Teaching Relativity in a College Physics course

[Herman Trivilino]

Of course a vertical distance measured by one observer may be seen as a horizontal distance by another observer, they just have to use coordinates systems rotated relative to one another!

That won't work because the two directions are not equivalent. In one direction a particle accelerates, in the other it doesn't. You'll never a get a plumb bob, for example, to hang in a horizontal direction. If you were to use different units to measure distances in these two directions it would emphasize this difference, but that's not necessarily a good reason for doing so.

Likewise, you'll never get a clock to measure distance and you'll never get a meter stick to measure time. If you were to use different units to measure intervals in these two quantities it would emphasize this difference, but that's not necessarily a good reason for doing so.

 

[nrqed]

That won't work because the two directions are not equivalent. In one direction a particle accelerates, in the other it doesn't. You'll never a get a plumb bob, for example, to hang in a horizontal direction. If you were to use different units to measure distances in these two directions it would emphasize this difference, but that's not necessarily a good reason for doing so.

Likewise, you'll never get a clock to measure distance and you'll never get a meter stick to measure time. If you were to use different units to measure intervals in these two quantities it would emphasize this difference, but that's not necessarily a good reason for doing so.

Ah you are not working in an isolated system. I was talking about an isolated system, you are assuming we are near the surface of the Earth, in which case of course there is a distinction. The post I was responding to was saying that in Euclidian geometry, a horizontal distance in one frame could never be seen as a vertical distance in another frame, by horizontal and vertical I though he/she meant x vs y, I did not realize what was meant was in a non isolated system.

But, in special relativity, a clock *can* in principle be used to measure distances and a clock can be used to measure time, because of the invariance of the speed of light. I agree with this and I thought this was everyone's point behind saying that "time and space are on equal footing". My only objection is that the value c=1 is just a random choice and has no deep physical meaning.

 

[nrqed]

No, this is just plain wrong. .

So you said that I am wrong when I say that there is a choice of units made when wet set c=1 (my point was that we really mean c = 1 unit of distance/1 unit of time).

Well, I must then have to relearn everything about physics. I thought that someone could use two synchronized clock at rest in an inertial frame, send a light signal from one to the other and use that to measure the speed of light. I thought that one would define the speed of light as being the distance between the two clocks divided by the time interval measured by the clocks. Since, I thought, the choice of units used for the distance is arbitrary and since the choice of unit of time is, I believed, arbitrary, I thought that the speed of light obtained could take any possible value (well, larger than zero), depending on the choice of units of distance and of time.

What I learned here is that either it is physically inconsistent to define the speed of light as distance over time (because the only correct value of c is 1!) OR one is not allowed to pick distance or time units as one desires.

I will have to go back to basics!

 

[Herman Trivilino]

Ah you are not working in an isolated system. I was talking about an isolated system, you are assuming we are near the surface of the Earth, in which case of course there is a distinction. The post I was responding to was saying that in Euclidian geometry, a horizontal distance in one frame could never be seen as a vertical distance in another frame, by horizontal and vertical I though he/she meant x vs y, I did not realize what was meant was in a non isolated system.

There is no way to define vertical and horizontal in such an isolated system. You need the presence of gravity in one direction (vertical) and an absence of gravity in the other (horizontal) to define the directions.

Consider any vertical plane and draw in it a diagonal line segment. That line has a vertical component and a horizontal component that lie in that plane. You can increase one at the expense of decreasing the other, and vice-versa, by rotating the line segment in that plane. Moreover you can show that the vertical and horizontal components can be combined in such a way that the length of the line segment remains the same regardless of how it's oriented in that plane. You can see how much harder it would be to demonstrate this invariance of length if distances in the vertical direction were measured in units that differ from the units used to measure distance in the horizontal direction.

Choosing different units for lengths in each direction wouldn't be "wrong" but it would "obscure" the geometrical relationship.

 

[Orodruin]

I thought that one would define the speed of light as being the distance between the two clocks divided by the time interval measured by the clocks.

This is wrong. The meter is defined as 1/299792458 s (which happens to be a very suitable unit for measuring the spatial size of many things), you never measure the speed of light.

What I learned here is that either it is physically inconsistent to define the speed of light as distance over time (because the only correct value of c is 1!) OR one is not allowed to pick distance or time units as one desires.

We never said it is wrong, it is just obscuring the geometry. By introducing and using ct everywhere you are essentially doing the same, you are just calling your time variable with a longer and more cumbersome name. And of course you can pick any units you like, just as you can chose to measure one spatial direction in feet and the other in light years. It just obstructs the symmetry and make the coordinate transformations awkward.

Then explain something to me: someone comes along and chooses to pick units such that c=2. You will say that this person is wrong, I guess. But based on what?? Based on the fact that then you do not find the equations pretty enough?

This is the same thing as selecting a non-normalised coordinate system in a Euclidean space. Of course you can do that, but introductory courses generally only deal with Minkowski coordinates, which are the equivalent of using a normalised Cartesian coordinate system in the Euclidean space.

This same kind of confusion arises when talking about mass and energy 'equivalence'. Mass and energy are very different concepts. Mass may be related to energy content of a body, but mass and energy are very different concepts. with very different units and very different physical attributes.

This is again a confusion. Mass and energy are very similar concepts, with mass simply being a measure of an objects rest energy. In relativity, it is very easy to see that the rest energy is also the inertia of the object in its rest frame. There is no other concept of mass in SR, the inertia of a moving object is a quantity that depends on the direction of acceleration.

 

[Andrew Mason]

This is wrong. The meter is defined as 1/299792458 s (which happens to be a very suitable unit for measuring the spatial size of many things), you never measure the speed of light.

1 m = ct where c = speed of light in a vacuum and t = (1/299792458) seconds. Since a second is defined as 9,192,631,770 periods of the radiation (one period = T) emitted by caesium-133 in the transition between the two hyperfine levels of the atom's ground state, with the atom at Earth sea-level and at rest (ie. at 0K), the definition of 1 m is not time, but is really a distance:

1 m = cT(9,192,631,770/299,792,458) = 30.66331898849837 times the wavelength of this radiation from the caesium-133 atom.

We never said it is wrong, it is just obscuring the geometry. By introducing and using ct everywhere you are essentially doing the same, you are just calling your time variable with a longer and more cumbersome name. And of course you can pick any units you like, just as you can chose to measure one spatial direction in feet and the other in light years. It just obstructs the symmetry and make the coordinate transformations awkward.

One could argue that by making the geometry pretty you are obscuring the physics. Every inertial observer has his own coordinate system where time and space are distinct physical quantities. Time is measured by clocks and distance is measured by the separation of the end points of sticks. The distinction between time and space is always maintained for each inertial observer.

It is just that they are not absolute: different inertial observers will disagree on time and space measurements between events because they disagree on simultanaeity of events.

This is again a confusion. Mass and energy are very similar concepts, with mass simply being a measure of an objects rest energy. In relativity, it is very easy to see that the rest energy is also the inertia of the object in its rest frame.

$m = E/c^2$ means that $m \propto E$. By setting units for c = 1 that still does not make m and E the same physical phenomena. That fact that mass or inertia can be converted into energy does not make them equal. Otherwise there would be no meaning to the "conversion".

There is no other concept of mass in SR, the inertia of a moving object is a quantity that depends on the direction of acceleration

?? This again is confusing. There is the concept of inertia and the concept of rest mass. Rest mass is constant for all observers. Inertia, or the ratio of $\vec{F}/\vec{a}$ is another concept of "mass", although it is generally frowned upon now because of the confusion with 'rest mass'.

AM

 

[Herman Trivilino]

I thought that one would define the speed of light as being the distance between the two clocks divided by the time interval measured by the clocks.

Actually, what that process does is define the distance as being one meter when the time interval is $\frac{1}{299\ 792\ 458}\ \mathrm{s}$. You don't measure the speed of light, you calibrate your meter stick. But this seems a tangential issue.

 

[Herman Trivilino]

$m = E/c^2$ means that $m \propto E$. By setting units for c = 1 that still does not make m and E the same physical phenomena.

Setting $c=1$ doesn't make them equivalent, nature does that, or at least as far as we can tell it does. Although it's the rest energy $E_o$ not the total energy $E$ that's proportional to $m$. More precisely, the thing that we measure and call $m$ is not distinguishable from the thing that we measure and call $E_o$. Measuring them in the same units is a matter of preference.

That fact that mass or inertia can be converted into energy does not make them equal. Otherwise there would be no meaning to the "conversion".
?? This again is confusing.

If you look at any of the processes that are called "conversions" what you see is that before the process what one is choosing to call $m$ is afterwards called $E_o$, or vice-versa.

There is the concept of inertia and the concept of rest mass. Rest mass is constant for all observers. Inertia, or the ratio of $\vec{F}/\vec{a}$ is another concept of "mass", although it is generally frowned upon now because of the confusion with 'rest mass'.

Calling the ratio $F/a$ the mass $m$ is valid only in the Newtonian approximation. The more general relation between $\vec{F}$, $m$, and $\vec{a}$ is $$\vec{a}=\frac{\vec{F}-(\vec{F}\cdot\vec{\beta})\vec{\beta}}{\gamma m}.$$ Note that the ratio $F/a$ is equal to $\gamma m$ only when $\vec{F}\cdot\vec{\beta}$ is zero. Identifying $m$ as the inertia has a meaning in the Newtonian approximation that I don't know how to generalize. (Certainly it's not $\gamma m$). Because of that, and the opinion that the concept of inertia clouds the true meaning of Newton's First Law, I try to avoid the term inertia when teaching Newtonian physics. And therefore when teaching relativity, too.

 

[Orodruin]

One could argue that by making the geometry pretty you are obscuring the physics.

I would argue that it is exactly the other way around. You are obstructing the actual physics by selecting a system of units that in relativity is obscure and not very natural. Physics do not depend on your choice of units.

By setting units for c = 1 that still does not make m and E the same physical phenomena.

Physics does not care if it makes sense to you or not. The fact of the matter is that the only mass you talk of in relativity is the rest energy, which in the non-relativistic limit is just the inertia of the object, there is no other mass concept.

There is the concept of inertia and the concept of rest mass.

But the point is that this is the only concept of mass and it is directly the same as the rest energy. Why do you want to introduce two quantities to describe the same thing? This is superfluous and confusing.

 

[nrqed]

I have given some more thoughts about how to explain my point about the choice c=1 and I will post a final comment...final, I promise;-)

The special rôle of the speed of light links space and time, I agree completely. But that does not assign a specific value to c, this is completely arbitrary.

There are three ways to think about this:A) One fixes (completely arbitrarily) a unit of distance.
and one fixes (completely arbitrarily) a unit of time. Then the speed of light is fixed (by a measurement) to some value in those units. The value of c can therefore take any value possible, it simply depends on the choice of units of time and distance, which is not fixed by physics!ORB) One fixes (completely arbitrarily) a unit of distance and one fixes (completely arbitrarily!) a value for the speed of light, for example c = 3 units of distance/unit of time. Then, using a light beam this fixes completely what the unit of time is: One unit of time is by definition how long it takes for light to cover 3 units of distance. Nothing forces one to use c=1 here, it is absolutely arbitrary to pick one value of c over another one. ORC)One fixes (completely arbitrarily) a unit own time and one fixes (completely arbitrarily!) a value for the speed of light, for example c = 3 units of distance/unit of time. Then, using a light beam this fixes completely what the unit of distance is: One unit of distance is by definition the distance traveled in one third of the unit of time.
My point is that in both B and C, the value chosen for c here is completely arbitrary! Nothing forces it to be 1. What I mean by this is that if two persons use different values of c (and the corresponding two different sets founts), they will agree on any physical result obtained by calculations.

I don’t know how to explain my point more clearly (sorry!) so I will definitely zip it :-)

 

[Herman Trivilino]

The special rôle of the speed of light links space and time, I agree completely. But that does not assign a specific value to c, this is completely arbitrary.

Right. I'm not sure how it's come about in this thread that anyone would think anyone is saying otherwise. The point under discussion, I thought, was whether or not it was of pedagogical value. Not whether it was right or wrong to choose $c=1$.

 

[vela]

A) One fixes (completely arbitrarily) a unit of distance.
and one fixes (completely arbitrarily) a unit of time. Then the speed of light is fixed (by a measurement) to some value in those units. The value of c can therefore take any value possible, it simply depends on the choice of units of time and distance, which is not fixed by physics!

OR

B) One fixes (completely arbitrarily) a unit of distance and one fixes (completely arbitrarily!) a value for the speed of light, for example c = 3 units of distance/unit of time. Then, using a light beam this fixes completely what the unit of time is: One unit of time is by definition how long it takes for light to cover 3 units of distance. Nothing forces one to use c=1 here, it is absolutely arbitrary to pick one value of c over another one.

OR

C)One fixes (completely arbitrarily) a unit own time and one fixes (completely arbitrarily!) a value for the speed of light, for example c = 3 units of distance/unit of time. Then, using a light beam this fixes completely what the unit of distance is: One unit of distance is by definition the distance traveled in one third of the unit of time.

My point is that in both B and C, the value chosen for c here is completely arbitrary! Nothing forces it to be 1. What I mean by this is that if two persons use different values of c (and the corresponding two different sets founts), they will agree on any physical result obtained by calculations.

I think the point you're missing is that it's completely unnecessary as far as the physics goes to have separate units for time and space. You can, of course, but at the cost of introducing a conversion factor, namely $c$, into the equations.

 

[FactChecker]

I wouldn't go into the Twins Paradox. It will only confuse things. Better to use that time on fundamentals, the experiments that forced the theory, and the relativity of simultaneity as the motivation of all that followed. (Maybe a brief mention of General Relativity at the end?)

 

[nrqed]

I think the point you're missing is that it's completely unnecessary as far as the physics goes to have separate units for time and space. You can, of course, but at the cost of introducing a conversion factor, namely $c$, into the equations.

I think you are missing my point. Yes, one can define

I think the point you're missing is that it's completely unnecessary as far as the physics goes to have separate units for time and space. You can, of course, but at the cost of introducing a conversion factor, namely $c$, into the equations.

A distance is still fundamentally different from a time. We can call a distance a "light-second" and conveniently drop the "light" and just call it a second but it is still a light-second. A distance cannot be measured with only a clock (and I do mean only a clock...not a clock plus a ruler or a clock plus a mirror etc etc).
So the speed of light can be given as one light-second per second, but to say that c = 1 (pure 1) is wrong. That was my whole point.

 

[nrqed]

Right. I'm not sure how it's come about in this thread that anyone would think anyone is saying otherwise. The point under discussion, I thought, was whether or not it was of pedagogical value. Not whether it was right or wrong to choose $c=1$.

Well, my point has always been that it is just a special choice of units that allows c=1, and that even with those units, we should say c=1unit off distance / 1 unit of time. We could equally well set c=2 or any other value. So pedagogically, as I said I think it is not helpful at all to students who are already struggling with understanding time dilation, etc. Good luck!

 

[vela]

A distance is still fundamentally different from a time.

I think this is the root of the disagreement. Sure, we perceive the dimensions differently, so for practical reasons, we measure them differently. From a mathematical and geometrical point of view, however, they're not so different, and that was the primary revelation of special relativity!

(Maybe a brief mention of General Relativity at the end?)

Another advantage of the geometrical approach to SR is that it sets you up for GR.

 

[nrqed]

Right. I'm not sure how it's come about in this thread that anyone would think anyone is saying otherwise. The point under discussion, I thought, was whether or not it was of pedagogical value. Not whether it was right or wrong to choose $c=1$.

Like I have mentioned before, you are right that it is useful to work with light-years or light-seconds or light-minutes. But instead of setting c=1 and calling these "years, seconds,minutes" and risking confusion when it can easily be avoided, what I do, and I think it is much simpler and pedagogically better (in my opinion), is to tell the student, to use 1 light-second = 1 second x c (which makes sense to them, it is the distance traveled by light in one second), 1 ly = 1 year x c and so on. Plugging these expressions in the equations works out nicely because the factors of c cancel out where they must,leaving time in years (or second or whatever) and leaving speeds in fractions of c. It works very nicely and I can focus on the physics of time dilation, etc.

 

[Herman Trivilino]

A distance cannot be measured with only a clock (and I do mean only a clock...not a clock plus a ruler or a clock plus a mirror etc etc).

Yes, it can. If I have a clock and know what time you're going to send me a light signal, I can use that clock to determine the distance between you and me.

 

[Herman Trivilino]

I wouldn't go into the Twins Paradox. It will only confuse things. Better to use that time on fundamentals, the experiments that forced the theory, and the relativity of simultaneity as the motivation of all that followed. (Maybe a brief mention of General Relativity at the end?)

I appreciate that. What would your reasons be for doing it? This is likely the last physics course these students will ever take. The fundamentals are covered before this lesson is presented, and the twin paradox is already a part of that textbook reading assignment. I would like for them to see this very interesting application of the theory, but perhaps that's just my personal preference.

 

[Andrew Mason]

I would argue that it is exactly the other way around. You are obstructing the actual physics by selecting a system of units that in relativity is obscure and not very natural. Physics do not depend on your choice of units.

I think this is the root of the disagreement. Sure, we perceive the dimensions differently, so for practical reasons, we measure them differently. From a mathematical and geometrical point of view, however, they're not so different, and that was the primary revelation of special relativity!

There is a reason that we use different units for space and time. Using the same units obscures the physical difference. A time interval in a given inertial reference frame is always a time interval to all observers in that reference frame. That same time interval may appear as an interval of space and time in another reference frame. But all observers make the distinction between time and space. If time and space were equal, it would be just a matter of changing the axes to make a time interval a space interval. But that cannot be done.

Physics does not care if it makes sense to you or not. The fact of the matter is that the only mass you talk of in relativity is the rest energy, which in the non-relativistic limit is just the inertia of the object, there is no other mass concept.

But the point is that this is the only concept of mass and it is directly the same as the rest energy. Why do you want to introduce two quantities to describe the same thing? This is superfluous and confusing.

I agree that matter contains potential energy and that the potential energy a body contains is proportional to its rest mass. I agree that a compressed spring has slightly more inertia or rest mass than an uncompressed spring, the difference in mass being $\Delta m = \frac{1}{2}kx^2/c^2$.

It is then a matter of semantics whether one wishes to say that the potential energy contained in a body and its mass are related by E = mc^2, or by choosing units of E and m such that E/m = 1, that potential energy and mass are equal. The problem with the latter is that the expression E = m is true ONLY if c=1 whereas E = mc^2 is always true.

Particle physicists often express the rest mass of particles in units of energy, such as meV, but it is always understood that this is just a short-hand. The units of mass are meV/c²

AM

 

[Herman Trivilino]

The problem with the latter is that the expression E = m is true ONLY if c=1 whereas E = mc^2 is always true.

There are systems of units where you write $E_o=kmc^2$ and then say $E_o=mc^2$ only for systems where k=1.

 

[nrqed]

This is wrong. The meter is defined as 1/299792458 s (which happens to be a very suitable unit for measuring the spatial size of many things), you never measure the speed of light.

I am not sure what you are saying... After choosing a system of units of time and distance, of course someone can then measure the speed of light.

The meter is *now* defined this way, but it has not always been defined that way, obviously. And before that definition was chosen, the speed of light *had* to be measured!

What you are talking about is exactly what I said in my earlier post: one can pick an arbitrary unit of time, then pick an arbitrary value for the speed of light and these two will then define a unique measure of distance. They defined the speed of light to be 2999792458 m/s just so that it would then give a unit of distance close to the previous one but they might as well have defined the speed of light to be 7.3891 zoobie per second, which would have defined the zoobie. Or they could have defined the speed to be 1 light-second per second, which would have defined the light-second. The number 1 is prettier than 7.3891 or 2999792 558 but that's just pure aesthetic.

We never said it is wrong, it is just obscuring the geometry. By introducing and using ct everywhere you are essentially doing the same, you are just calling your time variable with a longer and more cumbersome name. And of course you can pick any units you like, just as you can chose to measure one spatial direction in feet and the other in light years. It just obstructs the symmetry and make the coordinate transformations awkward.

Well you did say that it was wrong that c=1 really means c = 1 unit of distance over one unit of time. By the way, it seems like you feel that using different units for time and space is misleading because it introduces an artificial distinction between the two. Despite the two being connected in a deep way, they *are* different and this is not only because of a choice of units. We can move in all directions along the spatial coordinates but we cannot move in both direction in time (at least in SR). The time piece has a different sign than the three spatial pieces in the spacetime interval. We can stay at fixed x,y,z coordinates but not at a fixed time coordinate. Causality is related to whether or not $(\Delta t)^2$ is larger or smaller or equal to $(\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2$. In all these cases, time is clearly distinguished from the space coordinates. So insisting that because time and space are basically the same thing we should use the same units could also be considered as obscuring the different status of the two. And this difference is physical, it is not due to me using light-years for distance units and years for time units, say.But at least we agree on one thing: it is a matter of choice, not of physics, whether one uses c=1 or not. That's what I have tried to point out since the beginning. Weather something is more or less obscure is a matter of opinion, not of physics.

So now we can go back to the very beginning. What makes something more obscure to someone may make things less obscure to someone else, and vice versa. I agree that for advanced students, it can be interesting to be pointe out units in which c=1. But my point was that I think that for beginning students, it just obscures things by adding one layer of "newness" on top of all the subtleties of the Lorentz transformations. I think that they have enough in dealing with relativity of simultaneity, time dilation, etc etc without having to deal with a new set of units which *for them* will be new and confusing (they will not become comfortable with giving distances in seconds and giving speeds in pure fractions of c within a week or two of classes, while at the same time dealing with all the physics of the Lorentz transformations.

That's all I was trying to say.

 

[Orodruin]

My point is that in both B and C, the value chosen for c here is completely arbitrary! Nothing forces it to be 1. What I mean by this is that if two persons use different values of c (and the corresponding two different sets founts), they will agree on any physical result obtained by calculations.

But as I said before, this viewpoint is nothing new and the equivalent of selecting an orthonormal set of basis vectors in a Euclidean space. Sure, you can select a basis where the length of one of the vectors is 53, but why would you do that to yourself?

 

[Orodruin]

A time interval in a given inertial reference frame is always a time interval to all observers in that reference frame.

A timelike interval is always timelike. It is not the same as saying it is a time interval!

But all observers make the distinction between time and space. If time and space were equal, it would be just a matter of changing the axes to make a time interval a space interval. But that cannot be done.

But this is a property of the metric, not a property of the coordinates! By stating it otherwise you are missing fundamental relativistic insights.

I agree that matter contains potential energy and that the potential energy a body contains is proportional to its rest mass.

Seriously, it is not a potential energy! It is the rest energy and there is absolutely no other concept of mass in relativity and it exactly corresponds to the inertia in the classical limit. What more do you want from it to call it mass?

Particle physicists often express the rest mass of particles in units of energy, such as meV, but it is always understood that this is just a short-hand. The units of mass are meV/c2

I am a particle physicist ... And if you look into papers such as the announcements of the Higgs discovery, you will find that it quotes masses in GeV and not GeV/c^2. Of course, this is not wrong, it is simply using the fact that there is no other concept of mass than the rest energy and there is no point in inteoducing an arbitrary conversion constant.

I am not sure what you are saying... After choosing a system of units of time and distance, of course someone can then measure the speed of light.

So I challenge you to do this in our current definition of the units. You will not succeed. If anything, you will simply make a calibration measurement of your ruler.

You cannot measure something you have defined.

The meter is *now* defined this way, but it has not always been defined that way, obviously. And before that definition was chosen, the speed of light *had* to be measured!

Of course, but that was not the point. The point was that you cannot measure it in the current definition.

Despite the two being connected in a deep way, they *are* different and this is not only because of a choice of units. We can move in all directions along the spatial coordinates but we cannot move in both direction in time (at least in SR). The time piece has a different sign than the three spatial pieces in the spacetime interval. We can stay at fixed x,y,z coordinates but not at a fixed time coordinate.

Again, this is a property of the metric, not of the units you have used and this property also becomes clearer in units where c=1.

 

[Andrew Mason]

A timelike interval is always timelike. It is not the same as saying it is a time interval!

I was speaking of a time interval, ie. the difference in the time coordinates of two events. I was just saying that all observers in the same reference frame will agree on the magnitude and sign of the time interval. Observers in other reference frames will all agree that there is a time interval of some magnitude, and with the same sign, if the interval is time-like. The fact that intervals may be time-like or space-like means that time and space are distinct physical quantities.

But this is a property of the metric, not a property of the coordinates! By stating it otherwise you are missing fundamental relativistic insights.

I don't think I stated otherwise.

Seriously, it is not a potential energy! It is the rest energy and there is absolutely no other concept of mass in relativity and it exactly corresponds to the inertia in the classical limit. What more do you want from it to call it mass?

The coulomb potential energy between protons in a uranium nucleus contributes to the rest mass of the uranium nucleus. Do we stop calling it potential energy because it contributes to rest mass? When an atom absorbs a photon of energy $E = h\nu$ and an electron in the atom moves to a higher energy level, the rest mass of the atom increases by $\Delta m_0 = E/c^2$. Is it wrong to think of the atom in the higher energy state as having more potential energy?

AM

 

[Orodruin]

I was speaking of a time interval, ie. the difference in the time coordinates of two events. I was just saying that all observers in the same reference frame will agree on the magnitude and sign of the time interval.

Ok, I misread your statement as a statement about different frames. Being a statement about the same frame makes it moot instead. Of course people using the same frame will get the same time difference, they will get the same space-time difference to, which is a vector in Minkowski space. There is nothing strange going on here and again the physics becomes more transparent if you give all vector components the same dimension.

I don't think I stated otherwise.

But your statement that it is based on the dimensions of the coordinates seem to imply it. The distinction between a time-like and a space-like vector comes only from the metric tensor. It has absolutely nothing to do with the units used.

The coulomb potential energy between protons in a uranium nucleus contributes to the rest mass of the uranium nucleus. Do we stop calling it potential energy because it contributes to rest mass? When an atom absorbs a photon of energy E=hν and an electron in the atom moves to a higher energy level, the rest mass of the atom increases by Δm0=E/c2. Is it wrong to think of the atom in the higher energy state as having more potential energy?

No, what is wrong is to say that all mass is potential energy, which is what I understood from your statement. However, this just underlines my point. The only mass concept that is relevant (and a scalar) in special relativity is the rest energy. If you want to argue against this you need to provide a counter example where this is not the case.

 

[Andrew Mason]

Ok, I misread your statement as a statement about different frames. Being a statement about the same frame makes it moot instead. Of course people using the same frame will get the same time difference, they will get the same space-time difference to, which is a vector in Minkowski space. There is nothing strange going on here and again the physics becomes more transparent if you give all vector components the same dimension.

You can give the time dimension units of distance by multiplying by c. You can't do it by simply declaring the time dimension to be the same as a spatial dimension.

But your statement that it is based on the dimensions of the coordinates seem to imply it. The distinction between a time-like and a space-like vector comes only from the metric tensor. It has absolutely nothing to do with the units used.

Yes. But the three spatial dimensions are arbitrary (so long as they are mutually orthogonal) whereas the time dimension is not. In an inertial reference frame one can describe the space time coordinates of an event relative to the origin as (ct, x, y, z) where x, y and z can be anything so long as x²+y²+z² is always the same. But you cannot do that with t.

No, what is wrong is to say that all mass is potential energy, which is what I understood from your statement. However, this just underlines my point. The only mass concept that is relevant (and a scalar) in special relativity is the rest energy. If you want to argue against this you need to provide a counter example where this is not the case.

I will agree that speaking about mass as a form of potential energy is out of vogue. But there is nothing intrinsically wrong with saying that rest energy is a form of potential energy. The only way a particle's rest energy can be completely released is through annihilation with its anti-particle, and those are generally in short supply in our neck of the universe, so it is not really practical to treat rest energy as potential energy. But even Einstein in his 1907 essay on relativity spoke about rest energy as potential energy.

AM

 

[Herman Trivilino]

But the three spatial dimensions are arbitrary (so long as they are mutually orthogonal) whereas the time dimension is not.

The time dimension is orthogonal to the three spatial dimensions. By scaling them you form an orthonormal set of basis vectors. To do that each of the four basis vectors must be mutually orthogonal unit vectors.

In an inertial reference frame one can describe the space time coordinates of an event relative to the origin as (ct, x, y, z) where x, y and z can be anything so long as x²+y²+z² is always the same. But you cannot do that with t.

That's what he means about the metric being different. It's not $x^2+y^2+z^2$, it's $(ct)^2-x^2-y^2-z^2$ (timelike). As a result of the metric being different you can still exchange x, y, and z with each other, but you cannot exchange ct with x, y, or z.

None of that has anything to do with making the basis vectors orthonormal.

I will agree that speaking about mass as a form of potential energy is out of vogue. But there is nothing intrinsically wrong with saying that rest energy is a form of potential energy. The only way a particle's rest energy can be completely released is through annihilation with its anti-particle, and those are generally in short supply in our neck of the universe, so it is not really practical to treat rest energy as potential energy. But even Einstein in his 1907 essay on relativity spoke about rest energy as potential energy.

In the example you gave of an atom absorbing a photon look at the total energy of the system in its center of momentum frame before the absorption. In the low speed approximation it's $h \nu + Mc^2$; after the absorption it's $(M+m)c^2$, where $M$ is the mass of the atom before absorption and $(M+m)$ is the mass afterwards. (I think the approximation I'm using may also require that $Mc^2>>h \nu$, but either way it's satisfied!) The contribution to the rest energy made by the photon is $h \nu$, and it's equal to $mc^2$.

If you divide each term by $c^2$ you get the mass. That is, the mass of the system before the absorption is $h \nu/c^2 + M$ and afterwards it's $(M+m)$. The contribution to the mass made by the photon is $h \nu/c^2$, and it's equal to $m$.

So what is being converted here? The thing called the energy contribution, $h \nu$, is being converted into the thing called the mass contribution $m$, or is the thing called the mass contribution $h \nu/c^2$ being converted into the energy contribution $mc^2$? Now, this is all semantics of course. Rest energy and mass are two names for the same thing. The total mass of the system before the absorption equals the total mass afterwards. And the total rest energy of the system before the absorption equals the total rest energy afterwards.

The pedagogical point being made here is that the factor of $c^2$ is considered by some to obscure the physics. But regardless of anyone's opinion on that matter, rest energy and mass are equivalent. Note that to see the equivalence you must look at composite bodies, that is, systems that consist of entities.

Also note that you can do the same thing with energy and momentum vector components that you can with time and space vector components, respectively. When you do that the factor of $c$ gets in the way because, and I hope I'm saying this right, the basis vectors aren't orthonormal.

 

[Orodruin]

You can't do it by simply declaring the time dimension to be the same as a spatial dimension.

Yes you can. I am sorry, but stating otherwise is simply untrue. You don't have to, but you can.

Yes. But the three spatial dimensions are arbitrary (so long as they are mutually orthogonal) whereas the time dimension is not.

Again untrue. The time direction is quite arbitrary, just perform a Lorentz transformation. What stops you from completely exchanging time and space is the geometry (pseudo-Riemannian metric), not the units. The time direction is taken to be orthogonal to the spatial directions in standard Minkowski coordinates. Now, just as you don't have to use units where c=1, you of course do not need to use orthogonal coordinates. I can introduce light-cone coordinates where two coordinates are orthogonal to themselves.

But there is nothing intrinsically wrong with saying that rest energy is a form of potential energy.

The thing that is wrong with it is that you would be using a terminology which the rest of the world does not use. I can decide to go around calling kinetic energy potential energy (after all, it can be used to create new particles too!) but nobody will understand me.

 

[Andrew Mason]

Yes you can. I am sorry, but stating otherwise is simply untrue. You don't have to, but you can.

If you can it is certainly not going to be obvious and likely not understandable to a student who is learning this in a one week introduction. Apart from the units not working, it is conceptually not clear. In either case, it makes it more difficult to convey the physics. The physics is not that complicated: it all derives from the fact that c is the same in all inertial frames which directly results in the concept of simultaneity being relative rather than absolute. If you just wave your hands and say 'look at this neat geometry - it explains everything' I think you will not succeed in conveying anything except confusion.

Again untrue. The time direction is quite arbitrary, just perform a Lorentz transformation. What stops you from completely exchanging time and space is the geometry (pseudo-Riemannian metric), not the units. The time direction is taken to be orthogonal to the spatial directions in standard Minkowski coordinates. Now, just as you don't have to use units where c=1, you of course do not need to use orthogonal coordinates. I can introduce light-cone coordinates where two coordinates are orthogonal to themselves.

Since you cannot exchange time and space because of the geometry (I would say it is because of the physics), then it is confusing to say that time and space are the same. You seem to be saying that they are the same but different. BTW, I would be interested in seeing you use 'light-cone coordinates' for a reference frame that do not refer to any other reference frame.

The thing that is wrong with it is that you would be using a terminology which the rest of the world does not use. I can decide to go around calling kinetic energy potential energy (after all, it can be used to create new particles too!) but nobody will understand me.

I am not so sure about that. We speak about nuclear potential energy and electric potential energy in an atom which arises by virtue of the configuration of the parts of the atom's nucleus and electrons. So far as I can tell, these potential energies contribute significantly if not entirely to the rest energy of an atom. We speak about the energy contained within a system by virtue of its configuration (such as a compressed spring, compressed air, stretched elastics etc) as potential energy.

It seems to me that there is not a material difference between the use of potential energy in those contexts and the rest energy of an atom.

AM