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Albert Einstein was a revered scientist and mathematician, and for good reason. His developments in the fields of physics and mathematics paved the way for advanced topics that change the way we think our world works.
One such development was the theory of special relativity. Einstein’s theory of special relativity was arguably one of his most famous theories, and it correlates the dimensions of the universe with light and time; two things that were previously thought to be unrelated. Learning more about Einstein’s theories can help create new theories that may change our world further and may even help us solve new problems relating to space travel and time as the fourth dimension.
Einstein’s first theory, the theory of special relativity, relates speed, time, and distance at the lightspeed scale. Because much of our lives do not occur anywhere near lightspeed, some of these examples can be difficult to comprehend, but they will make sense in the end.
A train analogy is one of the best ways to represent special relativity because it depicts relative motion with an everyday example.
Let’s say that a man is on a train in a vacuum moving at three quarters of the speed of light. The first postulate of special relativity dictates that the laws of physics remain the same for all inertial reference frames. If the train was moving through empty space and the man saw a light in the distance, the man would not be able to differentiate whether he was moving towards the light or if the light was moving towards him. Luckily, through the first postulate, both scenarios obey the same laws of physics.
The second postulate states that light speed is always the same in a vacuum irrespective of the motion of the object that emits the light. For example, if the train was shining a light in front of it, the light’s speed would not be 1 ¾ times the speed of light; rather, it would still just be the speed of light. However, this constant state of lightspeed creates rather bizarre situations regarding time and distance. As per the equation d = vt, velocity is directly proportional to distance and inversely proportional to time.
However, when velocity is constant, time has to dilate and distance has to contract.
Let’s say that the man in the train flashes a light towards a mirror on the train one meter away from him, so the light would have traveled 2 meters in total. However, from an outsider’s perspective, the train is moving along with the man and the light. Thus, the light takes a diagonal path as shown in the image above. This diagonal path indicates that the light traveled a further distance from the outsider’s perspective, or a shorter distance from the man’s perspective. This is length contraction. But in that case, that would mean that the light was traveling for a longer period of time from the outsider’s perspective, which in turn means that time is moving ever-so-slightly slower for the man in the train. This concept is known as time dilation and its effects become more and more pronounced the closer the stationary observer gets to the speed of light.
The Lorentz factor is a crucial expression for understanding special relativity well, and it can be expressed as “the ratio between the time interval in the frame of reference with respect to which the object is moving to the time interval in the frame of reference in which the object is at rest” (Kataru, 2019).
Going back to the light example, we can represent the time taken for light to hit the mirror and return to the observer as t from the observer’s perspective and the time taken for light to do the same thing as t’ from the outsider’s perspective, caused by time dilation. We can also define the distance from the observer and the mirror as l and the train’s velocity as v, for generalization purposes. Thus, the distance that the light travels to the mirror is ct’/2 (where c is the speed of light), while the distance that the train travels in the same timespan is vt’/2. This creates a right triangle, with which we can use the Pythagorean Theorem to express l^2 as t'^2(c^2-v^2)/4. However, we also know that l can be defined as ct/2 from the observer’s perspective. Substituting that in, we achieve the equation c^2*t^2=t'^2(c^2-v^2). Using the definition of the Lorentz factor given above, we can rearrange the equation as a ratio between t’ and t, giving us the equation (t'^2/t^2)=((c^2)/(c^2-v^2)). Further simplifying, we get γ=1/(sqrt(1-v^2/c^2)), where γ (gamma) is t’/t. The Lorentz factor can be used in a variety of relativistic scenarios, including time dilation and length contraction. Additionally, the factor shows how the velocity of any object cannot be as fast as or faster than the speed of light because substituting such values into v in the equation would result in undefined or imaginary numbers.
Overall, the theory of special relativity is a very interesting one because of its wide applicability in many areas of science. Even Einstein observed this and expanded it to include gravity, another mysterious area of physics in his theory of general relativity, which will be saved for another time. Special relativity shows the future of lightspeed travel and if it is even possible in the near future. Through simplified examples, such complex areas of science can be applied and taught to everyone, proving the allure of physics and mathematics in our world.
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