Reaching speeds close to that of light (relativistic speeds) in traveling through space is predicted to have major effects. In Chapter 8 some of these effects have been mentioned. They are the result of the Theory of Special Relativity created by Einstein. According to the Theory of Special Relativity, there are no privileged frames of reference such as the famed "absolute inertial frame'' of classical physics. The fact is that the laws of dynamics appear the same in all frames of reference moving at constant velocity relative to each other (inertial but not absolute frames). This statement can be rephrased by saying that the laws of dynamics are "invariant" with respect to Galilean transformations, i.e., they remain the same in two frames of references in uniform motion (constant velocity) relative to each other. Experiments by Michelson and Morley also showed the speed of light is invariant with the frames of reference, i.e., does not increase or decrease due to the relative velocity between two inertial frames, a disconcerting and counterintuitive result that troubled many physicists. These two facts ultimately resulted in
Einstein's intuition that simultaneous events cannot exist. The second motivation for abandoning absolute frames of references and Galilean transformations was the need to make not only the laws of dynamics, but also the laws of electromagnetism invariant when changing frames of reference: in fact, contrary to the laws of dynamics, they change in a Galilean transformation. This mathematical result was unacceptable, amounting to the existence of different electromagnetism "physics" in different inertial frames. The work done by Larmor, Lorentz and Einstein himself convinced him that the Galilean transformations had to be replaced by the Lorentz transformations, in which the characteristic ratio between frame speed and the speed of light appear (see below). It is because of these new relationships between two inertial frames of reference that a clock on a spacecraft moving at constant velocity with respect to an Earth's observer would appear to him/her to run at a different speed than a clock on Earth. In other words Earth time is not spaceship time. The revolutionary character of Special Relativity stays in the fact that there cannot be a "third", or "impartial" observer capable of judging the "right" time between the two. The two frames in relative inertial motion are equally "right", each in its own frame, a consequence that alone can "explain" the twins paradox so often cited in connection to relativity. So, Earth time and ship time are different, but it is Earth time we must be concerned with, because that is the time in which the project team is living. H. David Froning has spent a career investigating deep space travel possibilities, and the authors wish to acknowledge his contribution to this section [Froning, 1980, 1981, 1985, 1986, 1989; Froning et al., 1998; Froning and Roach, 2002].
To recall, the Lorentz transformation of Special Relativity [Einstein, 1916; Lang, 1999] results in a time relationship for the Earth observer and for the spacecraft traveler as follows:
Note that in the Galilean transformations of classical physics the two times are assumed identical, that is tEarth Spacecraft (9-1a)
because the speed of light seemed at that time to be infinite. This classical result is in fact predicted by the Lorentz transformations when imposing c !i.
So as, the spacecraft approaches the speed of light, the crew's apparent time is shorter than the observer's apparent time on Earth. Both perceive that the event or journey has occurred over an equal duration. It is not until the spacecraft crew returns to Earth that the discrepancy in perceived times becomes apparent. Researchers have derived the relativistically correct equations for a spacecraft journey's duration (te) in an Earth-bound observer frame of reference, and for the journey duration (tsc) of that same spacecraft in its own moving reference [Froning, 1980]. For the simple case of one-dimensional rectilinear motion, Krause has derived the expressions for (te) and (tsc) for a spacecraft acceleration (asc) in its own moving frame during the initial half of the total journey distance (S) followed by a constant spacecraft deceleration (—asc) during the final half of the total journey [Krause, 1960; Maccone, 2008b]. The reader is warned that the relationships below can be easily derived and are valid only when the motion is rectilinear, i.e., when the space-time continuum is the so-called Rindler space-time (only two-dimensional), not a very realistic assumption but one that simplifies solution of this problem. In the fully four-dimensional space-time, or Minkowski's space, the effect of changing velocity (acceleration) is much more complex. There is in fact an important consequence with respect to changing velocity, because velocity is a vector. Even simply inverting direction invalidates the consequences of the Lorentz transformations, that are strictly valid between inertial frames, that is, with constant relative velocity. That is because velocity is defined by a magnitude (speed) and a direction. If either changes, then it had to be the result of acceleration. The most common concept of acceleration is a change in the magnitude of the speed. However, a constant speed turn is in fact an acceleration from a continuously varying direction. The direction of the acceleration is perpendicular to the flight path, and pointed at the center of (instantaneous) rotation. This acceleration is called centrifugal acceleration. Centrifugal acceleration is the result of any rotation of the velocity vector. Thus a spacecraft crew in orbit is under a constant acceleration, balanced of course by their gravitational weight. In space the thrust from a propulsion system is necessary to initiate any acceleration, whether positive or negative. Because there are no aerodynamic forces in space, any motion initiated will continue until it is negated by a counter propulsion force of equal magnitude and opposite direction. In the two-dimensional continuum assumed in the example by Krause the two times, crew's and Earth's, are given by the following equations:
These equations can be solved for a number of different destinations as a function of spacecraft acceleration, and their times compared. The life of a deep-space management team is probably about 20 Earth years. If we wish to travel farther into space, that is, faster relative to the Earth time frame of reference, then we must travel faster. But before discussing travel times, we need to establish the absolute limit, or boundary, posed by Special Relativity, that is, when spacecraft speed equals light speed. For such a flight profile, the maximum spacecraft velocity will be assumed to be reached at the journey midpoint only, see Figure 9.3. From the starting point to the midpoint the spacecraft has a continuous and constant positive acceleration. From the midpoint to the end point the spacecraft has a continuous and constant negative acceleration. Saenger derived the ratio of the spacecraft velocity (V) to light speed (c) at the journey midpoint, as given in equation (9.3) [Saenger, 1956].
1.0 "g" deceleration in ship frame of reference
1.0 "g" acceleration in ship frame of reference
1.0 "g" deceleration in ship frame of reference
Duration in Earth time
Duration in ship time
Earth to nearest star
Earth to Galaxy center
Earth to nearest spiral galaxy
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