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Michael Graczyk Simple Special Relativity
The concept of Special Relativity, introduced by Albert Einstein in 1905, revolutionized physics on all fronts. The idea that the laws behind Newtonian physics were only approximate estimations of actual physical properties was nothing new, but the scale and significance of this assertion was not fully understood until Einstein clearly laid out the theories and implications in his, On the Electrodynamics of Moving Bodies, 1905. The idea reconciled Maxwell’s equations describing the relative behavior pattern of the electromagnetic force. The paper showed in mathematics and thought experiments that the laws of mechanics dwindled in accuracy at speeds closer to the speed of light. The idea, working on the preexisting notion that light travels through an infinite and immovable medium known as aether, described light’s velocity as a universal constant. In fact, essential to the discussion of relativity is its description of light as having constant velocity unaffected by the emitter, though independent of a universal medium, which creates relative inertial frames of reference with different views of the same velocities relative to the constant speed of light. Special relativity must, then, describe a universe in which all the units of mechanics are not only describable only in terms of relationship to a specific frame of reference, but that the perception of properties such as mass and time are skewed based on the relative velocity of the observer. Generally, although the properties of physics remain constant inside specific frames of inertial reference, there exists no universal way to measure distance, mass, time, or energy between different inertial systems. This also disagrees with the previously accepted idea of a plane of aether, light’s wave medium, which created a privileged frame of reference. Special relativity does not, however, include the effects of gravity on the interaction between space and time. The theory of special relativity has several important parts. All uniform motion is relative; there is no universal determinant of velocity. The second part, Einstein’s addition, is that regardless of the frame of reference of the observer, the speed of light remains constant. The implications of the latter are best observed at high rates of speed. Although no universal frame of reference exists, speeds can be compared to their proximity to the speed of light. This makes time relative to the observer. Additionally, preservation of non-preferred frames of reference in a universe with constant speed of light means an inversely proportional relationship exists between the rate of speed relative to light and the amount of energy needed to accelerate a constant mass a given unit of speed. Thus, as an object accelerates closer to light speed, it becomes infinitely difficult to accelerate the object. There are many effects of this distortion in simple measurements. All are results are caused by Lorentz transformations, the product of a variable measurement of time, mass and space. The first and simplest result of relativity is time dilation. When an object moves faster relative to another frame of reference, the measure of time must slow to keep the speed of light constant. When a spaceship flying at half the speed of light emits a beam of light towards the earth, the ship will measure that beam of light as having the same velocity of an identical beam emitted from earth. Velocity is distance over time. If distance traveled by either beam remains constant when measured from both reference points, and the velocity of light is always constant, then the time observed to have passed on the space ship by Earth must be slow. So the time between two events, when measured using identical clocks, is variant based on the relative motion of either object. Another result of relativity is the variance of simultaneity. Two events separated by space may appear to be simultaneous to one observer, while separated when observed by another. Because the speed of light is constant in all inertial reference frames, a stationary object will see light emitted in a frame of reference in relative motion as moving with a constant velocity in all directions. Imagine an observer at the center of an enormous spherical mirror. He emits a flash of light in all directions, and can observe that the light reflects off all parts of the mirror at the same time, and return at the same time. Now another observer is moving past the sphere at a high rate of speed. The second observer will see the light emitted by the first observer as moving with the same velocity in all directions within his inertial frame of reference. So the light moving in parrallel with him will be moving at the universal speed of light, as will be the light moving away from him. His velocity does not counteract that of the emitter, and all light is measured at a constant speed. So, the light moving in motion parrallel to the second observers own velocity will reflect off the mirror first, before the light reflects of the opposite side of the mirror. Events that appeared simultaneous to one observer appeared separated in time to another. Again, the constancy of the speed of light creates paradoxes in which observers view space and time as variable to their frame of reference. Relativity also results in a distance contraction known as length contraction. As an object’s speed approaches the speed of light, its length becomes a quotient of the original length and the Lorentz factor, or the reciprocal of the square root of the difference between one and the ratio of the square of the velocity compared to the constant velocity of light. That means that the Lorentz factor increases as the velocity of any object approaches the velocity of light. So an object’s length will decrease as its velocity increases. This decrease is a result of a necessary decrease of length per measured time unit to create a constant speed of light. That means that as you view an accelerating object, the object’s back end will be closer to its front as velocity increases. Light reflecting off the front of the object moves at the same rate as light reflecting off the back end, so as the object’s velocity increases, the time interval between the front and the back occupying the same space decreases. Thus the observed length of the object decreases. Time dilation in a moving object means that a distance traveled in a specific amount of time will be less when compared to another frame of reference with a different velocity. Because each reference frame sees the other’s clock as running slowly, the length of the opposite objects will appear contracted. A fast moving object will appear shorter because its units of time are shorter, and time and distance are proportional quantities. Directly defying Newton’s laws of physics is the principle that velocity composition is not simply a sum. When at relatively low speeds, simply summing the vectors for velocity results in an accurate measurement of velocity. If there is car moving relative to the curb at 30 m/s, then launching a projectile at 60m/s off the front of the car will result in the projectile traveling with a velocity of 90 m/s relative to the curb. Relativity, however, tells us that the clock on the car runs slower when compared to the curbside clock. If the car is moving at a significant fraction of the speed of light, then the relationship between speeds becomes geometric rather than arithmetic. If the car is going half the speed of light relative to the curbside, and launches a projectile forward at the speed of light, then an addition of vectors would figure the projectile’s velocity to be the speed of light, which is impossibility. To the observer on the curb, the velocity of the projectile is the quotient of the velocity of the car, and the velocity of the projectile relative to the car, and the sum of one and the products of the ratios of the two relative velocities to the constant speed of light. What that means is that regardless of the frame of reference, or the sum of relative velocities, the velocity of and combination of successive propulsions has a limit at the speed of light. A stationary observer sees a ship moving half the speed of light, even though another ship moving half as quickly sees the first ship as moving substantially faster than a quarter the speed of light. Because the clock on the moving ships runs slower relative to the clock of the stationary observer, its measurements of velocity are smaller in comparison, and therefore the velocity is not simply a sum of successive velocities. Inertia is also redefined by relativity. The constancy of the speed of light creates a direct and interchangeable relationship between mass and energy, so the properties that affect mass also affect energy. Likewise, as a mass is accelerated from a stationary reference point, inertia of the energy of movement requires more force to accelerate the object the same amount at speeds closer to the speed of light. The relative mass of an object increases due to a variance in the measurement of time within the frame of reference. As viewed outside the frame of reference, this is projected as an increase in the mass of the object as the velocity increases. If a ball is moving at half the speed of light, it requires more energy to increase the ball’s velocity by the same factor than it did when the ball was in relative rest. Causality is also affected by relativity. The idea that light has a constant velocity through all reference frames is necessary to preserve universal causality. If light could move faster in a faster reference frame, it would be possible with reflections to view the outcomes of one’s actions before the action was performed, thereby breaking down causality. If a constant force on an object could continuously accelerate it compared to a stationary frame of reference, then the object could accelerate faster than light in the stationary frame of reference. This moving object could carry information, and therefore deliver the consequences of an action before it occurred. To preserve this causality, the clock of a faster moving object is slow proportional to the relative velocity of the observer. This results in light having the same measured speed in both frames of reference, but because time runs slower in the moving reference, even though the moving frame is already in motion. Relativity changes previous Newtonian Physics substantially, but does not negate its use because of the minimal effect of relativity at slow speeds that humans normally encounter in practical applications.
References:
Hamilton, Andrew. "Special Relativity." UC Boulder Astrophysics. 6 Feb. 1999. UC Boulder. 17 Dec. 2007 <http://casa.colorado.edu/~ajsh/sr/sr.shtml>.
http://www.fourmilab.ch/etexts/einstein/specrel/www/ On the Electrodynamics of Moving Bodies, A. Einstein, Annalen der Physik, 17:891, June 30, 1905 (in English translation)
G. A. Benford, D. L. Book, and W. A. Newcomb, The Tachyonic Antitelephone, Phys. Rev. D 2, 263 - 265 (1970) article.
Michael “Beast” Graczyk Nate “Beast #4” Claggett Egg Drop Paper All mass has the property to resist a change in a state of motion. The specific aspects and reasons behind this property, known as inertia, differ depending on whom you ask. Isaac Asimov would say that all matter is inherently “lazy,” while Einstein would say that the path of space-time curves around the gravitational field of the object. Newton disregarded explaining the phenomenon, preferring instead to simply describe the resulting mathematics, that it takes a force to accelerate a mass, and a force to decelerate it as well. Regardless of the specifics, the implication is the same; Massive objects resist relative changes in velocity. The inertia of an object is proportional to its mass. The manufacturers of safety seat belts know full and well the physics behind the life saving power of the safety belt. The idea behind a belt is to reduce the damage caused to the body as a result of the inertia of the body. When traveling in a car, your body’s inertia is overcome by accelerating your mass to the speed of the vehicle. Once the body gains momentum, the product of a velocity and a mass, a force is required to decrease the velocity of the mass. In the case of a person in a car, the key to reducing this force is increasing the interval over which a force of deceleration is applied. Since the change in momentum is equal to the force on the mass times the time over which it was applied, increasing the physical length of the distance over which the force is applied reduces the force on the body. Damage to a human body in a car accident is the result of forces on the body. A larger force is more likely to damage organs and harm the brain. When the distance of a momentum change is increased, these potentially fatal byproducts of deceleration decrease. When driving, high velocity differences between the vehicle and the surrounding environment exist. These differences can become problems if the vehicle collides with an object that exerts a strong normal force repelling the collision, and does not move. The entire momentum of the vehicle, rather than being transferred into the immovable object that it struck, exists in the bending of the metal of the frame, and the continued movement of objects not completely affixed to the vehicle. The passengers inside continue to move forward with the complete momentum of their entire mass. A severe problem not exists; how does the person stop once the vehicle has already stopped? If the person’s momentum is immediately met with a normal force, the resulting internal force would surely kill the soft human. The best conclusion is that the person must be stopped over an increased distance. Several problems are solved by the inclusion of a seat belt around the person traveling within the vehicle. First, the person cannot fly out of the car, where they could be met by unknown surfaces and hazards of the environments, whether it is a cliff, hard pavement, sharp objects, or even heat. Keeping the person involved in the accident inside the car is a large priority in the event of a collision. Second, the seat belt increases the distance over which the inevitable deceleration of the person’s body occurs. By providing a movable barrier to movement, or one that gives, the seatbelt decreases the force on the body by moving while it decelerates. This also prevents the body from impacting hard interior surfaces of the car, which would rapidly decelerate the body, causing internal damage as well.
Andréasson, Rune; Claes-Göran Bäckström (2000.). The Seat Belt : Swedish Research and Development for Global Automotive Safety. Stockholm: Kulturvårdskommittén Vattenfall AB, p. 12. ISBN 91-630-9389-8.
