Post by 1dave on Oct 12, 2020 8:48:44 GMT -7
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Now it’s time to turn to mass.
Fig. 1: For an object moving according to Einstein’s relativity, the relation between energy, momentum (times the universal speed limit c) and mass (times c squared) is analogous to the Pythagorean relationship between the hypotenuse of a right-angle triangle and its two sides. The velocity of the particle, divided by c, is the ratio of the length of the horizontal line to the length of the hypotenuse; this is also the sine of the angle marked alpha.
Unfortunately, there is a lot of confusion about mass, because shortly after Einstein’s work on relativity there were two notions of mass that coexisted for a time. Only one (the one which Einstein himself settled on, and which was sometimes called “invariant mass” or “rest mass” to distinguish it from the now-archaic term “relativistic mass”) is still used in particle physics today. I’ve explained this more carefully in a separate article.
The definition of mass m that I will be using throughout this website is the one that has a particular relation between energy and momentum. For an object that is moving on its own (that is, not interacting in any significant way with other objects), Einstein proposed (and countless experiments confirm) that its energy E, momentum p and mass m satisfy a simple Pythagorean relation
E2 = (p c)2 + (mc2)2 (equation #1)
[Remember Pythagoras, who said that a right-angle triangle with sides of length A and B and hypotenuse of length C has to satisfy C2 = A2 + B2? It’s the same type of relation: see Figure 1.] Here c is a constant speed that, as we will see in a moment, is the universal speed limit. We’ll also see in a minute why it is called “the speed of light.”
Fig. 2: As an object is made to move faster and faster, E and pc become larger and larger, and more and more similar in length, while m c-squared remains the same.
According to Einstein’s equations, the velocity of an object, divided by the speed limit c, is just the ratio of p c to E,
v/c = (p c) / E (equation #2)
i.e. the ratio of the length of the horizontal side of our triangle to the length of its hypotenuse. (This ratio is also equal to the sine of the angle α shown in Figure 1.) Wow! There it is, folks. Since the sides of a right-angle triangle are always shorter than its hypotenuse (i.e. the sine of any angle is always less than or equal to 1), no object’s velocity can be faster than c, the universal speed limit. As the velocity of the object increases (for fixed mass), both p and E become very large (Figure 2), but E is always bigger than p c, and so v is always less than c!
Fig. 3: Lessons from extreme triangles. (Left) A massive object that is not moving has E = m c-squared. (Right) A massless particle must have E = pc, and therefore v = c.
Fig. 4: Any right-angle triangle for which the vertical side has finite length always has a hypotenuse longer than its horizontal side, no matter how long the hypotenuse becomes; similarly, a massive object’s velocity can never exceed c, no matter how large its energy E is. (Here “>>” means “much greater than”.)
Meanwhile, for a massive particle, as shown in Figure 4, no matter how big you make the momentum and the energy, E is always a little bit bigger than p c, and so the velocity is always less than c.
Massless particles must travel at the speed limit; massive particles must travel below it.
At the other extreme, consider a slow massive object, moving very slowly compared to the speed of light, as in the case of a car. Then since its velocity v is much less than c, its momentum p times c is much less than E, and (as you can see from Figure 5) E is just a little bit bigger than m c2. Thus a slow object’s motion-energy E – mc2 is much smaller than its mass-energy mc2, while a fast object’s motion-energy can be made arbitrarily large, as we saw in Figure 4.
One tricky point I should mention: momentum is not just a number, it is a “vector”. That is, it has a size and a direction; it points in the direction the particle is moving. When I write “p” I’m just referring to its size. In many cases we have to keep track of the direction of the momentum too, but we don’t have to in equation #1 that relates momentum to energy and mass.
Fig. 5: An object moving much slower than the speed of light has momentum (times c) much less than its energy, and has just a little bit of motion-energy compared to its mass-energy. (Here “>>” and “<<” mean “much greater than” and “much less than”.)
A final tricky point: I’ve used triangles and a bit of ordinary trigonometry because everyone knows them from high school. But experts-to-be should beware: the right way to understand Einstein’s equations is using hyperbolic trigonometry, which most laypeople never encounter, but which is essential for understanding the structure of the theory, and makes important details such as how two velocities add, why lengths contract, etc., far more transparent. Non-experts can safely ignore this, though someday I might write a page explaining it, as it involves lovely mathematics.
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profmattstrassler.com/articles-and-posts/particle-physics-basics/mass-energy-matter-etc/mass-and-energy/
Matt Strassler [March 27, 2012]
A number of puzzling features of the world — including a number that my readers have asked about in comments — have everything to do with the nature of mass and energy (and also momentum.) We’ve all heard these words and many of us have a vague idea of what they mean. Of course the notions of “mass” and “energy” exist in English (and in other languages) with multiple definitions. Unfortunately, none of these definitions matches what physicists mean. So you have to leave those other definitions at the door and try to deal with these notions in the precise way that physicists do. Otherwise you’ll end up very confused.
Even before I start, there’s the potential for confusion. In bringing up “mass and energy”, I do not mean to call to your mind a different pairing of words: “matter and energy”, which you will often hear people refer to as though they are opposites, or partners, or mutually exclusive categories, or something meaningful in some other way as a two-some. Well, they’re not. Matter and energy don’t belong to the same categories; putting them together is like referring to apples and orangutans, or to heaven and earthworms, or to birds and beach balls. [Here’s an article going into more detail about why matter and energy are a false dichotomy.] Matter (no matter how you define it — and there are different definitions in different contexts) is a class of objects that you will find in the universe, while mass and energy are not objects; they are properties that every object in the universe can have. Mass and energy are in deep interplay, and they deserve to be discussed together.
Matt Strassler [March 27, 2012]
A number of puzzling features of the world — including a number that my readers have asked about in comments — have everything to do with the nature of mass and energy (and also momentum.) We’ve all heard these words and many of us have a vague idea of what they mean. Of course the notions of “mass” and “energy” exist in English (and in other languages) with multiple definitions. Unfortunately, none of these definitions matches what physicists mean. So you have to leave those other definitions at the door and try to deal with these notions in the precise way that physicists do. Otherwise you’ll end up very confused.
Even before I start, there’s the potential for confusion. In bringing up “mass and energy”, I do not mean to call to your mind a different pairing of words: “matter and energy”, which you will often hear people refer to as though they are opposites, or partners, or mutually exclusive categories, or something meaningful in some other way as a two-some. Well, they’re not. Matter and energy don’t belong to the same categories; putting them together is like referring to apples and orangutans, or to heaven and earthworms, or to birds and beach balls. [Here’s an article going into more detail about why matter and energy are a false dichotomy.] Matter (no matter how you define it — and there are different definitions in different contexts) is a class of objects that you will find in the universe, while mass and energy are not objects; they are properties that every object in the universe can have. Mass and energy are in deep interplay, and they deserve to be discussed together.
Part of what makes energy complicated to describe is that it can take many forms, not all of which are conceptually simple. Here are the three most commonly encountered:
First, energy can be locked away in an object’s mass; on this website I call this mass-energy (which is the famous E=mc2 energy associated with mass, and also called “rest-energy”, since it is the energy that an object has when it is at rest, i.e. not moving.)
Second, energy is associated with the motion of an object. On this website I call this motion-energy, whose technical name is “kinetic energy”; this kind of energy is rather intuitive, in that faster objects have more energy than slower ones, which is more or less what we would colloquially expect. Also, a heavier object has more motion-energy than a , confusing, and mostlight one, if the two are traveling at the same speed.
Third energy can be stored in the relationships among objects (and is typically called “potential energy”). It can be stored in a stretched spring, or in the water behind a dam, or in the gravitational interaction of the earth with the sun, or in the relationship among atoms in a molecule. In fact there are lots and lots of ways to store energy. This sounds very vague, but again, it’s just a failure of words; in every one of these cases, there are precise formulas for what the stored energy of system is, and there are clear and well-defined ways to measure it.
There’s one more thing about this third kind of energy, interaction-energy as I will call it, that is especially confusing at first. Unlike mass-energy and motion-energy, which are always greater than or equal to zero, interaction-energy can be either positive or negative (and often is negative in interesting situations). I’m not going to deal just yet with this fascinating and subtle issue. We’ll get back to it later. [In a previous version of this article I called this `relationship-energy’, but decided against this choice more recently.]
Energy is a very special quantity, of great importance in physics. The reason it is so essential [along with momentum, see below] is that it is “conserved” — read this as physics-language for preserved, or for maintained without change. What precisely does this mean?
First, energy can be locked away in an object’s mass; on this website I call this mass-energy (which is the famous E=mc2 energy associated with mass, and also called “rest-energy”, since it is the energy that an object has when it is at rest, i.e. not moving.)
Second, energy is associated with the motion of an object. On this website I call this motion-energy, whose technical name is “kinetic energy”; this kind of energy is rather intuitive, in that faster objects have more energy than slower ones, which is more or less what we would colloquially expect. Also, a heavier object has more motion-energy than a , confusing, and mostlight one, if the two are traveling at the same speed.
Third energy can be stored in the relationships among objects (and is typically called “potential energy”). It can be stored in a stretched spring, or in the water behind a dam, or in the gravitational interaction of the earth with the sun, or in the relationship among atoms in a molecule. In fact there are lots and lots of ways to store energy. This sounds very vague, but again, it’s just a failure of words; in every one of these cases, there are precise formulas for what the stored energy of system is, and there are clear and well-defined ways to measure it.
There’s one more thing about this third kind of energy, interaction-energy as I will call it, that is especially confusing at first. Unlike mass-energy and motion-energy, which are always greater than or equal to zero, interaction-energy can be either positive or negative (and often is negative in interesting situations). I’m not going to deal just yet with this fascinating and subtle issue. We’ll get back to it later. [In a previous version of this article I called this `relationship-energy’, but decided against this choice more recently.]
Energy is a very special quantity, of great importance in physics. The reason it is so essential [along with momentum, see below] is that it is “conserved” — read this as physics-language for preserved, or for maintained without change. What precisely does this mean?
Now it’s time to turn to mass.
Fig. 1: For an object moving according to Einstein’s relativity, the relation between energy, momentum (times the universal speed limit c) and mass (times c squared) is analogous to the Pythagorean relationship between the hypotenuse of a right-angle triangle and its two sides. The velocity of the particle, divided by c, is the ratio of the length of the horizontal line to the length of the hypotenuse; this is also the sine of the angle marked alpha.
Unfortunately, there is a lot of confusion about mass, because shortly after Einstein’s work on relativity there were two notions of mass that coexisted for a time. Only one (the one which Einstein himself settled on, and which was sometimes called “invariant mass” or “rest mass” to distinguish it from the now-archaic term “relativistic mass”) is still used in particle physics today. I’ve explained this more carefully in a separate article.
The definition of mass m that I will be using throughout this website is the one that has a particular relation between energy and momentum. For an object that is moving on its own (that is, not interacting in any significant way with other objects), Einstein proposed (and countless experiments confirm) that its energy E, momentum p and mass m satisfy a simple Pythagorean relation
E2 = (p c)2 + (mc2)2 (equation #1)
[Remember Pythagoras, who said that a right-angle triangle with sides of length A and B and hypotenuse of length C has to satisfy C2 = A2 + B2? It’s the same type of relation: see Figure 1.] Here c is a constant speed that, as we will see in a moment, is the universal speed limit. We’ll also see in a minute why it is called “the speed of light.”
Fig. 2: As an object is made to move faster and faster, E and pc become larger and larger, and more and more similar in length, while m c-squared remains the same.
According to Einstein’s equations, the velocity of an object, divided by the speed limit c, is just the ratio of p c to E,
v/c = (p c) / E (equation #2)
i.e. the ratio of the length of the horizontal side of our triangle to the length of its hypotenuse. (This ratio is also equal to the sine of the angle α shown in Figure 1.) Wow! There it is, folks. Since the sides of a right-angle triangle are always shorter than its hypotenuse (i.e. the sine of any angle is always less than or equal to 1), no object’s velocity can be faster than c, the universal speed limit. As the velocity of the object increases (for fixed mass), both p and E become very large (Figure 2), but E is always bigger than p c, and so v is always less than c!
Fig. 3: Lessons from extreme triangles. (Left) A massive object that is not moving has E = m c-squared. (Right) A massless particle must have E = pc, and therefore v = c.
Next, notice that if the object is not moving, so that its momentum p is zero, then the relation in equation #1 simplifies to
E2 = (mc2)2 , or in other words E=mc2 ,
Einstein’s famous relation that mass is associated with a fixed amount of energy (which is what I call mass-energy on this website) is just the statement that when the triangle becomes a vertical line, as in Figure 3 (left), its hypotenuse becomes the same length as its vertical side. But let me say that again, because it is so important: this relation E=mc2 does not mean that energy is always equal to mass times c2; only for an object that is not moving (and therefore has zero momentum) is this true.
Another interesting thing to note is that for a massless particle, the vertical side of the triangle is zero and the hypotenuse and horizontal side have the same length, as in Figure 3 (right). In such a case, E is inevitably equal to pc, which in turn means that v/c = 1, or in other words, v=c . Thus we see that a massless particle (such as a photon ) inevitably travels at the speed c. And so the speed of light is the same as the universal speed limit, c.
E2 = (mc2)2 , or in other words E=mc2 ,
Einstein’s famous relation that mass is associated with a fixed amount of energy (which is what I call mass-energy on this website) is just the statement that when the triangle becomes a vertical line, as in Figure 3 (left), its hypotenuse becomes the same length as its vertical side. But let me say that again, because it is so important: this relation E=mc2 does not mean that energy is always equal to mass times c2; only for an object that is not moving (and therefore has zero momentum) is this true.
Another interesting thing to note is that for a massless particle, the vertical side of the triangle is zero and the hypotenuse and horizontal side have the same length, as in Figure 3 (right). In such a case, E is inevitably equal to pc, which in turn means that v/c = 1, or in other words, v=c . Thus we see that a massless particle (such as a photon ) inevitably travels at the speed c. And so the speed of light is the same as the universal speed limit, c.
Fig. 4: Any right-angle triangle for which the vertical side has finite length always has a hypotenuse longer than its horizontal side, no matter how long the hypotenuse becomes; similarly, a massive object’s velocity can never exceed c, no matter how large its energy E is. (Here “>>” means “much greater than”.)
Meanwhile, for a massive particle, as shown in Figure 4, no matter how big you make the momentum and the energy, E is always a little bit bigger than p c, and so the velocity is always less than c.
Massless particles must travel at the speed limit; massive particles must travel below it.
At the other extreme, consider a slow massive object, moving very slowly compared to the speed of light, as in the case of a car. Then since its velocity v is much less than c, its momentum p times c is much less than E, and (as you can see from Figure 5) E is just a little bit bigger than m c2. Thus a slow object’s motion-energy E – mc2 is much smaller than its mass-energy mc2, while a fast object’s motion-energy can be made arbitrarily large, as we saw in Figure 4.
One tricky point I should mention: momentum is not just a number, it is a “vector”. That is, it has a size and a direction; it points in the direction the particle is moving. When I write “p” I’m just referring to its size. In many cases we have to keep track of the direction of the momentum too, but we don’t have to in equation #1 that relates momentum to energy and mass.
Fig. 5: An object moving much slower than the speed of light has momentum (times c) much less than its energy, and has just a little bit of motion-energy compared to its mass-energy. (Here “>>” and “<<” mean “much greater than” and “much less than”.)
A final tricky point: I’ve used triangles and a bit of ordinary trigonometry because everyone knows them from high school. But experts-to-be should beware: the right way to understand Einstein’s equations is using hyperbolic trigonometry, which most laypeople never encounter, but which is essential for understanding the structure of the theory, and makes important details such as how two velocities add, why lengths contract, etc., far more transparent. Non-experts can safely ignore this, though someday I might write a page explaining it, as it involves lovely mathematics.
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