Special relativity: Difference between revisions
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==Lorentz transformation== | ==Lorentz transformation== | ||
Time dilation specifies that time passes more slowly in an inertial frame that moves relative to our own, countering human perception of time's universally uniform passage. Time's passage, however, differs infinitesimally even well beyond the top speeds humans can achieve with the help of technology such as rocket propulsion. To calculate the magnitude of time dilation and the relativistic effect on length in the direction of motion, Special Relativity employs the [[Lorentz | Time dilation specifies that time passes more slowly in an inertial frame that moves relative to our own, countering human perception of time's universally uniform passage. Time's passage, however, differs infinitesimally even well beyond the top speeds humans can achieve with the help of technology such as rocket propulsion. To calculate the magnitude of time dilation and the relativistic effect on length in the direction of motion, Special Relativity employs the [[Inertial_frame_of_reference#Lorentz_frames_of_reference|Lorentz transformation]] first proposed by [[Hendrik Antoon Lorentz]]. | ||
To compare common-sense with the Lorentz transformation, consider first a common sense scenario. Suppose that we are motionless with our clock in an inertial frame (the ''reference frame'') as a car with its own clock passes by in its inertial frame (the ''primed frame'') at velocity ''v''. The car's length, which we measured beforehand as ''L'', is measured by its driver in the car's inertial frame as ''L' ''. If, according to our clock, we time an interval ''t'' after the car passes us, we know from experience the car's clock will also have passed the same time ''t''. If we measure the car's length as it goes by, our common sense and measuring technique also gives us length ''L''. The formulas ''t''' = ''t'' and ''L' '' = ''L'' follow from the [[Inertial_frame_of_reference#Galilean_frames_of_reference|Galilean transformation]]. | |||
As we have already seen, though, Special Relativity's effects elude our common sense perception, so to calculate the change in time and length we use formulas following from the Lorentz | As we have already seen, though, Special Relativity's effects elude our common-sense perception, so to calculate the change in time and length we now use formulas following from the Lorentz transformation: | ||
::<math>t^\prime = \frac{t}{\sqrt{1 - | ::<math>t^\prime = \frac{t}{\sqrt{1 - {v^2}/{c^2}}}</math> | ||
and | and | ||
::<math>L^\prime = L \ | ::<math>L^\prime = L \ \sqrt{1 - {v^2}/{c^2}}</math> | ||
Generally speaking, the Lorentz and Galilean | Generally speaking, the Lorentz and Galilean transformations specify two different mathematical techniques for mapping one [[Cartesian coordinate system]] onto another moving (primed) Cartesian system. We only apply them here to tell us how uniform motion relativistically induces time dilation and length contraction, also called [[Lorentz contraction]]. | ||
For human scaled velocities | For human scaled velocities – always infinitesimal relative to the speed of light – the fraction (''v''<sup>2</sup>/''c''<sup>2</sup>) is so close to zero that the quantity under the square root in both formulas is effectively ''1''. Since the square root of ''1'' is ''1'', we can see these formulas reduce to those conforming to common sense, which tells us time passes uniformly and object's length doesn't vary just from an object's uniform velocity, in accord with the formulas from the Galilean transformation. Should ''v'', however, become an appreciable fraction of ''c'', ''v''<sup>2</sup>/''c''<sup>2</sup> gets closer to ''1'' so √(''1'' - ''v''<sup>2</sup>/''c''<sup>2</sup>) gets close to zero. This implies that ''L' '' becomes only a small fraction of ''L'', that is, that length contracts in the primed frame. In the time formula, we invert this same fraction (so that it's now greater than ''1'') and move it to the other side of the equation to see that ''t' '' must be multiplied by some number greater than ''1'' in order to equal ''t''. This means more time ''t'' passes than time ''t' '', so time passes more slowly in the primed frame. | ||
==The twin paradox== | ==The twin paradox== | ||
Revision as of 14:34, 29 September 2011
The theory of special relativity was developed by Albert Einstein, and published in 1905. It describes the behaviour of objects travelling at very high speeds - close to the speed of light in vacuum - with respect to other objects. A prediction of the theory is that when the relative velocity of two objects is an appreciable fraction of light's speed, time passes at different rates for those objects. Distances also vary; for example, length shortens in the direction of motion.
An important postulate of the theory is that the speed of light in vacuum always is constant (its value is denoted by the symbol c by physicists, and by c0 in the SI system of units), no matter what the relative speed of the object emitting the light to that of the object receiving it. Unlike the 'classical' example of a bullet fired from a train (where the bullet's speed relative to the ground, or another train, depends on the muzzle velocity of the gun, and speed of the trains), the speed of a beam of light is always c, no matter what the velocity of the source, and the device measuring its speed.
These proposals utterly contradicted humans' intuitive view of the everyday world; in particular, our perception of time and distance, while quite correct in everyday life, improperly extends these intuitive ideas to high speeds. Because our intuitive understanding ultimately misunderstands them fundamentally, human perception fails to grasp what are thought to be the true nature of time and distance.
The theory was in accord with the paradoxical results of certain 19th century physical experiments which attempted to detect the universe's background ether, which was supposed to be the ultimate neutral background, or reference framework, against which the entire physical universe moved. Physicists had always assumed the ether's existence, but experiments--most notably the Michelson-Morley experiment of the 1880s--always failed to detect it.
By boldly refusing to assume the possibility of an ether and theorizing laws of motion without referring to an absolute background, Einstein's simple presumption of objects' "relativity" revolutionized the fundamental view of the physical universe. Astonishingly, Einstein developed the theory when he was only a twenty-six year old clerk in the Swiss patent office.
Einstein's assumptions
Einstein rested his theory on two uncontroversial postulates. He presumed physical experiments performed in any room moving at any constant speed in any constant direction, that is, in any inertial frame, must always produce the same results. In other words, all physical laws should take the same form in all inertial frames, including the laws of electromagnetism.[Note 1] This notion is shared by Newtonian mechanics and the mechanics of Galileo. Along with this proposal, called by Einstein the Special principle of relativity, Einstein proposed as well that the speed of light in vacuum should be the same for all inertial observers, which is the key postulate separating special relativity from earlier mechanics.[Note 2] This postulate was in accord with work published by experimental physicist Albert Michelson in 1881 and with greater accuracy in collaboration with Edward Morley in 1887, although these experiments may not have been a strong influence upon Einstein at the time.[Note 3] The Michelson-Morley experiment aimed to determine the speed of light relative to the background ether, which required detecting differences in light's speed depending on how it moved through that ether. Surprisingly, the experiment found that light moves with exactly the same speed all the time, regardless of the motion of the object from which the light emanates or is measured.
Aside from its basis in physicists' experimental results, assuming the constancy of light's speed also does not contradict human perception in any obvious way. In everyday life we experience light's speed as invariably infinite: turn on a light switch and a room is illuminated instantaneously. A simple thought experiment, however, reveals the strangeness of light's speed:
Imagine driving a car straight down a highway at 60 mph. An observer on the side of the road measures our speed at 60 mph. If another car comes toward us at 50 mph as measured by the observer on the side of the road, we inside our car would perceive it coming at us at (60 + 50) = 110 mph. Both cars and the outside observer are in inertial frames. From experience, we know that speeds simply add together. Now imagine that we turn on our headlights. Designating the speed of light in the traditional manner by the symbol c, we see the light beam travel away from us at light's constant speed c. We might also presume that the oncoming car's driver sees our light beam traveling at (c + 110) mph because experience tells us we must add the speed of our car and the oncoming car to the speed of our light beam. Our assumption that observers always measure light's speed as the same, however, means that the other car sees the light beam moving at speed c as well, and that the extra 110 mph makes no difference. The observer on the side of the road must also see our light beam traveling at speed c even though it emanates from a moving car. The cars' speeds make no difference. If both cars were traveling at half the speed of light, the oncoming car would still measure our light beam as traveling at speed c, not at c + (1/2) c + (1/2) c, regardless of the great speed of the two cars.
Time dilation
Consider another thought experiment. We travel in a train uniformly in one direction at speed v (that is, we're in an inertial frame). An observer stands motionless on the side of the tracks watching our train coach pass (that is, he's also in an inertial frame). We shoot a light beam from a flashlight straight up at a mirror on the train coach's ceiling a distance y from the flashlight. Inside the train we see the light beam go straight up, hit the mirror, and come straight back down, covering the distance y twice, which is a total distance of 2y. Let's call the short amount of time this experiment takes t. During time t the train travels a short distance.
Now consider what the observer on the side of the tracks sees. Because the train moves, he does not see the light beam go straight up and down, but sees it climb at an angle, hit the mirror, then travel back down at the same angle to hit the flashlight which has now moved a short distance. Each leg of the light beam's journey is a distance greater than y, so it has traveled a distance greater than the 2y we measured inside the train. We can easily prove this by imagining a line coming down from the mirror forming a right triangle one of whose sides is the mirror's height y and the other side half the distance traveled, say x. By the Pythagorean theorem, the hypotenuse must be greater than y, so the total distance traveled must be greater from the outside observer's point of view. Let's call the total distance we saw the light beam travel d and the outside observer's greater distance D. The time the experiment took for us is t, but let's designate the time passage for the outside observer as t'. The light beam's speed is the same for all observers, c.
Since distance = rate × time we now have
and
Because D > d and c is constant, we must conclude that t' > t, that is, more time passed during the experiment from the outside observer's point of view than it did from ours. In other words, from the outside observer's point of view, time passed more slowly inside the train than it did outside the train. The Theory of Special Relativity calls this effect time dilation.
Lorentz transformation
Time dilation specifies that time passes more slowly in an inertial frame that moves relative to our own, countering human perception of time's universally uniform passage. Time's passage, however, differs infinitesimally even well beyond the top speeds humans can achieve with the help of technology such as rocket propulsion. To calculate the magnitude of time dilation and the relativistic effect on length in the direction of motion, Special Relativity employs the Lorentz transformation first proposed by Hendrik Antoon Lorentz.
To compare common-sense with the Lorentz transformation, consider first a common sense scenario. Suppose that we are motionless with our clock in an inertial frame (the reference frame) as a car with its own clock passes by in its inertial frame (the primed frame) at velocity v. The car's length, which we measured beforehand as L, is measured by its driver in the car's inertial frame as L' . If, according to our clock, we time an interval t after the car passes us, we know from experience the car's clock will also have passed the same time t. If we measure the car's length as it goes by, our common sense and measuring technique also gives us length L. The formulas t' = t and L' = L follow from the Galilean transformation.
As we have already seen, though, Special Relativity's effects elude our common-sense perception, so to calculate the change in time and length we now use formulas following from the Lorentz transformation:
and
Generally speaking, the Lorentz and Galilean transformations specify two different mathematical techniques for mapping one Cartesian coordinate system onto another moving (primed) Cartesian system. We only apply them here to tell us how uniform motion relativistically induces time dilation and length contraction, also called Lorentz contraction.
For human scaled velocities – always infinitesimal relative to the speed of light – the fraction (v2/c2) is so close to zero that the quantity under the square root in both formulas is effectively 1. Since the square root of 1 is 1, we can see these formulas reduce to those conforming to common sense, which tells us time passes uniformly and object's length doesn't vary just from an object's uniform velocity, in accord with the formulas from the Galilean transformation. Should v, however, become an appreciable fraction of c, v2/c2 gets closer to 1 so √(1 - v2/c2) gets close to zero. This implies that L' becomes only a small fraction of L, that is, that length contracts in the primed frame. In the time formula, we invert this same fraction (so that it's now greater than 1) and move it to the other side of the equation to see that t' must be multiplied by some number greater than 1 in order to equal t. This means more time t passes than time t' , so time passes more slowly in the primed frame.
The twin paradox
Notice that the principle of relativity allows us to say that from the observer inside the car's point of view, he is at rest and it is the observer on the side of the road who moves. Since both observers are in inertial frames, physical experiments must always produce the same results, namely that the car's driver observes our time pass more slowly than his own. How is it possible for both frames to see the other as passing more slowly? Couldn't they stop, meet, and determine whose clock shows greater time passage? It could only be one clock or the other.
This is an example of the well known twin paradox:[Note 4]
- According to those on Earth, a twin taking a space trip at high speeds has a slower biological clock than an Earth-bound twin. The traveler therefore returns to Earth to find their stay-at-home sibling has aged in comparison. The age difference seems a paradox if one adopts the view that, to the traveler, the Earth-bound sibling appears to experience a high speed history, and so should age more slowly according to the traveler. That is, to the traveler upon return, the stay-at-home should be the younger twin, contradicting the Earth observers' expectations.
This tale is, however, no paradox: the Theory of Special Relativity does indeed conclude that observers in inertial frames always see clocks in other moving inertial frames passing time more slowly than their own--another of relativity's results showing the counterintuitive operations of the physical world. Despite the straightforwardness of the paradox's mathematical solution, which does indeed show that the two clocks will almost certainly differ when they meet, its detail goes beyond this article's scope. Suffice it to say the paradox arises from claiming that the two clocks stop and meet: since both clocks have to stop to meet, at least one of the clocks must have altered its velocity, that is, left its inertial frame. Recall that Special Relativity only regards the laws of physics in inertial frames. Once forces like acceleration (or deceleration) or gravity are introduced, one must turn to the Theory of General Relativity to explain motion's effects on length and the passage of time.
References
- ↑ Albert Einstein (1952). “The foundation of the general theory of relativity”, A Sommerfeld, ed: The Principle of Relativity: a collection of original memoirs on the special and general theory of relativity, Republication of 1923 translation by W Perret and GB Jefferey of original article. Courier Dover Publications, p. 111. ISBN 0486600815. “Special principle of relativity: If a system of coordinates K is chosen so that, in relation to it, physical laws hold good in their simplest form, the same laws hold good in relation to any other system of coordinates K' moving in uniform translation relatively to K.” Notice the emphasis upon simplest form for the laws, the key point that separates inertial frames from others.
- ↑ Albert Einstein (1952). “The foundation of the general theory of relativity”, A. Sommerfeld, editor: The Principle of Relativity: a collection of original memoirs on the special and general theory of relativity, Republication of 1923 translation by W Perret and GB Jefferey of original article. Courier Dover Publications, p. 111. ISBN 0486600815. “Thus, the special theory of relativity does not depart from classical mechanics through the postulate of relativity but through the postulate of the constancy of light in vacuo, from which, in combination with the special principle of relativity, there follow, in the well known way, the relativity of simultaneity, the Lorentzian transformation, and the related laws for the behaviour of moving bodies and clocks.”
- ↑ Jeremy Bernstein (1997). Albert Einstein and the frontiers of physics. Oxford University Press, p. 55. ISBN 0195120299. “On the other hand, how much did Einstein know of the Michelson-Morely experiment?...At various times he said that he had either heard of Michelson's work, or that he hadn't, or that if he had heard of it, it didn't matter.”
- ↑ The twin paradox has been studied experimentally by putting clocks on airplanes and comparing them with Earth-bound clocks. The faster traveling clocks develop a time lag. See Don Bernett Lichtenberg (2007). “§10.3 The twin paradox”, The universe and the atom. Springer, pp. 116 ff. ISBN 9812705619. A theoretical analysis is provided by Vesselin Petkov (2007). Relativity and the Nature of Spacetime, 2nd ed. Springer, pp. 146 ff. ISBN 3642019528.