1.1.1 Classical Physics

This chapter summarizes some very basic theorems of physics, mostly predating the theories of Special Relativity and of General Relativity.

Newton's Laws of Motion

Isaac Newton discovered the following laws that are still valid (meaning that they are a very goodapproximation) for speeds much slower than the speed of light.

1. An object at rest or in uniform motion in a straight line will remain at rest or in the same uniform motion unless acted upon by an unbalanced force. This is also known as the law of inertia.

2. The acceleration a of an object is directly proportional to the total unbalanced force F exerted on the object, and is inversely proportional to the mass m of the object (in other words, as mass increases, the acceleration has to decrease). The acceleration of an object moves in the same direction as the total force. This is also known as the law of acceleration.

Eq. 1.1

3. If one object exerts a force on a second object, the second object exerts a force equal in magnitude and opposite in direction on the object body. This is also known as the law of interaction.

Gravitation

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:

Eq. 1.2

G=6.67259*10-11m3kg-1s-2 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:

Eq. 1.3

Here g is an acceleration slightly varying throughout Earth's surface, with an average of 9.81ms-2.

Momentum Conservation

In an isolated system, the total momentum is constant. This fundamental law is not affected by the theoriesof Relativity. Energy conservation In an isolated system, the total energy is constant. This fundamental law is not affected by the theories ofRelativity.
 
 

Second Law of Thermodynamics

The overall entropy of an isolated system is always increasing. Entropy generally means disorder. Anexample 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.
 
 

Doppler Shift

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 sourceis moving with a speed v relative to the medium, mostly air, in which the sound is propagating at a speed w:

Eq. 1.4

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 thespeed of the receiver, then the following applies to the frequency:

Eq. 1.5

Here the plus sign denotes the case of an approching 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 lightpropagates and the two equations will merge to one relativistic Doppler shift.

Particle-Wave Dualism

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:

Eq. 1.6

The best-known example is the photon, a particle that represents electromagnetic radiation. The other way round, 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.
 
 

1.1.2 Special Relativity

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:

•  1. There is no invariant "fabric of space" relative to which an absolute speed could be defined or measured. The terms "moving" or "resting" make only sense if they refer to a certain other frame of reference. The perception of movement is always mutual; the starship pilot who leaves Earth could claim that he is actually resting while the solar system is moving away.
• 2. The speed of light, c=3*108m/s in the vacuum, is the same in all directions and in all frames of reference. This means that nothing is added or subtracted to this speed, as the light source apparently moves.

Frames of Reference

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 thereforegive 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.

Space-Time Diagram

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-dimensionaldiagram (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.

Fig. 1.1 Space-time diagram for a resting observer

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 showsthe 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 occuring 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* seesA&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 theangle 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=c, they will merge to one single axis, equivalent to the path of a light beam.

Time Dilation

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 pathsshow 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 therecontructed 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.

Fig. 1.4 Illustration of time dilation (right half) and length contraction (bottom half)

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 samefactor. 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:

Eq. 1.7

Note: When drawing the axes to scale in an x-t diagram, one has to acount 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 isfound, 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 cirsumvents all these problems in away that no such relativistic effects occur.

Length Contraction

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:

Eq. 1.8

Once again, note that when drawing the x*-axis to scale, a correction is necessary, a factor of cos alpha to the above formula.

Addition of Velocities

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.

Fig. 1.5 Non-relativistic addition of velocities

Fig. 1.6 Relativistic addition of velocities

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 projectilewith 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 observerwith any velocity (or in any frame of reference) would measure a light speed of exactly c=3*108m/s.

Space-time-diagrams allow to derive the addition theorem for relativistic velocities. The resulting speed u is given by:

Eq. 1.9

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 Increase

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 wholesystem 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:

Eq. 1.10

This function is plotted in Fig. 1.7.

Fig. 1.7 Mass increase for relativistic speeds

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 squareroot 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.

Mass-Energy Equivalence

Let us consider Eq. 1.10 again. It is possible to express it as follows:

Eq. 1.11

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!):

Eq. 1.12

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/antimatterreaction in which the two forms of matter annihilate each other, thereby transforming their whole mass into energy.

Light Cone

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.

Fig. 1.8 The light cone

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 Ais 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 isalso 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 willagree that there are certain, either affected (future) or affecting (past) events inside the light cone, and outside events they shouldn't bother about.