Metin Bektas's Blog: Metin's Media and Math - Posts Tagged "education"
Einstein’s Special Relativity – The Core Idea
It might surprise you that a huge part of Einstein’s Special Theory of Relativity can be summed up in just one simple sentence. Here it is:
“The speed of light is the same in all frames of references”
In other words: no matter what your location or speed is, you will always measure the speed of light to be c = 300,000,000 m/s (approximate value). Not really that fascinating you say? Think of the implications. This sentence not only includes the doom of classical physics, it also forces us to give up our notions of time. How so?
Suppose you watch a train driving off into the distance with v = 30 m/s relative to you. Now someone on the train throws a tennis ball forward with u = 10 m/s relative to the train. How fast do you perceive the ball to be? Intuitively, we simply add the velocities. If the train drives off with 30 m/s and the ball adds another 10 m/s to that, it should have the speed w = 40 m/s relative to you. Any measurement would confirm this and all is well.
Now imagine (and I mean really imagine) said train is driving off into the distance with half the light speed, or v = 0.5 * c. Someone on the train shines a flashlight forwards. Obviously, this light is going at light speed relative to the train, or u = c. How fast do you perceive the light to be? We have the train at 0.5 * c and the light photons at the speed c on top of that, so according to our intuition we should measure the light at a velocity of v = 1.5 * c. But now think back to the above sentence:
“The speed of light is the same in all frames of references”
No matter how fast the train goes, we will always measure light coming from it at the same speed, period. Here, our intiution differs from physical reality. This becomes even clearer when we take it a step further. Let’s have the train drive off with almost light speed and have someone on the train shine a flashlight forwards. We know the light photons to go at light speed, so from our perspective the train is almost able to keep up with the light. An observer on the train would strongly disagree. For him the light beam is moving away as it always does and the train is not keeping up with the light in any way.
How is this possible? Both you and the observer on the train describe the same physical reality, but the perception of it is fundamentally different. There is only one way to make the disagreement go away and that is by giving up the idea that one second for you is the same as one second on the train. If you make the intervals of time dependent on speed in just the right fashion, all is well.
Suppose that one second for you is only one microsecond on the train. In your one second the distance between the train and the light beam grows by 300 meter. So you say: the light is going 300 m / 1 s = 300 m/s faster than the train.
However, for the people in the train, this same 300 meter distance arises in just one microsecond, so they say: the light is going 300 m / 1 µs = 300 m / 0.000,001 s = 300,000,000 m/s faster than the train – as fast as it always does.
Note that this is a case of either / or. If the speed of light is the same in all frames of references, then we must give up our notions of time. If the light speed depends on your location and speed, then we get to keep our intiutive image of time. So what do the experiments say? All experiments regarding this agree that the speed of light is indeed the same in all frames of references and thus our everyday perception of time is just a first approximation to reality.
“The speed of light is the same in all frames of references”
In other words: no matter what your location or speed is, you will always measure the speed of light to be c = 300,000,000 m/s (approximate value). Not really that fascinating you say? Think of the implications. This sentence not only includes the doom of classical physics, it also forces us to give up our notions of time. How so?
Suppose you watch a train driving off into the distance with v = 30 m/s relative to you. Now someone on the train throws a tennis ball forward with u = 10 m/s relative to the train. How fast do you perceive the ball to be? Intuitively, we simply add the velocities. If the train drives off with 30 m/s and the ball adds another 10 m/s to that, it should have the speed w = 40 m/s relative to you. Any measurement would confirm this and all is well.
Now imagine (and I mean really imagine) said train is driving off into the distance with half the light speed, or v = 0.5 * c. Someone on the train shines a flashlight forwards. Obviously, this light is going at light speed relative to the train, or u = c. How fast do you perceive the light to be? We have the train at 0.5 * c and the light photons at the speed c on top of that, so according to our intuition we should measure the light at a velocity of v = 1.5 * c. But now think back to the above sentence:
“The speed of light is the same in all frames of references”
No matter how fast the train goes, we will always measure light coming from it at the same speed, period. Here, our intiution differs from physical reality. This becomes even clearer when we take it a step further. Let’s have the train drive off with almost light speed and have someone on the train shine a flashlight forwards. We know the light photons to go at light speed, so from our perspective the train is almost able to keep up with the light. An observer on the train would strongly disagree. For him the light beam is moving away as it always does and the train is not keeping up with the light in any way.
How is this possible? Both you and the observer on the train describe the same physical reality, but the perception of it is fundamentally different. There is only one way to make the disagreement go away and that is by giving up the idea that one second for you is the same as one second on the train. If you make the intervals of time dependent on speed in just the right fashion, all is well.
Suppose that one second for you is only one microsecond on the train. In your one second the distance between the train and the light beam grows by 300 meter. So you say: the light is going 300 m / 1 s = 300 m/s faster than the train.
However, for the people in the train, this same 300 meter distance arises in just one microsecond, so they say: the light is going 300 m / 1 µs = 300 m / 0.000,001 s = 300,000,000 m/s faster than the train – as fast as it always does.
Note that this is a case of either / or. If the speed of light is the same in all frames of references, then we must give up our notions of time. If the light speed depends on your location and speed, then we get to keep our intiutive image of time. So what do the experiments say? All experiments regarding this agree that the speed of light is indeed the same in all frames of references and thus our everyday perception of time is just a first approximation to reality.
How Statistics Turned a Harmless Nurse Into a Vicious Killer
Let’s do a thought experiment. Suppose you have 2 million coins at hand and a machine that will flip them all at the same time. After twenty flips, you evaluate and you come across one particular coin that showed heads twenty times in a row. Suspicious? Alarming? Is there something wrong with this coin? Let’s dig deeper. How likely is it that a coin shows heads twenty times in a row? Luckily, that’s not so hard to compute. For each flip there’s a 0.5 probability that the coin shows heads and the chance of seeing this twenty times in a row is just 0.5^20 = 0.000001 (rounded). So the odds of this happening are incredibly low. Indeed we stumbled across a very suspicious coin. Deep down I always knew there was something up with this coin. He just had this “crazy flip”, you know what I mean? Guilty as charged and end of story.
Not quite, you say? You are right. After all, we flipped 2 million coins. If the odds of twenty heads in a row are 0.000001, we should expect 0.000001 * 2,000,000 = 2 coins to show this unlikely string. It would be much more surprising not to find this string among the large number of trials. Suddenly, the coin with the supposedly “crazy flip” doesn’t seem so guilty anymore.
What’s the point of all this? Recently, I came across the case of Lucia De Berk, a dutch nurse who was accused of murdering patients in 2003. Over the course of one year, seven of her patients had died and a “sharp” medical expert concluded that there was only a 1 in 342 million chance of this happening. This number and some other pieces of “evidence” (among them, her “odd” diary entries and her “obsession” with Tarot cards) led the court in The Hague to conclude that she must be guilty as charged, end of story.
Not quite, you say? You are right. In 2010 came the not guilty verdict. Turns out (funny story), she never commited any murder, she was just a harmless nurse that was transformed into vicious killer by faulty statistics. Let’s go back to the thought experiment for a moment, imperfect for this case though it may be. Imagine that each coin represents a nurse and each flip a month of duty. It is estimated that there are around 300,000 hospitals worldwide, so we are talking about a lot of nurses/coins doing a lot of work/flips. Should we become suspicious when seeing a string of several deaths for a particular nurse? No, of course not. By pure chance, this will occur. It would be much more surprising not to find a nurse with a “suspicious” string of deaths among this large number of nurses. Focusing in on one nurse only blurs the big picture.
And, leaving statistics behind, the case also goes to show that you can always find something “odd” about a person if you want to. Faced with new information, even if not reliable, you interpret the present and past behavior in a “new light”. The “odd” diary entries, the “obsession” with Tarot cards ... weren’t the signs always there?
Be careful to judge. Benjamin Franklin once said he should consider himself lucky if he’s right 50 % of the time. And that’s a genius talking, so I don’t even want to know my stats.
Not quite, you say? You are right. After all, we flipped 2 million coins. If the odds of twenty heads in a row are 0.000001, we should expect 0.000001 * 2,000,000 = 2 coins to show this unlikely string. It would be much more surprising not to find this string among the large number of trials. Suddenly, the coin with the supposedly “crazy flip” doesn’t seem so guilty anymore.
What’s the point of all this? Recently, I came across the case of Lucia De Berk, a dutch nurse who was accused of murdering patients in 2003. Over the course of one year, seven of her patients had died and a “sharp” medical expert concluded that there was only a 1 in 342 million chance of this happening. This number and some other pieces of “evidence” (among them, her “odd” diary entries and her “obsession” with Tarot cards) led the court in The Hague to conclude that she must be guilty as charged, end of story.
Not quite, you say? You are right. In 2010 came the not guilty verdict. Turns out (funny story), she never commited any murder, she was just a harmless nurse that was transformed into vicious killer by faulty statistics. Let’s go back to the thought experiment for a moment, imperfect for this case though it may be. Imagine that each coin represents a nurse and each flip a month of duty. It is estimated that there are around 300,000 hospitals worldwide, so we are talking about a lot of nurses/coins doing a lot of work/flips. Should we become suspicious when seeing a string of several deaths for a particular nurse? No, of course not. By pure chance, this will occur. It would be much more surprising not to find a nurse with a “suspicious” string of deaths among this large number of nurses. Focusing in on one nurse only blurs the big picture.
And, leaving statistics behind, the case also goes to show that you can always find something “odd” about a person if you want to. Faced with new information, even if not reliable, you interpret the present and past behavior in a “new light”. The “odd” diary entries, the “obsession” with Tarot cards ... weren’t the signs always there?
Be careful to judge. Benjamin Franklin once said he should consider himself lucky if he’s right 50 % of the time. And that’s a genius talking, so I don’t even want to know my stats.
Earth’s Magnetic Field
You have been in a magnetic field all your life. The Earth, just like any other planet in the solar system, spawns its own magnetic field. The strength of the field is around B =0.000031 T, but research has shown that this value is far from constant. Earth’s magnetic field is constantly changing. How do we know this? When rocks solidify, they store the strength and direction of the magnetic field. Hence, as long it is possible to figure out the orientation of a rock at the time of solidification, it will tell us what the field was like back then
For rocks that are billions of years old, deducing the original orientation is impossible. Continental drifting has displaced and turned them too often. But thanks to the incredibly low speed of drifting continents, scientists were able to recreate the magnetic field of Earth for the past several million years. This revealed quite a bit.
For one, the poles don’t stand still, but rather wander across the surface with around 50 km per year. The strength of the field varies from practically zero to 0.0001 T (about three times the current strength). And even more astonishingly: the polarity of the field flips every 300,000 years or so. The north pole then turns into the south pole and vice versa. The process of pole reversal takes on average between 1000 and 5000 years, but can also happen within just 100 years. There is no indication that any of these changes had a noticeable impact on plants or animals.
Where does the magnetic field come from? At present there’s no absolute certainty, but the Parker Dynamo Model developed in the sixties seems to be provide the correct answer. The inner core of Earth is a sphere of solid iron that is roughly equal to the Moon in size and about as hot as the surface of the Sun. Surrounding it is the fluid outer core. The strong temperature gradient within the outer core leads to convective currents that contain large amounts of charged particles. According to the theory, the motion of these charges is what spawns the field. Recent numerical simulations on supercomputers have shown that this model is indeed able to reproduce the field in most aspects. It explains the intensity, the dipole structure, the wandering of the poles (including the observed drifting speed) and the pole reversals (including the observed time spans).
It is worth noting that the pole which lies in the geographic north, called the North Magnetic Pole, is actually a magnetic south pole. You might recall that we defined the north pole of a magnet as the pole which will point northwards when the magnet is allowed to turn freely. Since unlike poles attract, this means that there must be a magnetic south pole in the north (talk about confusing). By the same logic we can conclude that the Earth’s South Magnetic Pole is a magnetic north pole.
More interesting physics for non-physicists can be found here:
Physics! In Quantities and Examples
For rocks that are billions of years old, deducing the original orientation is impossible. Continental drifting has displaced and turned them too often. But thanks to the incredibly low speed of drifting continents, scientists were able to recreate the magnetic field of Earth for the past several million years. This revealed quite a bit.
For one, the poles don’t stand still, but rather wander across the surface with around 50 km per year. The strength of the field varies from practically zero to 0.0001 T (about three times the current strength). And even more astonishingly: the polarity of the field flips every 300,000 years or so. The north pole then turns into the south pole and vice versa. The process of pole reversal takes on average between 1000 and 5000 years, but can also happen within just 100 years. There is no indication that any of these changes had a noticeable impact on plants or animals.
Where does the magnetic field come from? At present there’s no absolute certainty, but the Parker Dynamo Model developed in the sixties seems to be provide the correct answer. The inner core of Earth is a sphere of solid iron that is roughly equal to the Moon in size and about as hot as the surface of the Sun. Surrounding it is the fluid outer core. The strong temperature gradient within the outer core leads to convective currents that contain large amounts of charged particles. According to the theory, the motion of these charges is what spawns the field. Recent numerical simulations on supercomputers have shown that this model is indeed able to reproduce the field in most aspects. It explains the intensity, the dipole structure, the wandering of the poles (including the observed drifting speed) and the pole reversals (including the observed time spans).
It is worth noting that the pole which lies in the geographic north, called the North Magnetic Pole, is actually a magnetic south pole. You might recall that we defined the north pole of a magnet as the pole which will point northwards when the magnet is allowed to turn freely. Since unlike poles attract, this means that there must be a magnetic south pole in the north (talk about confusing). By the same logic we can conclude that the Earth’s South Magnetic Pole is a magnetic north pole.
More interesting physics for non-physicists can be found here:
Physics! In Quantities and Examples
