $m$ replaced by the relativistic mass $\gamma m$.^[1]

Center of momentum coordinate system¶

The classical relations for handling the kinematics of colliding objects, carry over to special relativity when the relativistic definition of linear momentum, Equation 17.21, is assumed. That is, one can continue to apply conservation of linear momentum. However, there is one important conceptual difference for relativistic dynamics in that the center of mass no longer is a meaningful concept due to the interrelation of mass and energy. However, this problem is eliminated by considering the center of momentum coordinate system which, as in the non-relativistic case, is the frame where the total linear momentum of the system is zero. Using the concept of center of momentum incorporates the formalism of classical non-relativistic kinematics.

Force¶

Newton’s second law $\mathbf{F} = \dfrac{d\mathbf{p}}{ dt}$ is covariant under a Galilean transformation. In special relativity this definition also applies using the relativistic definition of momentum $\mathbf{p}$. The fact that the relativistic momentum $\mathbf{p}$ is conserved in the force-free situation, leads naturally to using the definition of force to be

Then the relativistic momentum is conserved if $\mathbf{F} = 0$.

Energy¶

The classical definition of work done is defined by

Assume $T_1 = 0$, let $d\mathbf{r} = \mathbf{u}dt$ and insert the relativistic force relation in Equation 17.23, gives

Integrate by parts, followed by algebraic manipulation, gives

Define the rest energy $E_0$

and total relativistic energy $E$

then Equation 17.25 can be written as

This is the famous Einstein relativistic energy that relates the equivalence of mass and energy. The total relativistic energy $E$ is a conserved quantity in nature. It is an extension of the conservation of energy and manifestations of the equivalence of energy and mass occur extensively in the real world.

In nuclear physics we often convert mass to energy and back again to mass. For example, gamma rays with energies greater than 1.022 $MeV$, which are pure electromagnetic energy, can be converted to an electron plus positron both of which have rest mass. The positron can then annihilate a different electron in another atom resulting in emission of two 511 $keV$ gamma rays in back to back directions to conserve linear momentum. A dramatic example of Einstein’s equation is a nuclear reactor. One gram of material, the mass of a paper clip, provides $E = 9 \times 10^{13}$ joules. This is the daily output of a 1 ${GWatt}$ nuclear power station or the explosive power of the Nagasaki or Hiroshima bombs.

As the velocity of a particle $v$ approaches $c$, then $\gamma$ and the relativistic mass $\gamma m$ both approach infinity. This means that the force needed to accelerate the mass also approaches infinity, and thus no particle can exceed the velocity of light. The energy continues to increase not by increasing the velocity but by increase of the relativistic mass. Although the relativistic relation for kinetic energy is quite different from the Newtonian relation, the Newtonian form is obtained for the case of $u \ll c$ in that

An especially useful relativistic relation that can be derived from the above is

This is useful because it provides a simple relation between total energy of a particle and its relativistic linear momentum plus rest energy.

17.5: Geometry of Space-time¶

Four-Dimensional Space-Time¶

In 1906 Poincaré showed that the Lorentz transformation can be regarded as a rotation in a 4-dimensional Euclidean space-time introduced by adding an imaginary fourth space-time coordinate $ict$ to the three real spatial coordinates. In 1908 Minkowski reformulated Einstein’s Special Theory of Relativity in this 4-dimensional Euclidean space-time vector space and concluded that the spatial variables $q_i$, where $(i = 1, 2, 3)$, plus the time $q_0 = ict$ are equivalent variables and should be treated equally using a covariant representation of both space and time. The idea of using an imaginary time axis $ict$ to make space-time Euclidean was elegant, but it obscured the non-Euclidean nature of space-time as well as causing difficulties when generalized to non-inertial accelerating frames in the General Theory of Relativity. As a consequence, the use of the imaginary $ict$ has been abandoned in modern work. Minkowski developed an alternative non-Euclidean metric that treats all four coordinates $(ct, x, y, z)$ as a four-dimensional Minkowski metric with all coordinates being real, and introduces the required minus sign explicitly.

Analogous to the usual 3-dimensional cartesian coordinates, the displacement four vector $d\mathbf{s}$ is defined using the four components along the four unit vectors in either the unprimed or primed coordinate frames.

The convention used is that greek subscripts (covariant) or superscripts (contravariant) designate a four vector with $0 \leq \mu \leq 3$. The covariant unit vectors $\mathbf{\hat{e}}_{\mu}$ are written with the subscript $\mu$ which has 4 values $0 \leq \mu \leq 3$. As described in appendix 19.5.3, using the Einstein convention the components are written with the contravariant superscript $dx^{\mu}$ where the time axis $x^0 = ct$, while the spatial coordinates, expressed in cartesian coordinates, are $x^1 = x$, $x^2 = y$, and $x^3 = z$. With respect to a different (primed) unit vector basis $\mathbf{\hat{e}}^{\prime}_{\mu}$, the displacement must be unchanged as given by Equation 17.31. In addition, Equation 17.43 shows that the magnitude $|ds|^2$ of the displacement four vector is invariant to a Lorentz transformation.

The most general Lorentz transformation between inertial coordinate systems $S$ and $S^{\prime}$, in relative motion with velocity $\mathbf{v}$, assuming that the two sets of axes are aligned, and that their origins overlap when $t = t^{\prime} = 0$, is given by the symmetric matrix $\lambda$ where

This Lorentz transformation of the four vector $\mathbb{X}$ components can be written in matrix form as

Assuming that the two sets of axes are aligned, then the elements of the Lorentz transformation $\lambda_{ \mu \nu }$ are given by

where $\beta = \frac{v}{ c}$ and $\gamma = \frac{1}{\sqrt{ 1−\beta^2}}$ and assuming that the origin of $S$ transforms to the origin of $S^{\prime}$ at $(0, 0, 0, 0)$.

For the case illustrated in Figure 17.2.1, where the corresponding axes of the two frames are parallel and in relative motion with velocity $v$ in the $x_1$ direction, then the Lorentz transformation matrix 17.34 reduces to

This Lorentz transformation matrix is called a standard boost since it only boosts from one frame to another parallel frame. In general a rotation matrix also is incorporated into the transformation matrix $\lambda$ for the spatial variables.

Four-vector scalar products¶

Scalar products of vectors and tensors usually are invariant to rotations in three-dimensional space providing an easy way to solve problems. The scalar, or inner, product of two four vectors is defined by

The correct sign of the inner product is obtained by inclusion of the Minkowski metric $g$ defined by

that is, it can be represented by the matrix

The sign convention used in the Minkowski metric, Equation 17.38, has been chosen with the time coordinate $(ct)^2$ positive which makes $(ds)^2 > 0$ for objects moving at less than the speed of light and corresponds to $ds$ being real.^[2]

The presence of the Minkowski metric matrix, in the inner product of four vectors, complicates General Relativity and thus the Einstein convention has been adopted where the components of the contravariant four-vector $\mathbb{X}$ are written with superscripts $X^{\mu}$. See also appendix 19.6. The corresponding covariant four-vector components are written with the subscript $X_{\mu}$ which is related to the contravariant four-vector components $X^{\nu}$ using the $\mu\nu$ component of the covariant Minkowski metric matrix $\mathbf{g}$. That is

The contravariant metric component $g^{\mu\nu}$ is defined as the $\mu\nu$ component of the inverse metric matrix $\mathbf{g}^{−1}$ where

where $\mathbf{I}$ is the four-vector identity matrix. The contravariant components of the four vector can be expressed in terms of the covariant components as

Thus equations 17.39 and 17.41 can be used to transform between covariant and contravariant four vectors, that is, to raise or lower the index $\mu$.

The scalar inner product of two four vectors can be written compactly as the scalar product of a covariant four vector and a contravariant four vector. The Minkowski metric matrix can be absorbed into either $\mathbb{X}$ or $\mathbb{Y}$ thus

If this covariant expression is Lorentz invariant in one coordinate system, then it is Lorentz invariant in all coordinate systems obtained by proper Lorentz transformations.

The scalar inner product of the invariant space-time interval is an especially important example.

This is invariant to a Lorentz transformation as can be shown by applying the Lorentz standard boost transformation given above. In particular, if $S^{\prime}$ is the rest frame of the clock, then the invariant space-time interval $ds$ is simply given by the proper time interval $d\tau$.

Minkowski Space-Time¶

Figure 17.5.1 illustrates a three-dimensional $( ct, x^1, x^2)$ representation of the 4−dimensional space-time diagram where it is assumed that $x^3 = 0$. The fact that the velocity of light has a fixed velocity leads to the concept of the light cone defined by the locus of $|x| = ct$.

Inside the light cone¶

The vertex of the cones represent the present. Locations inside the upper cone represent the future while the past is represented by locations inside the lower cone. Note that $(ds)^2 = c^2 (dt)^{\mathbf{2}} − (d\mathbf{r})^2 > 0$ inside both the future and past light cones. Thus the space-time interval $c\Delta t$ is real and positive for the future, whereas it is real and negative for the past relative to the vertex of the light cone. A world line is the trajectory a particle follows is a function of time in Minkowski space. In the interior of the future light cone $\Delta t > 0$ and, since it is real, it can be asserted unambiguously that any point inside this forward cone must occur later than at the vertex of the cone, that is, it is the absolute future. A Lorentz transformation can rotate Minkowski space such that the axis $x_0$ goes through any point within this light cone and then the “world line” is pure time like. Similarly, any point inside the backward light cone unambiguously occurred before the vertex, i.e. it is absolute past.

Outside the Light Cone¶

Outside of the light cone, has

and thus $\Delta s$ is imaginary and is called space like. A spacelike plane hypersurface in spatial coordinates is shown for the present time in the unprimed frame. A rotation in Minkowski space can be made to $s^{\prime}$ such that the space-like hypersurface now is tilted relative to the hypersurface shown and thus any point $P$ outside the light cone can be made to occur later, simultaneous, or earlier than at the vertex depending on the orientation of the space-like hypersurface. This startling situation implies that the time ordering of two points, each outside the others light cone, can be reversed which has profound implications related to the concept of simultaneity and the notion of causality.

For the special case of two events lying on the light cone:

and thus these events are separated by a light ray travelling at velocity $c$. Only events separated by time-like intervals can be connected causally. The world line of a particle must lie within its light cone. The division of intervals into space-like and time-like, because of their invariance, is an absolute concept. That is, it is independent of the frame of reference.

The concept of proper time can be expanded by considering a clock at rest in frame $S^{\prime}$ which is moving with uniform velocity $v$ with respect to a rest frame $S$. The clock at rest in the $S^{\prime}$ frame measures the proper time$\tau$, then the time observed in the fixed frame can be obtained by looking at the interval $ds$. Because of the invariance of the interval, $ds^2$ then

That is,

that is $dt = \gamma d\tau$ which satisfies the normal expression for time dilation, $(17.3.5)$.

Momentum-energy four vector¶

The previous four-vector discussion can be elegantly exploited using the covariant Minkowski space-time representation. Separating the spatial and time of the differential four vector gives

Remember that the square of the four-dimensional space-time element of length $(ds)^2$ is invariant 17.43, and is simply related to the proper time element $d\tau$. Thus the scalar product

Thus the proper time is an invariant.

The ratio of the four-vector element $d\mathbb{X}$ and the invariant proper time interval $d\tau$, is a four-vector called the four-vector velocity $\mathbb{U}$ where

where $\mathbf{u}$ is the particle velocity, and $\gamma_u = \frac{1}{\sqrt{(1 - \frac{u^2}{c^2})}}$.

The four-vector momentum $\mathbb{P}$ can be obtained from the four-vector velocity by multiplying it by the scalar rest mass $m$

However,

thus the momentum four vector can be written as

where the vector $\mathbf{p}$ represents the three spatial components of the relativistic momentum. It is interesting to realize that the Theory of Relativity couples not only the spatial and time coordinates, but also, it couples their conjugate variables linear momentum $\mathbf{p}$ and total energy, $\frac{E}{c}$.

An additional feature of this momentum-energy four vector $\mathbb{P}$, is that the scalar inner product $\mathbb{P} \cdot \mathbb{P}$ is invariant to Lorentz transformations and equals $(mc)^2$ in the rest frame

which leads to the well-known equation

The Lorentz transformation matrix $\lambda$ can be applied to $\mathbb{P}$

The Lorentz invariant four-vector representation is illustrated by applying the Lorentz transformation shown in Figure 17.2.1, which gives, $p^{\prime}_1 = \gamma \left(p_1 - (\frac{v}{c})^2 E \right)$, $p^{\prime}_2 = p_2$, $p^{\prime}_3 = p_3$, and $E^{\prime} = \gamma (E − vp_1)$.

17.6: Lorentz-Invariant Formulation of Lagrangian Mechanics¶

Parametric Formulation¶

The Lagrangian and Hamiltonian formalisms in classical mechanics are based on the Newtonian concept of absolute time $t$ which serves as the system evolution parameter in Hamilton’s Principle. This approach violates the Special Theory of Relativity. The extended Lagrangian and Hamiltonian formalism is a parametric approach, pioneered by Lanczos[La49], that introduces a system evolution parameter $s$ that serves as the independent variable in the action integral, and all the space-time variables $q_i (s), t(s)$ are dependent on the evolution parameter $s$. This extended Lagrangian and Hamiltonian formalism renders it to a form that is compatible with the Special Theory of Relativity. The importance of the Lorentz-invariant extended formulation of Lagrangian and Hamiltonian mechanics has been recognized for decades.[La49, Go50, Sy60] Recently there has been a resurgence of interest in the extended Lagrangian and Hamiltonian formalism stimulated by the papers of Struckmeier[Str05, Str08] and this formalism has featured prominently in recent textbooks by Johns[Jo05] and Greiner[Gr10]. This parametric approach develops manifestly-covariant Lagrangian and Hamiltonian formalisms that treat equally all $2n+1$ space-time canonical variables. It provides a plausible manifestly-covariant Lagrangian for the one-body system, but serious problems exist extending this to the $N$-body system when $N > 1$. Generalizing the Lagrangian and Hamiltonian formalisms into the domain of the Special Theory of Relativity is of fundamental importance to physics, while the parametric approach gives insight into the philosophy underlying use of variational methods in classical mechanics.^[3]

In conventional Lagrangian mechanics, the equations of motion for the $n$ generalized coordinates are derived by minimizing the action integral, that is, Hamilton’s Principle.

where $L(\mathbf{q}(t), \mathbf{\dot{q}}(t),t)$ denotes the conventional Lagrangian. This approach implicitly assumes the Newtonian concept of absolute time $t$ which is chosen to be the independent variable that characterizes the evolution parameter of the system. The actual path $[\mathbf{q}(t), \mathbf{\dot{q}}(t)]$ the system follows is defined by the extremum of the action integral $S(\mathbf{q}, \mathbf{\dot{q}},t)$ which leads to the corresponding Euler-Lagrange equations. This assumption is contrary to the Theory of Relativity which requires that the space and time variables be treated equally, that is, the Lagrangian formalism must be covariant.

Extended Lagrangian¶

Lanczos[La49] proposed making the Lagrangian covariant by introducing a general evolution parameter $s$, and treating the time as a dependent variable $t(s)$ on an equal footing with the configuration space variables $q^i (s)$. That is, the time becomes a dependent variable $q_0(s) = ct(s)$ similar to the spatial variables $q_{\mu} (s)$ where $1 \leq \mu \leq n$. The dynamical system then is described as motion confined to a hypersurface within an extended space where the value of the extended Hamiltonian and the evolution parameter $s$ constitute an additional pair of canonically conjugate variables in the extended space. That is, the canonical momentum $p_0$, corresponding to $q_0 = ct$, is $p_0 = \frac{E}{c}$ similar to the momentum-energy four vector, equation $(17.5.21)$.

An extended Lagrangian $\mathbb{L}(\mathbf{q}(s), \frac{d\mathbf{q}(s) }{ds },t(s), \frac{dt(s)}{ ds} )$ can be defined which can be written compactly as $\mathbb{L}(q^{\mu} (s), \frac{dq^{\mu }(s)}{ ds} )$ where the index $0 \leq \mu \leq n$ denotes the entire range of space-time variables.

This extended Lagrangian can be used in an extended action functional $\mathbb{S}(\mathbf{q}, \frac{d\mathbf{q} }{ds} ,t, \frac{dt}{ ds} )$ to give an extended version of Hamilton’s Principle^[4]

The conventional action $S$, and extended action $\mathbb{S}$, address alternate characterizations of the same underlying physical system, and thus the action principle implies that $\delta S = \delta \mathbb{S} = 0$ must hold simultaneously. That is,

As discussed in chapter 9.3, there is a continuous spectrum of equivalent gauge-invariant Lagrangians for which the Euler-Lagrange equations lead to identical equations of motion. Equation 17.57 is satisfied if the conventional and extended Lagrangians are related by

where $\Lambda (\mathbf{q},t)$ is a continuous function of $\mathbf{q}$ and $t$ that has continuous second derivatives. It is acceptable to assume that $\frac{d\Lambda (\mathbf{q},t)}{ ds} = 0$, then the extended and conventional Lagrangians have a unique relation requiring no simultaneous transformation of the dynamical variables. That is, assume

Note that the time derivative of $\mathbf{q}$ can be expressed in terms of the $s$ derivatives by

Thus, for a conventional Lagrangian with $n$ variables, the corresponding extended Lagrangian is a function of $n + 1$ variables while the conventional and extended Lagrangians are related using equations 17.59, and 17.60.

The derivatives of the relation between the extended and conventional Lagrangians lead to

where $1 \leq \mu \leq n$ since the $\mu = 0$ time derivatives are written explicitly in equations 17.62, 17.64.

Equations 17.63 — 17.64, summed over the extended range $0 \leq \mu \leq n$ of time and spatial dynamical variables, imply

Equation 17.65 can be written in the form

If the extended Lagrangian $\mathbb{L}(\mathbf{q}, \frac{d\mathbf{q}}{ds},t, \frac{dt}{ds} )$ is homogeneous to first order in the $n+1$ variables $\frac{dq^{\mu}}{ds}$, then Euler’s theorem on homogeneous functions trivially implies the relation given in Equation 17.66. Struckmeier[Str08] identified a subtle but important point that if $\mathbb{L}$ is not homogeneous in $\frac{dq^{\mu}}{ds}$, then Equation 17.66 is not an identity but is an implicit equation that is always satisfied as the system evolves according to the solution of the extended Euler-Lagrange equations. Then Equation 17.59 is satisfied without it being a homogeneous form in the $n+1$ velocities $\frac{dq^{\mu}}{ds}$. This introduces a new class of non-homogeneous Lagrangians. The relativistic free particle, discussed in example 17.6.1, is a case of a non-homogeneous extended Lagrangian.

Extended generalized momenta¶

The generalized momentum is defined by

Assume that the definitions of the extended Lagrangian $\mathbb{L}$, and the extended Hamiltonian $\mathbb{H}$, are related by a Legendre transformation, and are based on variational principles, analogous to the relation that exists between the conventional Lagrangian $L$ and Hamiltonian $H$. The Legendre transformation requires defining the extended generalized (canonical) momentum-energy four vector $\mathbb{P}(s)= ( \frac{\mathbb{E}(s)}{ c} ,\mathbf{p}(s))$. The momentum components of the momentum-energy four vector $\mathbb{P}(s)= ( \frac{\mathbb{E}(s)}{ c} , \mathbf{p}(s))$ are given by the $1 \leq \mu \leq n$ components using Equation 17.63.

The $\mu = 0$ component of the momentum-energy four vector can be derived by recognizing that the right-hand side of Equation 17.64 is equal to $−H(p_{\mu }, q^{\mu }, t)$. That is, the corresponding generalized momentum $p_0$, that is conjugate to $q_0 = ct$, is given by

Extended Lagrange equations of motion¶

By direct analogy with the non-relativistic action integral 17.55, the extremum for the relativistic action integral $S(\mathbf{q}, \frac{d\mathbf{q}}{ds},t, \frac{dt}{ds} )$ is obtained using the Euler-Lagrange equations derived from Equation 17.56 where the independent variable is $s$. This implies that for $0 \leq \mu \leq n$

where the extended generalized force $\mathbb{Q}^{EX}_{\mu}$ shown on the right-hand side of Equation 17.70, accounts for all forces not included in the potential energy term in the Lagrangian. The extended generalized force $\mathbb{Q}^{EX}_{\mu}$ can be factored into two terms as discussed in chapter 6, equation $(6.5.12)$. The Lagrange multiplier term includes $1 \leq k \leq m$ holonomic constraint forces where the $m$ holonomic constraints, which do no work, are expressed in terms of the $m$ algebraic equations of holonomic constraint $g_k$. The $Q^{EXC}_{\mu}$ term includes the remaining constraint forces and generalized forces that are not included in the Lagrange multiplier term or the potential energy term of the Lagrangian.