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27.8.13
“The truth is not for all men, but only for those who seek it.” - Ayn Rand quotes
Ayn Rand quotes
Quote by Ayn Rand: Reason is not automatic. Those who deny it cann...
Quote by Ayn Rand: Reason is not automatic. Those who deny it cann...
― Ayn RandRead more quotes from Ayn Rand
Ayn Rand > Quotes
“Reason is not automatic. Those who deny it cannot be conquered by it. Do not count on them. Leave them alone.”
― Ayn Rand
18.8.13
Twin paradox, a thought experiment in special relativity - Wikipedia
Twin paradox - Wikipedia, the free encyclopedia
In physics, the twin paradox is a thought experiment in special relativity involving identical twins, one of whom makes a journey into space in a high-speed rocket and returns home to find that the twin who remained on Earth has aged more. This result appears puzzling because each twin sees the other twin as traveling, and so, according to a naive application of time dilation, each should paradoxically find the other to have aged more slowly. However, this scenario can be resolved within the standard framework of special relativity (because the twins are not equivalent; the space twin experienced additional, asymmetrical acceleration when switching direction to return home), and therefore is not a paradox in the sense of a logical contradiction.
Starting with Paul Langevin in 1911, there have been numerous explanations of this paradox, many based upon there being no contradiction because there is no symmetry—only one twin has undergone acceleration and deceleration, thus differentiating the two cases. Max von Laue argued in 1913 that since the traveling twin must be in two separate inertial frames, one on the way out and another on the way back, this frame switch is the reason for the aging difference, not the acceleration per se.[1] Explanations put forth by Albert Einstein and Max Born invoked gravitational time dilation to explain the aging as a direct effect of acceleration.[2]
The twin paradox has been verified experimentally by precise measurements of atomic clocks flown in aircraft and satellites. For example, gravitational time dilation and special relativity together have been used to explain the Hafele–Keating experiment.[3][4]
In physics, the twin paradox is a thought experiment in special relativity involving identical twins, one of whom makes a journey into space in a high-speed rocket and returns home to find that the twin who remained on Earth has aged more. This result appears puzzling because each twin sees the other twin as traveling, and so, according to a naive application of time dilation, each should paradoxically find the other to have aged more slowly. However, this scenario can be resolved within the standard framework of special relativity (because the twins are not equivalent; the space twin experienced additional, asymmetrical acceleration when switching direction to return home), and therefore is not a paradox in the sense of a logical contradiction.
Starting with Paul Langevin in 1911, there have been numerous explanations of this paradox, many based upon there being no contradiction because there is no symmetry—only one twin has undergone acceleration and deceleration, thus differentiating the two cases. Max von Laue argued in 1913 that since the traveling twin must be in two separate inertial frames, one on the way out and another on the way back, this frame switch is the reason for the aging difference, not the acceleration per se.[1] Explanations put forth by Albert Einstein and Max Born invoked gravitational time dilation to explain the aging as a direct effect of acceleration.[2]
The twin paradox has been verified experimentally by precise measurements of atomic clocks flown in aircraft and satellites. For example, gravitational time dilation and special relativity together have been used to explain the Hafele–Keating experiment.[3][4]
Contents
- 1 History
- 2 Specific example
- 3 Resolution of the paradox in special relativity
- 4 Transfer of clock reading in a twin paradox trip
- 5 The equivalence of biological aging and clock time-keeping
- 6 What it looks like: the relativistic Doppler shift
- 7 Calculation of elapsed time from the Doppler diagram
- 8 Viewpoint of the traveling twin
- 9 Difference in elapsed time as a result of differences in twins' spacetime paths
- 10 Difference in elapsed times: how to calculate it from the ship
- 11 A rotational version
- 12 Explanation in terms of Mach's principle
- 13 See also
- 14 Primary sources
- 15 Secondary sources
- 16 Further reading
- 17 External links
Collected Quotes from Albert Einstein
Collected Quotes from Albert Einstein
- "My religion consists of a humble admiration of the illimitable superior spirit who reveals himself in the slight details we are able to perceive with our frail and feeble mind."
- "Yes, we have to divide up our time like that, between our politics and our equations. But to me our equations are far more important, for politics are only a matter of present concern. A mathematical equation stands forever."
- "The release of atom power has changed everything except our way of thinking...the solution to this problem lies in the heart of mankind. If only I had known, I should have become a watchmaker."
- "Great spirits have always found violent opposition from mediocrities. The latter cannot understand it when a man does not thoughtlessly submit to hereditary prejudices but honestly and courageously uses his intelligence."
- "The most beautiful thing we can experience is the mysterious. It is the source of all true art and all science. He to whom this emotion is a stranger, who can no longer pause to wonder and stand rapt in awe, is as good as dead: his eyes are closed."
- "A man's ethical behavior should be based effectually on sympathy, education, and social ties; no religious basis is necessary. Man would indeeded be in a poor way if he had to be restrained by fear of punishment and hope of reward after death."
- "The further the spiritual evolution of mankind advances, the more certain it seems to me that the path to genuine religiosity does not lie through the fear of life, and the fear of death, and blind faith, but through striving after rational knowledge."
- "Now he has departed from this strange world a little ahead of me. That means nothing. People like us, who believe in physics, know that the distinction between past, present, and future is only a stubbornly persistent illusion."
- "You see, wire telegraph is a kind of a very, very long cat. You pull his tail in New York and his head is meowing in Los Angeles. Do you understand this? And radio operates exactly the same way: you send signals here, they receive them there. The only difference is that there is no cat."
- "One had to cram all this stuff into one's mind for the examinations, whether one liked it or not. This coercion had such a deterring effect on me that, after I had passed the final examination, I found the consideration of any scientific problems distasteful to me for an entire year."
- "...one of the strongest motives that lead men to art and science is escape from everyday life with its painful crudity and hopeless dreariness, from the fetters of one's own ever-shifting desires. A finely tempered nature longs to escape from the personal life into the world of objective perception and thought."
- "He who joyfully marches to music rank and file, has already earned my contempt. He has been given a large brain by mistake, since for him the spinal cord would surely suffice. This disgrace to civilization should be done away with at once. Heroism at command, how violently I hate all this, how despicable and ignoble war is; I would rather be torn to shreds than be a part of so base an action. It is my conviction that killing under the cloak of war is nothing but an act of murder."
- "A human being is a part of a whole, called by us _universe_, a part limited in time and space. He experiences himself, his thoughts and feelings as something separated from the rest... a kind of optical delusion of his consciousness. This delusion is a kind of prison for us, restricting us to our personal desires and to affection for a few persons nearest to us. Our task must be to free ourselves from this prison by widening our circle of compassion to embrace all living creatures and the whole of nature in its beauty."
- "Not everything that counts can be counted, and not everything that can be counted counts." (Sign hanging in Einstein's office at Princeton)
13.8.13
Most Recent Common Ancestors: Interesting Mental Kung Fu
Most Recent Common Ancestors: Interesting Thing of the Day
June 25, 2004
Eve, Charlemagne, and you
If you asked me to name any three subjects at random about which I know very little, I might very well mention biology, statistics, and genealogy. Somehow I managed to get through high school, college, and graduate school without ever taking a biology course. My mathematical studies never progressed much beyond garden-variety trigonometry. And when it comes to genealogy, I still haven’t been able to work out which one of the Baltic states my paternal great-grandfather hailed from. So you can imagine my dismay when a friend of mine told me about the notion of a Most Recent Common Ancestor (MRCA). Apparently some Very Smart People had determined based on genetic evidence and statistical calculations that everyone of European ancestry now alive descended from Charlemagne. At first I found this claim utterly ridiculous, but as I began reading up on it my incredulity softened to mere incomprehension. For me, at least, this is a rather challenging idea to wrap my brain around, but to the extent I’ve been able to understand it, I find the implications truly fascinating.
Let’s start with a few uncontroversial facts. You were directly descended from two parents, four grandparents, eight great-grandparents, and so on. So as you go back in time, the number of your ancestors increases. Go back far enough and you’ll find millions or perhaps trillions of forebears in your personal lineage. The same, of course, is true for everyone else. But as you go back in time the total population of the world decreases, meaning that the farther back you look, the more likely you are to encounter someone who was a common ancestor of any two random people alive today. So far so good? OK, the question then becomes: How far must you go back in time to find someone who is a common ancestor of everyone alive today? And for bonus points, who might that have been?
Madam, I’m Adam
You might be tempted to think you have to go back all the way to the beginning of the human race, however you wish to define that. Of course, some of the earliest humans were ancestors of us all—but it turns out that we also share common ancestors who lived much more recently. There are a couple of ways to figure this out.
One approach is to look at your DNA. DNA that live in the parts of cells known as the mitochondria are normally passed on to children only by the mother. This means that your mother, her mother, her grandmother, and all her female ancestors share the same mitochondrial DNA. This DNA is relatively stable, but tends to undergo tiny mutations at predictable intervals over thousands of years. Since scientists know how often such changes occur, they can calculate, based on any two samples of mitochondrial DNA, how many generations ago the two individuals had a common ancestor. By examining hundreds of samples from all over the world, researchers have determined how recently a woman could have lived who is a direct ancestor of everyone tested, which by extension implies nearly everyone in the world. Their estimate is 150,000 years, give or take 50,000—and they’ve dubbed this hypothetical woman “Mitochondrial Eve.” This is not to say anyone believes she was the first woman, or even the only woman alive at the time. She may have had tens of thousands of contemporaries. But she was the only one alive then with living offspring today.
Of course, this only tells you about our most recent common maternal ancestor. What about Adam? Well, just as mitochondrial DNA is passed on relatively intact only from mother to child, the Y chromosome is passed on only from fathers to sons. Using the same sort of “molecular clock” that indicated how far back a common female ancestor was, geneticists have determined that the most recent male ancestor of everyone currently alive—called “Y-chromosomal Adam”—lived about 60,000 years ago (again, give or take a few tens of thousands of years).
Let’s start with a few uncontroversial facts. You were directly descended from two parents, four grandparents, eight great-grandparents, and so on. So as you go back in time, the number of your ancestors increases. Go back far enough and you’ll find millions or perhaps trillions of forebears in your personal lineage. The same, of course, is true for everyone else. But as you go back in time the total population of the world decreases, meaning that the farther back you look, the more likely you are to encounter someone who was a common ancestor of any two random people alive today. So far so good? OK, the question then becomes: How far must you go back in time to find someone who is a common ancestor of everyone alive today? And for bonus points, who might that have been?
Madam, I’m Adam
You might be tempted to think you have to go back all the way to the beginning of the human race, however you wish to define that. Of course, some of the earliest humans were ancestors of us all—but it turns out that we also share common ancestors who lived much more recently. There are a couple of ways to figure this out.
One approach is to look at your DNA. DNA that live in the parts of cells known as the mitochondria are normally passed on to children only by the mother. This means that your mother, her mother, her grandmother, and all her female ancestors share the same mitochondrial DNA. This DNA is relatively stable, but tends to undergo tiny mutations at predictable intervals over thousands of years. Since scientists know how often such changes occur, they can calculate, based on any two samples of mitochondrial DNA, how many generations ago the two individuals had a common ancestor. By examining hundreds of samples from all over the world, researchers have determined how recently a woman could have lived who is a direct ancestor of everyone tested, which by extension implies nearly everyone in the world. Their estimate is 150,000 years, give or take 50,000—and they’ve dubbed this hypothetical woman “Mitochondrial Eve.” This is not to say anyone believes she was the first woman, or even the only woman alive at the time. She may have had tens of thousands of contemporaries. But she was the only one alive then with living offspring today.
Of course, this only tells you about our most recent common maternal ancestor. What about Adam? Well, just as mitochondrial DNA is passed on relatively intact only from mother to child, the Y chromosome is passed on only from fathers to sons. Using the same sort of “molecular clock” that indicated how far back a common female ancestor was, geneticists have determined that the most recent male ancestor of everyone currently alive—called “Y-chromosomal Adam”—lived about 60,000 years ago (again, give or take a few tens of thousands of years).
Diminishing Returns
DNA, however, doesn’t tell the whole story. For one thing, DNA studies can produce MCRA estimates based on female ancestors or male ancestors, but not both at the same time. And although each of your parents gave you half of their DNA, you may be aware that for any given gene, there is less than 100% probability that it will be passed on to one’s children. Thus, you cannot be at all certain that you got exactly one-eighth of your DNA from each of your great-grandparents. The farther back in time you go, the greater the probability that any given gene of an ancestor was not passed on to you. In other words, you may have any number of ancestors with whom you share no DNA at all.
A completely different approach neatly sidesteps these issues and uses simple math (well, actually, rather complicated math) to show statistically how long ago a Most Recent Common Ancestor existed—based on both male and female lines. Yale statistician Joseph Chang wrote a 1999 paper entitled “Recent Common Ancestors of All Present-Day Individuals.” If his many pages of equations, theorems, and proofs are to be believed—and even Chang says his figures probably don’t account for all the facts—we could have a most recent common ancestor who lived as recently as A.D. 1200. More recent research and computer simulations have pushed that date back to perhaps A.D. 300, but that’s still a far cry from 60,000 years ago.
Looking for a Date
However, Chang’s paper proves something much more surprising. According to his calculations, there was a date in the not-too-distant past at which all individuals were either ancestors of everyone alive today, or ancestors of no one alive today. This date varies depending on what portion of the population you look at, but for Europe, it would seem to be in the neighborhood of A.D. 800—the year Charlemagne, king of the Franks, became emperor of Rome. And because Charlemagne is known to be the ancestor of some people alive today, that must mean he was the ancestor of all people of European descent. Of course, we pick on Charlemagne because he’s so well known, but he doesn’t have any special status as a common ancestor. In reality, about 80% of the people living in Europe at Charlemagne’s time were also ancestors of everyone from the West.
If this is a bit mind-boggling, it’s just the tip of the iceberg. Dr. Mark Humphrys, a professor at the School of Computing, Dublin City University, has analyzed the results of Chang’s work as well as numerous other academic papers and come up with some startling conclusions. According to Humphrys, everyone in the world is descended from the Egyptian royal house (ca. 1600 B.C.), and almost everyone in the world is descended from Confucius (ca. 500 B.C.). Moreover, he claims, nearly every Muslim, Jew, and Westerner is descended from the Prophet Muhammad (ca. A.D. 600). I don’t have the mathematical kung fu to evaluate these claims thoroughly, but they certainly are thought-provoking. Even more scary: the thought that thousands of years from now, I could be the ancestor of everyone alive.
—Joe Kissell
DNA, however, doesn’t tell the whole story. For one thing, DNA studies can produce MCRA estimates based on female ancestors or male ancestors, but not both at the same time. And although each of your parents gave you half of their DNA, you may be aware that for any given gene, there is less than 100% probability that it will be passed on to one’s children. Thus, you cannot be at all certain that you got exactly one-eighth of your DNA from each of your great-grandparents. The farther back in time you go, the greater the probability that any given gene of an ancestor was not passed on to you. In other words, you may have any number of ancestors with whom you share no DNA at all.
A completely different approach neatly sidesteps these issues and uses simple math (well, actually, rather complicated math) to show statistically how long ago a Most Recent Common Ancestor existed—based on both male and female lines. Yale statistician Joseph Chang wrote a 1999 paper entitled “Recent Common Ancestors of All Present-Day Individuals.” If his many pages of equations, theorems, and proofs are to be believed—and even Chang says his figures probably don’t account for all the facts—we could have a most recent common ancestor who lived as recently as A.D. 1200. More recent research and computer simulations have pushed that date back to perhaps A.D. 300, but that’s still a far cry from 60,000 years ago.
Looking for a Date
However, Chang’s paper proves something much more surprising. According to his calculations, there was a date in the not-too-distant past at which all individuals were either ancestors of everyone alive today, or ancestors of no one alive today. This date varies depending on what portion of the population you look at, but for Europe, it would seem to be in the neighborhood of A.D. 800—the year Charlemagne, king of the Franks, became emperor of Rome. And because Charlemagne is known to be the ancestor of some people alive today, that must mean he was the ancestor of all people of European descent. Of course, we pick on Charlemagne because he’s so well known, but he doesn’t have any special status as a common ancestor. In reality, about 80% of the people living in Europe at Charlemagne’s time were also ancestors of everyone from the West.
If this is a bit mind-boggling, it’s just the tip of the iceberg. Dr. Mark Humphrys, a professor at the School of Computing, Dublin City University, has analyzed the results of Chang’s work as well as numerous other academic papers and come up with some startling conclusions. According to Humphrys, everyone in the world is descended from the Egyptian royal house (ca. 1600 B.C.), and almost everyone in the world is descended from Confucius (ca. 500 B.C.). Moreover, he claims, nearly every Muslim, Jew, and Westerner is descended from the Prophet Muhammad (ca. A.D. 600). I don’t have the mathematical kung fu to evaluate these claims thoroughly, but they certainly are thought-provoking. Even more scary: the thought that thousands of years from now, I could be the ancestor of everyone alive.
—Joe Kissell
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