There would be a witty or insightful quote here if your browser had JavaScript enabled!
There would be a cool link here if your browser had JavaScript enabled!
Written by Bernd Schneider
Webmaster's Note: This page makes extensive use XHTML, MathML, and SVG to present this article. If the equations and figures do not appear properly, please consult the original version published by Bernd Schneider.
Note: You may need to install additional fonts for MathML to properly display all equations.
This chapter summarizes some very basic theorems of physics, mostly predating the theories of Special Relativity and of General Relativity.
Isaac Newton discovered the following laws that are still valid (meaning that they are a very good approximation) for speeds much slower than the speed of light.
Two objects with a mass of m₁ and m₂, respectively, and a distance of r between the centers of mass attract each other with a force F of:
is Newton's constant of gravitation. If an object with a mass m of much less than Earth's mass is close to Earth's surface, it is convenient to approximate Eq. 1.2 as follows:
Here g is an acceleration slightly varying throughout Earth's surface, with an average of .
In an isolated system, the total momentum is constant. This fundamental law is not affected by the theories of Relativity. Energy conservation In an isolated system, the total energy is constant. This fundamental law is not affected by the theories of Relativity.
The overall entropy of an isolated system is always increasing. Entropy generally means disorder. An example is the heat flow from a warmer to a colder object. The entropy in the colder object will increase more than it will decrease in the warmer object. This why the reverse process, leading to lower entropy, would never take place spontaneously.
If the source of the wave is moving relative to the receiver or the other way round, the received signal will have a different frequency than the original signal. In the case of sound waves, two cases have to be distinguished. In the first case, the signal source is moving with a speed v relative to the medium, mostly air, in which the sound is propagating at a speed w:
f is the resulting frequency, f₀ the original frequency. The plus sign yields a frequency decrease in case the source is moving away, the minus sign an increase if the source is approaching. If the receiver is moving relative to the air, the equations are different. If v is the speed of the receiver, then the following applies to the frequency:
Here the plus sign denotes the case of an approaching receiver and an according frequency increase; the minus sign applies to a receiver that moves away, resulting in a lower frequency.
The substantial difference between the two cases of moving transmitter and moving receiver is due to the fact that sound needs air in order to propagate. Special Relativity will show that the situation is different for light. There is no medium, no "ether" in which light propagates and the two equations will merge to one relativistic Doppler shift.
In addition to Einstein's equivalence of mass and energy, de Broglie unified the two terms in that any particle exists not only alternatively but even simultaneously as matter and radiation. A particle with a mass m and a speed v was found to be equivalent to a wave with a wavelength lambda. With h, Planck's constant, the relation is as follows:
The best-known example is the photon, a particle that represents electromagnetic radiation. The other way around, electrons, formerly known to have particle properties only, were found to show a diffraction pattern which would be only possible for a wave. The particle-wave dualism is an important prerequisite to quantum mechanics.
Special Relativity (SR) doesn't play a role in our daily life. Its impact becomes apparent only for speed differences that are considerable fractions of the speed of light, c. I will henceforth occasionally refer to them as "relativistic speeds". The effects were first measured as late as towards the end of the 19th century and explained by Albert Einstein in 1905.
There are many approaches in literature and in the web to explain Special Relativity. Please refer to the appendix. A very good reference from which I have taken several suggestions is Jason Hinson's article on Relativity and FTL Travel. You may wish to read his article in parallel.
The whole theory is based on two postulates:
In order to explain Special Relativity, it is necessary to introduce frames of reference. Such a frame of reference is basically a point-of-view, something inherent to an individual observer who sees an event from a certain angle. The concept is in some way similar to the trivial spatial parallax where two or more persons see the same scene from different spatial angles and therefore give different descriptions of it. However, the following considerations are somewhat more abstract. "Seeing" or "observing" will not necessarily mean a sensory perception. On the contrary, the observer is assumed to account for every "classic" measurement error such as signal delay or Doppler shift.
Aside from these effects that can be rather easily handled there is actually one even more severe restriction. The considerations on Special Relativity require inertial frames of reference. According to the definition in the General Relativity chapter, this would be a floating or free falling frame of reference. Any presence of gravitational or acceleration forces would not only spoil the measurement,but even question the validity of the SR. One provision for the following considerations is that all observers should float within their starships in space so that they can be regarded as local inertial frames. Basically, every observer has their own frame of reference; two observers are in the same frame if their relative motion to a third frame is the same, regardless of their distance.
The concept of a four-dimensional space-time has already been briefly explained in the GR chapter. Since in an inertial frame all the cartesic spatial coordinates x,y and z are equivalent (for instance, there is no "up" and "down" in space), we may replace the three axes with one generic horizontal space (x-) axis. Together with the vertical time (t-) axis we obtain a two-dimensional diagram (Fig. 1.1). It is very convenient to give the distance in light years and the time in years. Irrespective of the current frame of reference, the speed of light always equals c and would be exactly 1ly per year in our diagram, according to the second postulate. The beam will therefore always form an angle of either 45° or -45° with the x-axis and the t-axis, as indicated by the yellow lines.
A resting observer O draws a perpendicular x-t-diagram. The x-axis is equivalent to t=0 and is therefore a line of simultaneity, meaning that for O everything located on this line is simultaneous. This applies to every parallel line t=const. likewise. The t-axis and every line parallel to it denote x=const. and therefore no movement in this frame of reference. If O is to describe the movement of another observer O* with respect to himself, O*'s time axis t* is sloped, and the reciprocal slope indicates a certain speed v=x/t. Fig. 1.2 shows the O's coordinate system in gray, and O*'s in white. At the first glance it seems strange that O*'s x*-axis is sloped into the opposite direction than his t*-axis.
The x*-axis can be explained by assuming two events A and B occurring at t*=0 in some distance to the two observers, as depicted in Fig. 1.3. O* sees them simultaneously, whereas O sees them at different times. Since the two events are simultaneous in O*'s frame, the line A-0-B defines his x*-axis. A and B might be located anywhere on the 45-degree light paths traced back from the point "O* sees A&B", so we need further information to actually localize A and B. Since O* is supposed to *see* them at the same time (and not only date them back to t*=0), we also know that the two events A and B must have the same distance from the origin of the coordinate system. Now A and B are definite, and connecting them yields the x*-axis. Some simple trigonometry would reveal that actually the angle between x* and x is the same as between t* and t, only the direction is opposite.
The faster the moving observer is, the closer will the two axes t* and x* move to each other. It is obvious that finally, at v=ac, they will merge to one single axis, equivalent to the path of a light beam.
The above space-time diagrams don't have a scale on the x*- and t*-axes so far. The method of determining the t*-scale is illustrated in the left half of Fig. 1.4. When the moving observer O* passes the resting observer O, they both set their clocks to t*=0 and t=0, respectively. Some time later, O's clock shows t=3, and at a yet unknown instant O*'s clock shows t*=3. The yellow light paths show when O will actually *see* O*'s clock at t*=3, and vice versa. If O is smart enough, he may calculate the time when this light was emitted (by tracing back the -45° yellow line to O*'s t*-axis). His lines of simultaneity are exactly horizontal (red line), and there constructed event "t*=3" will take place at some yet unknown time on his t-axis. The quotation marks distinguish O's reconstruction of "t*=3" and O*'s direct reading t*=3. O* will do the same by reconstructing the event "t=3" (green line). Since O*'s x*-axis and therefore the green line is sloped, it is impossible that the two events t=3 and "t*=3" and the two events t*=3 and "t=3" are simultaneous on the respective axis.
If there is no absolute simultaneity, at least one of the to observers would see the other one's time dilated (slow motion) or compressed (fast motion). Now we have to apply the first postulate, the principle of relativity. There must not be any preferred frame of reference, all observations have to be mutual. This means that either observer would see the other one's time dilated by the same factor. In our diagram the red and the green line have to cross, and the ratio "t*=3" to t=3 has to be equal to "t=3" to t*=3. Some further calculations yield the following time dilation:
Note: When drawing the axes to scale in an x-t diagram, one has to account for the inherently longer hypotenuse t* and multiply the above formula with an additional factor cos alpha to "project" t* on t.
Note that the time dilation would be the square root of a negative number (imaginary), if we assume an FTL speed v>c. Imaginary numbers are not really forbidden, on the contrary, they play an important role in the description of waves. Anyway, a physical quantity such as the time doesn't make any sense once it gets imaginary. Unless a suited interpretation or a more comprehensive theory is found, considerations end as soon as a time (dilation) that has to be finite and real by definition would become infinitely large, infinitely small or imaginary. The same applies to the length contraction and mass increase. Warp theory circumvents all these problems in away that no such relativistic effects occur.
The considerations for the scale of the x*-axis are similar as those for the t*-axis. They are illustrated in the bottom portion of Fig. 1.4. Let us assume that O and O* both have identical rulers and hold their left ends x=x*=0 when they meet at t=t*=0. Their right ends are at x=l on the x-axis and at x*=l on the x*-axis, respectively. O and his ruler rest in their frame of reference. At t*=0 (which is not simultaneous with t=0 at the right end of the ruler!) O* obtains a still unknown length "x=l" for O's ruler (green line). O* and his ruler move along the t*-axis. At t=0, O sees an apparent length "x=l" of O*'s ruler (red line). Due to the slope of the t*-axis, it is impossible that the two observers mutually see the same length l for the other ruler. Since the relativity principle would be violated in case one observer saw two equal lengths and the other one two different lengths, the mutual length contraction must be the same.Note that the geometry is virtually the same as for the time dilation, so it's not astounding that length contraction is determined by the factor gamma too:
Once again, note that when drawing the x*-axis to scale, a correction is necessary, a factor of cos alpha to the above formula.
One of the most popular examples used to illustrate the effects of Special Relativity is the addition of velocities. It is obvious that in the realm of very slow speeds it's possible to simply add or subtract velocity vectors from each other. For simplicity, let's assume movements that take place in only one dimension so that the vector is reduced to a plus or minus sign along with theabsolute speed figure, like in the space-time diagrams. Imagine a tank that as a speed of v compared to the ground and to an observer standing on the ground (Fig. 1.5). The tank fires a projectile, whose speed is determined as w by the tank driver. The resting observer, on the other hand, will register a projectile speed of v+w relative to the ground. So far, so good.
The simple addition (or subtraction, if the speeds have opposite directions) seems very obvious, but it isn't so if the single speeds are considerable fractions of c. Let's replace the tank with a starship (which is intentionally a generic vessel, no Trek ship), the projectile with a laser beam and assume that both observers are floating, one in open space and one in his uniformly moving rocket, at a speed of c/2 compared to the first observer (Fig. 1.6). The rocket pilot will see the laser beam moving away at exactly c. This is still exactly what we expect. However, the observer in open space won't see the light beam travel at v+c=1.5c but only at c. Actually, any observer with any velocity (or in any frame of reference) would measure a light speed of exactly .
Space-time-diagrams allow to derive the addition theorem for relativistic velocities. The resulting speed u is given by:
For v,w<<c we may neglect the second term in the denominator and obtain u=v+w, as we expect it for small speeds. If vw gets close to c2, we get a speed u that is close to, but never equal to or even faster than c. Finally, if either v or w equals c, u is equal to c as well. There is obviously something special to the speed of light. c always remains constant, no matter where in which frame and which direction it is measured. c is also the absolute upper limit of all velocity additions and can't be exceeded in any frame of reference.
Mass is a property inherent to any kind of matter. One may distinguish two forms of mass, one that determines the force that has to be applied to accelerate an object (inert mass) and one that determines which force it experiences in a gravitational field (heavy mass). At latest since the equivalence principle of GR they have been found to be absolutely identical.
However, mass is apparently not an invariant property. Consider two identical rockets that started together at t=0 and now move away from the launch platform in opposite directions, each with an absolute speed of w. Each pilot sees the launch platform move away at w, while Eq. 1.9 shows us that the two ships move away from each other at a speed u<2w. The "real" center of mass of the whole system of the two ships would be still at the launch platform, however, each pilot would see a center of mass closer to the other ship than to his own. This may be interpreted as a mass increase of the other ship to m compared to the rest mass m0 measured for both ships prior to the launch:
This function is plotted in Fig. 1.7.
So each object has a rest mass m0 and an additional mass m-m0 due to its speed as seen from another frame of reference. This is actually a convenient explanation for the fact that the speed of light cannot be reached. The mass increases more and more as the object approaches c, and so would the required momentum to propel the ship.
Finally, at v=c, we would get an infinite mass, unless the rest mass m0 is zero. The latter must be the case for photons which actually move at the speed of light, which even define the speed of light. If we assume an FTL speed v>c, the denominator will be the square root of a negative number, and therefore the whole mass will be imaginary. As already stated for the time dilation, there is not yet a suitable theory how an imaginary mass could be interpreted. Anyway, warp theory circumvents this problem in that the mass neither gets infinite nor imaginary.
Let us consider Eq. 1.10 again. It is possible to express it as follows:
It is obvious that we may neglect the third order and the following terms for slow speeds. If we multiply the equation with c2 we obtain the familiar Newtonian kinetic energy *m0v2 plus a new term m0c2. Obviously already the resting object has a certain energy content m0c2. We get a more general expression for the complete energy E contained in an object with a rest mass m0 and a moving mass m, so we may write (drumrolls!):
Energy E and mass m are equivalent; the only difference between them is the constant factor c2. If there is an according energy to each given mass, can the mass be converted to energy? The answer is yes, and Trek fans know that the solution is a matter/antimatter reaction in which the two forms of matter annihilate each other, thereby transforming their whole mass into energy.
Let us have a look at Fig. 1.1 again. There are two light beams running through the origin of the diagram, one traveling in positive and one in negative x direction. The slope is 1ly per year and equals c. If nothing can move faster than light, then every t*-axis of a moving observer and every path of a message sent from one point to another in the diagram must be steeper than these two lines.This defines an area "inside" the two light beams for possible signal paths originating at or going to (x=0,t=0). This area is marked dark green in Fig. 1.8. The black area is "outside" the light cone. The origin of the diagram marks "here (x=0)" and "now (t=0)" for the resting observer.
The common-sense definition would be that "future" is any event at t>0, and past is any event at t<0. Special Relativity shows us a different view of these two terms. Let us consider the four marked events which could be star explosions (novae), for instance. Event A is below the x-axis and within the light cone. It is possible for the resting observer O to see or to learn about the event in the past, since a -45° light beam would reach the t-axis at about one and a half years prior to t=0. Therefore this event belongs to O's past. Event B is also below the x-axis, but outside the light cone. The event has no effect on O in the present, since the light would need almost another year to reach him. Strictly speaking, B is not in O's past. Similar considerations are possible for the term "future". Since his signal wouldn't be able to reach the event C, outside the light cone, in time, O is not able to influence it. It's not in his future. Event D, on the other hand, is inside the light cone and may therefore be cause or influenced by the observer.
What about a moving observer? One important consequence of the considerations in this whole chapter was that two different observers will disagree about where and when a certain event happens. The light cone, on the other hand, remains the same, irrespective of the frame of reference. So even if two observers meeting at t=0 have different impressions about simultaneity, they will agree that there are certain, either affected (future) or affecting (past) events inside the light cone, and outside events they shouldn't bother about.
The considerations about the time dilation in Special Relativity had the result that the terms "moving observer" and "resting observer" are interchangeable as are their space-time diagrams. If there are two observers with a speed relative to each other, either of them will see the other one move. Either observer will see the other one's clock ticking slower. Special Relativity necessarily requires that the observations are mutual, since it forbids a preferred, absolutely resting frame of reference. Either clock is slower than the other one? How is this possible?
Specifically, the twin paradox is about twins of whom one travels to the stars at a relativistic speed while the other one stays on Earth. It is obvious that the example assumes twins, since it would be easier to see if one of them actually looks older than the other one when they meet again. Anyway, it should work with unrelated persons as well. What happens when the space traveler returns to Earth? Is he the younger one, or maybe his twin on Earth, or are they equally old?
The following example for the twin paradox deliberately uses the same figures as Jason Hinson's excellent treatise on Relativity and FTL Travel, to increase the chance of understanding it.
To anticipate the result, the space traveler will be the younger one when he returns. The solution is almost trivial. Time dilation only remains the same, as long as both observers stay in their respective frames of reference. However, if the two observers want to meet again, one of them or both of them have to change their frame(s) of reference. In this case it is the space traveler who has to decelerate, turn around, and accelerate his starship in the other direction. It is important to note that the whole effect can be explained without referring to any General Relativity effects. Time dilation attributed to acceleration or gravity will change the result, but it will not play a role in the following discussion. The twin paradox is no paradox, it can be solved, and this is best done with a space-time diagram.
Fig. 1.9 shows the first part of the FTL travel. O is the "resting" observer who stays on Earth the whole time. Earth is subsequently regarded as an approximated inertial frame. Strictly speaking, O would have to float in Earth's orbit, according to the definition in General Relativity. Once again, however, it is important to say that the following considerations don't need General Relativity at all. I only refer to O as staying in an inertial frame so as to exclude any GR influence.
The moving observer O* is supposed to travel at a speed of 0.6c relative to Earth and O. When O* passes by O (x=x*=0), they both set their clocks to zero (t=t*=0). So the origin of their space-time diagrams is the same, and the time dilation will become apparent in the different times t* and t for simultaneous events. As outlined above, t* is sloped, as is x* (see also Fig. 1.3). The measurement of time dilation works as outlined in Fig. 1.4. O's lines of simultaneity are parallel to his x-axis and perpendicular to his t-axis. He will see that a5 years on his t-axis correspond with only 4 years on the t*-axis (red arrow), because the latter is stretched according to Eq. 1.7. Therefore O*'s clock is ticking slower from O's point-of-view. The other way round, O* draws lines of simultaneity parallel to his sloped x*-axis and he reckons that O's clock is running slower, 4 years on his t*-axis compared to 3.2 years on the t-axis (green arrow). It is easy to see that the mutual dilation is the same, since 5/4 equals 4/3.2. Who is correct? Answer: Both of them, since they are in different frames of reference, and they stay in these frames. The two observers just see things differently; they wouldn't have to care whether their perception is "correct" and the other one is actually aging slower, unless they wanted to meet again.
Now let us assume that O* stops his starship when his clock shows t*=4 years, maybe to examine a phenomenon or to land on a planet. According to Fig. 1.10 he is now resting in space relative to Earth and his new x**-t** coordinate system is parallel to the x-t system of O on Earth. O* is now in the same frame of reference as O. And this is exactly the point: O*'s clock still shows 4 years, and he notices that not 3.2 years have elapsed on Earth as briefly before his stop, but 5 years, and this is exactly what O says too. Two observers in the same frame agree about their clock readings. O* has been in a different frame of reference at 0.6c for 4 years of his time and 5 years of O's time. This difference becomes a permanent offset when O* enters O's frame of reference. Paradox solved.
It is obvious that the accumulative dilation effect will become the larger the longer the travel duration is. Note that O's clock has always been ticking slower in O*'s moving frame of reference. The fact that O's clock nevertheless suddenly shows a much later time (namely5 instead of 3.2 years) is solely attributed to the fact that O* is entering a frame of reference in which exactly these 5 years have elapsed.
Once again, it is crucial to annotate that the process of decelerating would only change the result qualitatively, since there could be no exact kink, as O* changes from t* to t**. Deceleration is no sudden process, and the transition from t* to t** should be curved. Moreover,the deceleration itself would be connected with a time dilation according to GR, but the paradox is already solved without taking this into account.
Let us assume that at t*=4 years, O* suddenly gets homesick and turns around instead of just resting in space. His relative speed is v=-0.6c during his travel back to Earth, the minus sign indicating that he is heading to the negative x direction. It is obvious that this second part of his travel should be symmetrical to the first part at +0.6c in Fig. 1.9, the symmetry axis being the clock comparison at t=5 years and t**=4 years. This is exactly the moment when O* has covered both half of his way and half of his time.
Fig. 1.11 demonstrates what happens to O*'s clock comparison. Since he is changing his frame of reference from v=0.6c to v=-0.6c relative to Earth, the speed change and therefore the effect is twice as large as in Fig. 1.10. Assuming that O* doesn't stop for a clock comparison as he did above, he would see that O's clock directly jumps jumps from 3.2 years to 6.8 years. Following O*'s travel back to Earth, we see that the end time is t**=8 years (O*'s clock) and t=10 years (O's clock). The traveling twin is actually two years younger.
We could imagine several other scenarios in which O might catch up with the traveling O*, so that O is actually the younger one. Alternatively, O* could stop in space, and O could do the same travel as O*, so that they would be equally old when O reaches O*. The analysis of the twin paradox shows that the simple statement "moving observers age slower" is not sufficient. The statement has to be modified in that "moving observers age slower as seen from a different frame of reference, and they notice it when they enter this frame themselves".
As already stated further above, two observers in different frames of reference will disagree about the simultaneity of certain events (see Fig. 1.3). The same event might be in one observer's future, but in another observer's past when they meet each other. This is not a problem in Special Relativity, since no signal is allowed to travel faster than light. Any event that could be theoretically influenced by one observer, but has already happened for the other one, is outside the light cone depicted in Fig. 1.8. Causality is preserved.
Fig. 1.12 depicts the space-time diagrams of two observers with a speed relative to each other. Let us assume the usual case that the moving observer O* passes by the resting observer O at t=t*=0. They agree about the simultaneity of this passing event, but not about any other event at t<>0 or t*<>0. Event A is below the t-axis, meaning that it occurs in O's past, but above the t*-axis and therefore in O*'s future. This doesn't matter as long as they can send and receive only STL signals. Event A is outside the light cone, and the argumentation would be as follows: A is in O*'s future, but he has no means of influencing it at t*=0, since his signal couldn't reach it in time. A is in O's past, but it doesn't play a role, since he can't know of it at t=0.
What would be different if either FTL travel or FTL signal transfer were possible? In this case we would be allowed to draw signal paths of less than 45 degrees steepness in the space-time diagram. Let us assume that O* is able to send an FTL signal to influence or to cause event A in the first place, just when the two observers pass each other. Note that this signal would travel at v>c in any frame of reference, and that it would travel back in time in O's frame, since it runs into negative t-direction in O's orthogonal x-t coordinate system, to an event that is in O's past. If O* can send an FTL signal to cause the event A, then a second FTL signal can be sent to O to inform him of A as soon as it has just happened. This signal would run at v>c in positive t-direction for O, but in negative t*-direction for O*. So the situation is exactly inverse to the first FTL signal. Now O is able to receive a message from O*'s future.
The paradox occurs when O, knowing about the future, decides to prevent A from happening. Maybe O* is a bad guy, and event A is the death of an unfortunate victim, killed because of his FTL message. O would have enough time to hinder O*, to warn the victim or to take other precautions, since it is still t<0 when he receives the message, and O* has not yet caused event A.
The sequence of events (in logical rather than chronological order) would be as follows:
This is obviously another version of the well-known grandfather paradox. Note that these considerations don't take into account which method of FTL travel or FTL signal transfer is used. Within the realm of Special Relativity, they should apply to any form of FTL travel. Anyway, if FTL travel is feasible, then it is much like time travel. It is not clear how this paradox can be resolved. The basic suggestions are the same as for generic time travel and are outlined in my time travel article.
Rocket propulsion (as a generic term for any drive using accelerated particles) can be described by momentum conservation, resulting in the following simple equation:
The left side represents the infinitesimal speed increase (acceleration) dv of the ship with a mass m, the right side is the mass decrease -dm of the ship if particles are thrusted out at a speed w. This would result in a constant thrust and therefore in a constant acceleration, at least in the range of ship speeds much smaller than c. Eq. 1.13 can be integrated to show the relation between an initial mass m0, a final mass m1 and a speed v1 to be achieved:
The remaining mass m1 at the end of the flight, the payload, is only a fraction of the total mass m0, the rest is the necessary fuel. The achievable speed v1 is limited by the speed w of the accelerated particles, i.e. the principle of the drive, and by the fuel-to-payload ratio.
Let us assume a photon drive as the most advanced conventional propulsion technology, so that w would be equal to c, the speed of light. The fuel would be matter and antimatter in the ideal case, yielding an efficiency near 100%, meaning that according to Eq. 1.14 almost the complete mass of the fuel could contribute to propulsion. Eq. 1.13 and Eq. 1.14 would remain valid, with w=c. If relativistic effects are not yet taken into account, the payload could be as much as 60% of the total mass of the starship, if it's going to be accelerated to 0.5c. However, the mass increase at high sublight speeds as given in Eq. 1.10 spoils the efficiency of any available propulsion system as soon as the speed gets close to c, since the same thrust will effect a smaller acceleration. STL examples will be discussed in section 1.1.7.
Eq. 1.13 shows that the achievable speed is limited by the momentum (speed and mass) of the accelerated particles, provided that a conventional rocket drive is used. The requirements of such a drive, e.g. the photon drive outlined above, are that a considerable amount of particles has to be accelerated to a high speed at a satisfactory efficiency.
Even more restrictive, the human body simply couldn't sustain accelerations of much more than g=9.81ms-2, which is the acceleration on Earth's surface. Accelerations of several g are taken into account in aeronautics and astronautics only for short terms, with critical peak values of up to 20g. Unless something like Star Trek's IDF (inertial damping field) will be invented [Ste91], it is probably the most realistic approach to assume a constant acceleration of g from the traveler's viewpoint during the whole journey. This would have the convenient side effect that an artificial gravity equal to Earth's surface would be automatically created.
