Now substitute p = 7mv and perform the integration, which is left as an exercise to the reader. The result is then
This kinetic energy equation contains the famous E = mc2 result. This is used in the invariant interval for the momentum-energy vector P = (E, p) for p = 0, PTP = (mc2)2. This interval remains invariant under all Lorentz transformations, and so
At this stage we have the Lorentz transformations of special relativity, the nature of the invariant interval and now the invariant momentum-energy interval.
To extend this theory, the interval needs to be cast in a differential form. A straight line in spacetime defines a particle without the action of any force upon it, which conforms to the spacetime extension of Newton's first law. However, in order to generalize Newton's second and third laws it is necessary to consider ds expressed according to dt, dx, dy, dz as ds2 = —(cdt)2 + dx2 + dy2 + dz2 . (6.21)
For a particle on a spacetime path in the presence of a force, s = ds gives the invariant proper time on the particle world line. Since ds is an infinitesimal element of the invariant interval, and length of a path in spacetime, this is the appropriate time to be used in a Newton's second law calculation. The four dimensional force vector on a particle that is accelerating is dP
Because ds is an invariant, this expresses Newton's second law of motion so that it has the same form in any inertial reference frame. To transform from one inertial frame to another, the force is merely Lorentz transformed accordingly F' = MTFM.
This leads to the nature of accelerations in spacetime. The relativistic form of Newton's second law is obviously
where A is a vector in the four dimensions of spacetime. To understand this acceleration in spacetime return to the line element with ds2 = —(cdt)2 + dx2 + dy2 + dz2 = dxadxa , (6.24)
where the last term implies a summation over each dxa, dx1 = cdt, dx2 = dx, dx3 = dy, and dx4 = dz. This implied sum is referred to as the Einstein summation convention. Now divide both sides of this equation by ds2
for Ua = dxa/ds the four velocity vector. Now differentiate this by d/ds to find that a
This curious result shows that in spacetime the direction of the velocity of a particle is perpendicular to the acceleration, and by Newton's second law it is perpendicular to the force. The magnitude of the acceleration is g defined by g2 = AaAa. For acceleration along the x-axis this gives
Solving At and Ax according to Ut and Ux results in the differential equations dUt dUX
For those unfamiliar with differential equations, these are equations built from derivatives of an unknown function, where functions are the set of solutions to the differential equation. The solutions are the functions
or equivalently that t = g-1 sinh(gs) and x = g-1 cosh(gs) [6.3]. The acceleration of a particle in spacetime appears in Figure 6.4. The asymptotes to the hyperbolic curves define a particle horizon. Any spacetime point to the future of the upper asymptote is unable to send a signal to the accelerating particle. This is important in future discussion on relativistic starcraft.
Special relativity and its dynamics are similar to Newton's original laws of motion. The space of Newtonian mechanics is extended to spacetime. The three laws of motion are modified accordingly.
Fig. 6.4. Hyperbolic nature of acceleration in spacetime.
Velocity = ± c asymptotes
Fig. 6.4. Hyperbolic nature of acceleration in spacetime.
The first law of motion is:
• The world line of a particle in spacetime in the absence of any force is a time-like line with constant intervals of proper time along this line.
This defines a reference frame as inertial with an invariant speed of light. An appropriate observer in spacetime is on an inertial reference frame in the absence of any accelerating force. An inertial observer is able to observe the dynamics of particles and their interactions in spacetime consistent with the dynamical equation given by the second law of motion.
The second law of motion is:
• The acceleration of a particle in spacetime Ab = dUb/ds, for ds an infinitesimal element of the invariant interval or proper time, with Ub = dxb/ds, is proportional to a force in four dimensions Fb according to
This dynamical equation can only be properly applied for an observer on an inertial reference frame that satisfies the first law. An accelerated observer is not able to properly apply the second law of motion for particles under forces.
The third law of motion is: • Whenever one body exerts a four dimensional force on a second body, the second body exerts a four-force with components of equal magnitude on the first, where the spatial components are in the opposite direction and with the same temporal force F1. This is exactly the same statement for the Newtonian case, but with the term force replaced by a four-force. Further, the reactive force is only
opposite in its spatial direction. The time part, AE = dE/dsAs term is the same on both bodies. This changes the physics so that if one body changes its momentum by AP = (AE, Ap) by exerting a force on a second body, the second body changes its momentum by AP = (AE, - Ap). This change in the third law reflects that the symmetry of spacetime is isotropic in spatial direction, as well as homogeneous, but that there is an additional transformation principle given by the Lorentz transformations, called the Lorentz group. The retention of the translation and rotation invariance seen in the standard Newton's third law with the Lorentz group defines the Poincare group. This is now the fundamental symmetry of spacetime. It is interesting to note that for small velocities v ^ c this structure recovers Newton's laws.
Just as with the standard Newton's second law the dynamics of special relativity is deterministic. Since the differential equation is second order in proper time the dynamics are completely deterministic. Given a particle subjected to a force its trajectory is completely determined into the future and into the past. The major difference is that a force can't be instantaneously imparted to a particle at a distance. Special relativity was arrived at to fix problems with electrodynamics, so that a fast moving particle in an electric field in the lab frame interacts with transformed electric field plus a magnetic field on its frame. This suggests some inconsistency with Newtonian gravity, an issue fixed by general relativity. However, special relativity recovers the property of deterministic dynamics of Newton's second law, since the dynamical equation is second order an invariant under s ^ -s.
These dynamics will form the basis for relativistic space flight. In particular the equations for accelerations will be important. With this machinery at hand it is possible to explore the physics required for sending a spacecraft to another star.
A discussion of special relativity should include some mention of general relativity. This will not be explored in great detail, but the later discussion on exotic space propulsion methods does require some description of general relativity. What makes special relativity "special" is that it assumes that an inertial reference frame coordinate system can extend throughout spacetime. This is the case where spacetime is flat, and there is no gravitating body that curves the spacetime. If spacetime is curved a flat spacetime configuration only exists in a sufficiently small region where deviations from flatness are negligible. In general spacetime is curved, which is the cause for the adjective "general" in general relativity. This subject is vast and deep, with a massive accumulation of literature since Einstein advanced general relativity in 1916.
General relativity was motivated by the existence of gravity. Those annoying invisible lines of force in Newton's law of gravitation implied an instantaneous propagation of the gravity field. So if one gravitating body is moved by some means to another region of spacetime changing lines of gravitating force would instantaneously propagate to any other body. Yet special relativity implies that causal propagation can only be on a null or timelike path. So Newtonian gravity is inconsistent with special relativity.
A crucial ingredient in the formulation of general relativity is the equivalence principle. An inertial reference frame in flat space, with no gravity, is one where all bodies will by the first law of motion remain at rest or in a constant state of motion with respect to each other. Similarly a reference frame that is falling in a gravity field is one where bodies will remain at rest or in a constant state of motion with respect to each other. Astronauts in the space shuttle orbiting the Earth are weightless because they are falling with the shuttle and its contents at the same rate. Yet the craft is moving fast enough to keep missing the Earth which curves away from the falling path. Hence if this falling reference frame is small enough so that the radial divergence of the gravity field may be ignored the two situations, an inertial reference frame in free space and a reference frame freely falling in a gravity field, are completely equivalent. In effect the freely falling observer cancels out gravity. This equivalence principle extends to accelerated frames as well. An observer on a rocketship accelerating with a constant acceleration, say one gee, observes the same physics as an observer here on Earth. An observer on the rocket who drops a mass sees that mass drop to the floor according to v = at, where indeed the mass is inertial and the floor of the rocket accelerates towards the ball. Yet within the rocketship frame this observation is no different from what an observer on Earth sees with dropping a mass.
That the freely falling reference frame has to be small suggests that an inertial reference frame in a gravity field is small in extent. Gravity must in some ways involve a "patching together" of many of these local regions that can be considered as local inertial reference frames. Einstein considered Riemannian geometry, devised by Bernhard Riemann a half century earlier, as a way to approach this problem. This is a "deformation", where special relativity is a deformation of Newton's laws.
The first law is:
• The world line of a particle in spacetime in the absence of any force is a time-like curve, or geodesic, with constant intervals of proper time along this line. Since this particle moves through local Lorentz inertial reference frames the condition for zero acceleration is dUa ds bc
where the rabcUbUc is the correction term needed to glue local Lorentz inertial frames together the inertial particle passes through.
In the case that spacetime is flat rabcUbUc vanishes and special relativity is recovered. This connection coefficient is involved with the computation of spacetime curvatures. This is the geodesic equation and also defines the appropriate frames from which dynamics is studied. In the case that this connection coefficient is nonzero this equation defines the proper inertial reference frame as freely falling in a gravity field. For a weak gravity field r*ooU0U0 = —GMri/r3, for the index i over spatial dimensions, recovers Newton's law of gravitation. This indicates that the gravity force is no longer a force. The force associated with gravity is due to other forces which prevent geodesic motion of a body in curved spacetime. The second law is:
• A body under a force will experience an acceleration defined by
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