General Scientific Principles
Written: 1998.08.29
Last Revised: 1998.12.19
Upper/Lower Limits | Efficiency | Force, Energy, and Power | Newton's Laws | Gravity | Relativity | Philosophy | Process Analysis
There appears to be considerable confusion about the meaning of upper and lower limits.
An upper limit is a figure for the absolute highest possible value of a given figure (ie- the figure cannot possibly be higher than this limit, but it can be much lower). As an example, if a matter/antimatter reactor consumes 100kg of antimatter, then an upper limit for its energy output would be 2·100·c² = 1.8E19 joules. In other words, it is physically impossible for a matter/antimatter reactor to produce more than 1.8E19 joules from 100kg of antimatter fuel. In fact, the costs of operating the reactor and various process inefficiencies would most likely reduce such a reactor's true power output dramatically, usually to a small fraction of the upper limit. Typically, trekkies calculate upper limits and then treat those limits as order-of-magnitude estimates rather than limits!
A lower limit is a figure for the absolute lowest theoretically possible value of a given figure. As an example, if a 1kg block of steel is heated from 300K to 400K with no other changes, the change in its enthalpy is roughly 60kJ. It is impossible to heat it thusly without expending at least 60 kJ, because the energy state of the system has increased by 60kJ. In reality the energy expenditure will be higher due to process inefficiencies, and perhaps much higher. As with upper limits, lower limits are routinely misinterpreted by trekkies as estimates or even upper limits! Again, this routine misinterpretation may be due to either sheer ignorance or quasi-religious fanaticism.
The laws of physics, particularly the Second Law of Thermodynamics, prevent any process from ever being perfect. Nothing is 100% efficient anywhere in the universe. Nothing can be measured or controlled to 100% exact precision. No process can occur in zero elapsed time. These concepts are simple, easily understood, and obvious to any scientifically trained person. However, since most trekkies lack scientific training and more importantly, they lack the scientific mindset, they seem unable to grasp these simple concepts. Trekkie "analyses" of their technology invariably include phrases like "perfect targeting", "instant reaction", "100% efficiency", etc. even though any such assumptions inevitably lead to highly unrealistic upper limits rather than reasonable estimates.
The concepts of force, energy, and power are fundamental building blocks of science. No one can claim to have even a remote familiarity with science unless he or she has a firm grasp on these concepts- anyone without a firm grasp on these concepts cannot understand the more complex scientific principles.
Force: the Physics definition of Force (as opposed to the various unscientific colloquial definitions of force or The Force) is defined in Webster's New World Dictionary as "the cause, or agent, that puts an object at rest into motion or alters the motion of a moving object." That's probably as good as definition as any- a fundamental concept like force is difficult to define accurately without resorting to equations or circular definitions involving synonyms. In equation format, F=ma as implied by Newton's Second Law of Motion. In other words, the amount of force required to accelerate an object is proportional to the rate of acceleration multiplied by the mass of the object.
Energy
Definition: Energy is perhaps the most fundamental concept in science. All events and processes can be analyzed strictly in terms of energy balances, from obvious examples like power plants, drive motors, and refridgeration systems to less obvious examples like the biological processes in the human body and the severity of collisions. Energy balances can be used to determine whether a process is possible at all, and the extent to which a process is reversible. Energy can take many forms- mechanical, kinetic, electrical, thermal, electromagnetic, chemical, entropic, etc. A full description of all the various types of energies and the various methods by which they can be related to various situations and processes is obviously beyond the scope of this document. However, we can say that the units of energy have the dimensions of force multiplied by distance, although there are many other ways to combine units to result in energy.
The First Law of Thermodynamics: This law is the most fundamental concept of physics: it states that the total amount of energy in a closed system is constant. A more popular way of describing it is to say that energy can neither be created or destroyed, but I personally prefer the former definition. The meaning is the same in either case, but it is important to remind people of the importance of the closed system in energy analyses.
Energy Balances: Basically, energy balances involve "before and after" states of energy. An example is a 10kg block of steel resting on the ground compared to the same block suspended 10m above the ground. The suspended block has ~981J more potential energy than the block sitting on the ground, so at least 981J of energy will be required to go from one state to the other. The height change cannot be accomplished without adding 981J of energy (in some form or other) into the system because that would mean that some of its new potential energy came "out of nowhere", a clear violation of the First Law of Thermodynamics. If far more energy is poured into the system (say, 2kJ), then the excess energy must go somewhere- the block has only gained 981J of potential energy, so the extra 1019J must become something else, eg. thermal energy in the steel or the surrounding air. Note: it is perfectly possible to expend energy without adding energy to your intended target: if you push against a concrete wall with all of your might, you will expend large quantities of chemical energy within your body without adding any significant energy to the wall. This is not a violation of energy balances or the First Law of Thermodynamics- you expended energy in your body, but you never added any energy to the wall. In the overall scheme of things, chemical energy was transformed into thermal energy in your body but no mechanical energy was added to the system.
Case Study: the Genesis Device: The concept of energy balances is deceptively simple in definition, but it is misunderstood with frightening regularity, which is very ironic considering that people are talking about science fiction and claim to understand science. One example is the Genesis Device seen in ST2. Trekkies have advanced ludicrous energy estimates for the Genesis Device (the author of the Daystrom Institute Technical Library claims that a "conservative" estimate for the Genesis Device energy output is 2E48 J!), but in reality, their claims merely serve as good illustrations of how not to perform an energy balance:
The erroneous trekkie "analysis" goes as follows: the mass of the planet must collapse quickly to form a planet (actually, the author of the aforementioned propaganda claims that it formed an entire star system even though the entire process took place inside an existing star system). Therefore, we can generate some arbitrary figures for the size of the nebula (eg. Mr. Kennedy claims that the nebula is 1.5 light years wide even though it is small enough to fit inside a star system, reachable from a nearby planet within minutes at sublight velocities). Once we have generated these large and impressive (albeit totally incorrect) numbers, we can assume that the entire collapse process occurs in seconds (although the actual elapsed time is totally unknown). Since we have exaggerated this combination of nebula size and elapsed time to the point where it necessitates superluminal collapse speeds, we can then conclude that the matter is warp-driven (perhaps they put tiny warp nacelles on all of the little molecules). Therefore, we can use the warp-power chart in the TM to guesstimate the energy required to raise the speed of the matter by the necessary amount to form an entire solar system (even though only a planet was formed). Even if we disregard the various logical and dimensional errors in this analysis, we can easily determine that it violates the First Law of Thermodynamics for two reasons:
The change in energy states is negative, not positive! When a planet or even an entire star system is formed from a nebula, the energy state of the newly-formed planet (or star system, if one chooses to accept the author's non-canon assumptions about creating an entire star system) is actually lower than the energy state of the pre-existing nebula. In fact, given enough time, it is entirely possible that gravitational forces would have eventually drawn the nebula together into a single mass naturally! While the Death Star must take a planet in a low gravitational potential energy state and bring its entire mass to a high gravitational potential energy state, the Genesis Device did the exact opposite: it took a nebula with a high gravitational potential energy state and brought it to a low gravitational potential energy state. This means that the process of drawing a nebula together into a planet and/or star will actually tend to release energy rather than consuming it.
Where does all the excess energy go? If the Genesis Device actually releases 2E48 J of energy, it has to go somewhere! If it expends all of that energy to accelerate the matter to incredible speeds, what happens when all of that matter reaches the target? Does the kinetic energy simply disappear? It cannot simply disappear, due to the First Law of Thermodynamics. 2E48 J of energy is equivalent to the simultaneous explosion of tens of millions of supernovae- such a massive release of energy would have obliterated the entire Regula star system and most likely eliminated all life on nearby star systems as well. There is no way to perform work on an object without raising its energy state: accelerated objects gain kinetic energy, deformed objects gain thermal energy, etc. Federation cultists ask us to believe that an input of 2E48 J actually lowered the energy state of the Mutara Nebula!
We can clearly see that the trekkie claims regarding the Genesis Device are based on a deplorable misunderstanding of the First Law of Thermodynamics; instead of comparing the "before" and "after" energy states, as anyone with thermodynamics or physics training would do, they make uneducated guesses about what happens during the process itself, and don't particularly care whether it makes any sense from the point of view of overall energy balances. To summarize, don't make the same mistakes made by trekkies: remember that "energy after" - "energy before" = "energy input". Thus endeth the lesson.
Power: Power is the rate at which energy is expended. A knowledge of both power and energy output is required to determine the destructive power of a weapon, because a small amount of energy can be released at a very high rate, thus resulting in a large power output, or a large amount of energy can be released at a very low rate, thus resulting in a low power output. In either case, the destructive capabilities of the weapon would be far lower than a high-power, high-energy weapon. A high-power, low-energy weapon will simply not be able to do a lot of work on its target, so the power level is only impressive from a mathematical standpoint. A low-power high-energy weapon may be able to perform a lot of work on its target, but if the power level is extremely low then various energy-dissipation mechanisms will come into play, preventing the energy from concentrating in a single location to the point where it can be dangerous.
Inertia: "Every body persists in its state of rest or of uniform motion in a straight line unless it is compelled to change that state by forces imposed on it."
Force and acceleration: "The rate of change of momentum of a body is proportional to the resultant force acting on the body and is in the direction of that force." This law is often expressed as the resultant equation, F=ma.
Action/reaction: "To every action there is always opposed an equal reaction; or, the mutual attractions of two bodies upon each other are always equal, and directed to contrary parts." This concept is very poorly understood by a large number of people. Let's suppose your mass is 70kg. As you sit in your chair under 1g conditions reading this page, you are therefore exerting roughly 690N of force upon the chair. The direct implication of Newton's third law is that the chair is exerting 690N of force back up against your posterior. Similarly, when an aircraft moves through the atmosphere at constant speed, its engines push it forward with (for example) 250kN of force. The aerodynamic drag of the aircraft must push back with the same force.
Newton's law of universal gravitation is as follows: "The force between any two particles having masses m1 and m2 separated by a distance r is an attraction acting along the line joining the particles and has the magnitude F=Gm1m2/r² where G is a universal constant having the same value for all pairs of particles." The value of G has been experimentally determined (see the Constants page), and r is the distance between the two particles.
The above equation is often applied to large objects by idealizing those objects into point-masses where the entire mass of the object is considered to be located at its centre of gravity. Note that this idealization is invalid when one object is inside another (eg- a small object buried deep within the Earth), but it can be used when the objects are separate.
The gravitational potential energy can be determined by integrating the gravitational force equation from r to infinity. The result is that the gravitational potential energy of one object with respect to another works out to U=-Gm1m2/r. The negative sign is not a mistype; the potential energy is zero at infinite distances and decreases as the separation distance decreases. This is not intuitive but many scientific concepts are not instantly intuitive, in spite of Federation cultists' concerted efforts to rewrite science to conform to their intuitive beliefs. As the distance decreases, the potential energy must also decrease. If the sign were reversed, we would find that the potential energy would continually increase as the two objects approach. In other words, the potential energy of an object falling toward the Earth would increase as it falls! This is obviously a ludicrous solution, so hopefully it should be obvious why the sign is negative.
A common exercise is to determine an object's escape velocity from a celestial object such as a planet, moon, or star. Escape velocity is defined as the velocity at which an object's kinetic energy is equal to the magnitude of its gravitational potential energy. In other words, 0.5mv²=GmM/r where m is the small object's mass, M is the celestial body's mass, and r is the initial distance between their centres of gravity. We can easily rearrange the equation to find that v²=2GM/r; note that the escape velocity is independent of the smaller object's mass. For an object on Earth's surface, we can easily substitute G=6.67E-11, M=5.97E24, and r=6.37E6 to find that v=1.12E4 m/s.
Another exercise is to determine the binding energy of a system of particles (such as a planet being destroyed by the Death Star). The basic principle is that the binding energy of a system of particles is equal to the sum total of the magnitudes of the gravitational potential energy existing between any two particles in that system. For a system of only three particles A, B, and C, the binding energy would be the sum total of the magnitude of the gravitational potential energy between A+C added to the potential energy between B+C added to the potential energy between A+C.
The 20th century Earth scientist known as Einstein was the first of his species to strike upon two concepts:
The laws of physics are the same in all inertial frames. No preferred inertial frame exists.
The speed of light is the same in all inertial frames.
From these two basic postulates the entire field of General Relativity appeared. Numerous conclusions were drawn about the relationship between matter and energy based on the above two basic postulates, such as the following:
Matter-Energy equivalence: E = mc². This formula describes the relationship between energy and mass- if a given amount of mass is completely converted to energy, it will release a quantity of energy equal to its mass multiplied by the speed of light squared, which is 9E16 m²/s².
Mass dilation:
This formula describes the relationship between rest-mass and true mass, as velocity increases. For a 1kg object traveling at 0.99c, its relativistic mass will be more than 7 kg.Time dilation:
This formula assigns quantitative values to the basic concept that "moving clocks run slow". As speeds increase, the perception of time's passage slows down. An observer traveling at 0.99c would experience the passage of 0.14 seconds for every second that passes from the point of view of a stationary observer.Length dilation:
This formula describes the increase in length associated with increasing speeds. In other words, a fast object becomes longer and conversely, a fast-moving observer will perceive slow-moving objects to be shorter. An observer traveling at 0.99c would perceive a 1m long object as being only 0.14m long.Kinetic energy: The kinetic energy of an object moving at relativistic speeds is (m-mo)·c². If one substitutes the mass increase equation into that formula, it can be determined that when v is small, the equation simplifies to the classical equation, KE=0.5mv². For a 1kg object traveling at 0.99c, Newtonian physics predicts a kinetic energy of 4.41E16 J, while general relativity predicts kinetic energy of 5.48E17 J, more than 12 times larger. Note that this is actually higher than the energy yield that would come from annihilating the 1kg mass entirely into energy.
Momentum: The momentum of an object moving at relativistic speeds is calculated by simply substituting the relativistic mass into the classical p=mv equation. A 1kg object traveling at 0.99c will have 2.97E8 kg·m/s of linear momentum according to Newtonian physics and 2.11E9 kg·m/s of linear momentum according to general relativity.
The philosophy of science is a subject for much discussion and a real discussion is beyond the scope of this document. However, the basic philosophy of science is that it exists to describe the physical universe. The key word is "describe." It does not exist to promulgate belief systems, support or deny any particular set of beliefs, or create technology. Its profound impact on society and technology is a side-effect, not its intended purpose.
The basic scientific method is quite simple: analyze measured data, formulate theories which fit the data, and then perform experiments designed to disprove the theories. If those experiments fail to disprove those theories, the theories gain weight. Notice here the important distinction between failing to disprove a theory and proving the theory. It is impossible to prove a scientific theory.
Suppose we have numerous theories which we have failed to disprove, and which fit the facts. How do we choose between them? We use Occam's Razor, which states that when faced with numerous theories that all fit the facts, the simplest theory will always be the correct one. This may sound like an arbitrary decision, but it is not: this is the way the universe works. Without Occam's Razor, we could potentially generate an infinite number of theories, of progressively increasing complexity, to explain any given phenomenon.
An interesting tendency of trekkies is their propensity for discussion of process rather than measurement of results. Anyone wishing to maintain any remote adherence to science must remember that science and engineering are both based on the measurement of results, not discussion of process. Without measured data, scientists would never be able to sort nonsensical theories out from valid theories. Energy balances, for example, are entirely based on the measurement of energy states before and after a process, rather than examining what happens inside the process itself. If we wish to understand the process, we always perform the energy balance first, and then attempt to derive any process analysis from that foundation.
How does this apply to trekkies? One example is the Federation "warp core." Instead of discussing the known capabilities and limitations of warp cores (ie- what they have and have not been able to do), or describing their fuel consumption as a basis for upper limits, the vast majority of trekkies assume that the upper limit is in fact the energy output! If a modern-day automotive engineer attempted to estimate automobile engine horsepower based on fuel consumption, he would be laughed out of work. Yet, this is the precise method that Federation cultists use to "estimate" warp core output!
If we were to analyze automobile engines based on their intake manifold fluid mechanics, fuel delivery system design, crankshaft geometry, camshaft profile, lifter design, valve geometry, air intake system design, fuel consumption, combustion chamber geometry, spark timing, etc., we would be expending an incredible amount of effort in a hopelessly futile endeavour. Depending on the set of assumptions and approximations we use, we would be able to arbitrarily calculate any of a wide range of horsepower figures, particularly if we base our analyses on superficial diagrams and descriptions (like the information in the TM) rather than complete engineering specifications. Even with complete engineering specifications, automotive engineers never know how powerful an engine will be until they actually hook it up to a dynamometer and measure its horsepower. Measurement of results rather than discussion of process; that is how it is done in the real world, and for good reason.
If trekkies want to generate realistic lower limits for any given technology, they need to perform energy balances to determine what they have and have not accomplished. For example, we know that Federation shuttlecraft, Imperial starfighters, Victory-class Star Destroyers, etc. can all easily escape the gravity well of a planet, so they are all capable of easily generating 6E7 J/kg for their mass. We know from official sources (the TM, BTM, and the ROTJ novelization) that both Federation and Imperial starships are capable of travelling at a significant fraction of the speed of light, and even exceeding the speed of light, but it is doubtful that their actual kinetic energy is commensurate with their apparent speed because it would require the annihilation of most of their mass into energy. In the case of Federation vessels, they claim to use a subspace "mass-lightening field" to achieve high speeds without expending large amounts of energy, and similarly extenuating circumstances may apply to Imperial starships. There is, of course, no scientific basis for a "mass-lightening" field, but we have no choice but to accept certain technobabble devices in science fiction series, otherwise the series collapse and almost none of the plot occurences are possible.
However, regardless of what disagreements might result from the measurement and discussion of phenomena, at least we would be discussing actual results rather than meaningless discussion of process (based of superficial diagrams and descriptions of technology which is not fully explained). Unlike measurement of results which generates established and undeniable capabilities, discussion of process merely generates huge volumes of discussion without meaningful conclusions.