Teaching Relativity in a College Physics course

[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

 

[Mister T]

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.

 

[Mister T]

$m = E/c^2$ 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 :-)

 

[Mister T]

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.

 

[Mister T]

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.

 

[Mister T]

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

 

[Mister T]

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

 

[Mister T]

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

 

[Mister T]

I am not so sure about that. We speak about nuclear potential energy, and electric potential energy. 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 that context and the rest energy of an atom.

The kinetic energy of electrons makes a positive contribution to the mass (rest energy) of an atom. Researchers are finding that this is making a significant contribution to the shielding provided by the inner electrons in the heaviest elements. There was an article on it recently in Physics Today.

 

[Orodruin]

Apart from the units not working, it is conceptually not clear.

Yes it is. It is very clear. I am sorry if you do not see this.

In either case, it makes it more difficult to convey the physics.

On the contrary, it makes the actual physics easier to convey as you do not have to worry about unit conversions. There is a reason we do not use metric units in one spatial direction and imperial in another.

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.

No, this is not the main physical point in relativity. Simultaneity is a convention, nothing else. Applying that convention gives different result in different Minkowski coordinates, but you could have chosen any other simultaneity convention as well. For example, in a FRW space-time, there is a natural simultaneity convention in terms of the comoving time. The main physics results are the geometry of space-time, the proper time being the pseudo-Riemannian length of a time-like world line, and the division of a space-time into the future, past, and elsewhere for a given event. Things such as length contraction and time dilation are all coordinate dependent statements that rely on an arbitrary definition of simultaneity.

I will agree that this is not the way we usually teach students, but the way we teach students may be wrong. I have seen what happens to students (and PF visitors!) struggling with understanding time dilation, length contraction, and other "classical" concepts in SR when they are confronted with the actual physics. They all want to base their understanding on this arbitrary definition of simultaneity and it can go terribly wrong.

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.

But you can! All you need to do is to introduce a curvilinear coordinate system where the basis varies continuously. Then you have coordinates, let us call them $\xi$ and $\zeta$ where $\xi$ may be time-like at one point and $\zeta$ at another (take polar coordinates on a 1+1 dimensional space-time - there is nothing stopping you from doing this). You seem to fixate on using a set of Minkowski coordinates, but again that is a special case and unless you can convince me that spherical coordinates are useless when dealing with three spatial dimensions, you will not be able to convince me that Minkowski coordinates hold any kind of special role (apart from the fact that the metric is always diag(1,-1,-1,-1)). In particular not if you insist on using a set of Minkowski coordinates which is not normalised.

Physics is not coordinates, physics is things which you can compute and then go to your laboratory and measure.

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.

Take a two-dimensional affine space and introduce the coordinates $\xi$ and $\zeta$. Introduce the metric $ds^2 = 2 d\xi d\zeta$. Done.

Of course you can easily show that this is equivalent to 1+1-dimensional Minkowski space, but that is not the point. The physics will work in exactly the same way!

I am not so sure about that. We speak about nuclear potential energy, and electric potential energy. 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 that context and the rest energy of an atom.

This might work until you get down to the level of elementary particles. The mass of the elementary particles is an intrinsic property and has nothing to do with an internal field configuration (it has to do with expanding the theory around a vacuum which does not respect gauge symmetry, but that is another matter).

 

[Andrew Mason]

The kinetic energy of electrons makes a positive contribution to the mass (rest energy) of an atom. Researchers are finding that this is making a significant contribution to the shielding provided by the inner electrons in the heaviest elements. There was an article on it recently in Physics Today.

Good point. But I expect that the electron kinetic energy is small in comparison to the coulomb potential energy of the protons in the nucleus, let alone the nuclear potential energy of the neutrons and protons.

In any event, it depends on how deep one looks. Here is an example where we think of kinetic energy as potential energy: When we say that a can of compressed air (ideal gas) has potential energy we are really referring to its (potential) ability to do PV work on its surroundings. And that ability to do work is due to the kinetic energy of the molecules inside the container.

AM

 

[Andrew Mason]

Yes it is. It is very clear. I am sorry if you do not see this.

Don't feel sorry for me. Feel sorry for the student in this one-week course who you are expecting to understand how time = space and c is dimensionless ! Saying it is clear doesn't make it any clearer.

Simultaneity is a convention, nothing else.

The definition of 'simultaneous' is hardly arbitrary and not really a convention. It is a perfectly understandable word in normal use that means 'occurring at the same time'. Relativity accepts that definition: two events are simultaneous to an inertial observer if the observer's measurements give the same time co-ordinate for each event.

Things such as length contraction and time dilation are all coordinate dependent statements that rely on an arbitrary definition of simultaneity.

I will agree that this is not the way we usually teach students, but the way we teach students may be wrong. I have seen what happens to students (and PF visitors!) struggling with understanding time dilation, length contraction, and other "classical" concepts in SR when they are confronted with the actual physics. They all want to base their understanding on this arbitrary definition of simultaneity and it can go terribly wrong.

But we implicitly use simultanaeity to measure lengths. It is hardly arbitrary. The length of a stick is the spatial separation between two simultaneous observations ie. the observations of the location of each end of the stick. That is a very simple thing to convey to a new student. So with c being absolute (i.e the speed of a light signal is measured to be the same in all inertial reference frames) it is easy to show that simultaneous events to one observer are not simultaneous to an inertial observer in an other reference frame, and that explains length contraction. Time dilation simply follows from absolute c and length contraction.

AM

 

[Orodruin]

Don't feel sorry for me. Feel sorry for the student in this one-week course who you are expecting to understand how time = space and c is dimensionless ! Saying it is clear doesn't make it any clearer.

I would say we have long since transcended talking about the one-week course. This discussion ensued from you claiming it to be unequivocally wrong to use the same units for time and space coordinates, which I find outright misleading.

The definition of 'simultaneous' is hardly arbitrary and not really a convention.

Yes it is. It comes down to an arbitrary choice of coordinates - or if you prefer calling a set of Minkowski coordinates a frame - an arbitrary choice of frame. There is a multitude of other possibilities of defining simultaneities as space-like foliations of space-time. In particular, the arbitrariness also becomes clearer in GR where it is not necessarily possible to define a global simultaneity.

Let us study the 1+1 dimensional FRW metric $ds^2 = c^2 dt^2 - a(t)^2 dx^2$ (I inserted the $c$ just for you). Would you agree that selecting comoving time $t$ is a good definition of something being simultaneous? A comoving observer is an observer for which $dx = 0$ and this observer will measure a proper time progressing at the same rate as the time-like coordinate $t$.

Relativity accepts that definition: two events are simultaneous to an inertial observer if the observer's measurements give the same time co-ordinate for each event.

This is just one of many different possible definitions of simultaneity. I will give you that it is the most common one in SR, but that does not make it the only one. The problem arises when you realize that $t$ is just a coordinate.

But we implicitly use simultanaeity to measure lengths.

This is how you define lengths. As such, it is intrinsically dependent on the definition of simultaneity - not the other way around.

and that explains length contraction.

But length contraction is a coordinate effect, as is time-dilation. It appears because you have defined simultaneity in a particular way - it is a result of the commonly used definition of simultaneity, not something which the definition of simultaneity explains. They are really not the fundamental thing in relativity, regardless of what introductory textbooks would have you believe. They are both an artefact of the coordinate systems used, which has led to many a student obsessing over the symmetry of time dilation when it really is nothing but an effect of rotating the coordinate axes. An effect which you have also in Euclidean space (see, e.g, my Insight on this subject).

 

[Andrew Mason]

I would say we have long since transcended talking about the one-week course. This discussion ensued from you claiming it to be unequivocally wrong to use the same units for time and space coordinates, which I find outright misleading.

I don't think I said it was wrong to use the same units for the time and space coordinates. If the time co-ordinate is ct, it is not a problem. ct is a distance. And since c is a constant, the ct coordinate is always proportional to time (as measured in that inertial reference frame). What I objected to was simply stating that time and space are the same physical phenomena.

Yes it is. It comes down to an arbitrary choice of coordinates - or if you prefer calling a set of Minkowski coordinates a frame - an arbitrary choice of frame. There is a multitude of other possibilities of defining simultaneities as space-like foliations of space-time. In particular, the arbitrariness also becomes clearer in GR where it is not necessarily possible to define a global simultaneity.

Let us study the 1+1 dimensional FRW metric $ds^2 = c^2 dt^2 - a(t)^2 dx^2$ (I inserted the $c$ just for you). Would you agree that selecting comoving time $t$ is a good definition of something being simultaneous? A comoving observer is an observer for which $dx = 0$ and this observer will measure a proper time progressing at the same rate as the time-like coordinate $t$.

You've lost me there. We are talking about Special Relativity - inertial reference frames.

This is how you define lengths. As such, it is intrinsically dependent on the definition of simultaneity - not the other way around.

How would you define length?

But length contraction is a coordinate effect, as is time-dilation. It appears because you have defined simultaneity in a particular way - it is a result of the commonly used definition of simultaneity, not something which the definition of simultaneity explains. They are really not the fundamental thing in relativity, regardless of what introductory textbooks would have you believe. They are both an artefact of the coordinate systems used, which has led to many a student obsessing over the symmetry of time dilation when it really is nothing but an effect of rotating the coordinate axes. An effect which you have also in Euclidean space (see, e.g, my Insight on this subject).

I think time dilation is a bit more than a coordinate effect, whatever you mean by that. The effect is real and readily observed. The direction of photons emitted from relativistic electrons in a synchrotron is a direct result of time dilation.

AM

 

[Orodruin]

You've lost me there. We are talking about Special Relativity - inertial reference frames.

But SR is not only about inertial frames! Of course we start by teaching it like that just as we do not start vector analysis in parabolic coordinates, but it is a trivial matter to do SR in a general coordinate system. This is not what separates SR from GR. A very common example of a curvilinear coordinate system in SR is Rindler coordinates.

Also, there is a point to my question about the 1+1-dimensional FRW space-time so I would like you to answer it.

What I objected to was simply stating that time and space are the same physical phenomena.

Of course they are not, but this is a geometrical effect and not a dimensional one. The fact remains that one person's pure time direction has space components in a different person's frame.

How would you define length?

The same way you do, the distance in the surface of simultaneity between the end points. But this length definition is going to depend on the simultaneity convention.

The effect is real and readily observed.

No, this is again just plain wrong. What is observed is that, e.g., muons from the upper atmosphere reach the ground. The reason for this is that their proper time between the creation and hit the ground events is short enough for them not to decay. This will be described differently in different coordinate systems and you may refer to it using whatever coordinates you select. But "time dilation" is the effect of the proper time of an observer being shorter than the difference in time coordinates. Obviously this is a coordinate dependent statement. Just the fact that the wording of the description in different frames is different should tell you that the description is coordinate dependent. The physical observable is the proper time and its value depends only on the world line and the geometry of space-time.

The direction of photons emitted from relativistic electrons in a synchrotron is a direct result of time dilation.

Time dilation may be used to describe it in the lab frame, but the effect itself is based on the geometry of space-time. You could go to any other coordinate system and the result will be the same.

 

[Andrew Mason]

But SR is not only about inertial frames! Of course we start by teaching it like that just as we do not start vector analysis in parabolic coordinates, but it is a trivial matter to do SR in a general coordinate system. This is not what separates SR from GR. A very common example of a curvilinear coordinate system in SR is Rindler coordinates.

Also, there is a point to my question about the 1+1-dimensional FRW space-time so I would like you to answer it.

As I said, you lost me there. Sounds like GR. I am restricting my comments to SR and inertial reference frames.

Of course they are not, but this is a geometrical effect and not a dimensional one. The fact remains that one person's pure time direction has space components in a different person's frame.

I am really not sure what that means. I am quite certain a student in a one-week course on SR would feel the same. The challenge is not to explain things in a way that appears elegant to a mathematician. The challenge is to make it into a readable colouring book.

It seems to me that the physics is what it is whether or not we apply a mathematical construct to the real world. Dimensions, on the other hand, are physical. For example, collisions of bodies occur or do not occur for dimensional reasons (time being one of those dimensions).

As a general comment, I don't think we disagree on essential points but we have, obviously, a different approach to teaching the subject. As I say, our universe is what it is. While there are various approaches one may take in fitting mathematical models to SR, so long as they are consistent and fit the evidence, they are all valid. The question is which one should be taught and when.

No, this is again just plain wrong. What is observed is that, e.g., muons from the upper atmosphere reach the ground. The reason for this is that their proper time between the creation and hit the ground events is short enough for them not to decay. This will be described differently in different coordinate systems and you may refer to it using whatever coordinates you select. But "time dilation" is the effect of the proper time of an observer being shorter than the difference in time coordinates. Obviously this is a coordinate dependent statement. Just the fact that the wording of the description in different frames is different should tell you that the description is coordinate dependent. The physical observable is the proper time and its value depends only on the world line and the geometry of space-time.

This is very much like the twin paradox: the muon ages less during its journey than the stationary observer on the earth, which is the reference frame in which the muon began and ended its journey. So it seems natural to use the coordinate system of the Earth to analyse what is happening. Such an event cannot happen if the muon's clock ran at the same rate as that of an inertial observer on the earth.

Time dilation may be used to describe it in the lab frame, but the effect itself is based on the geometry of space-time. You could go to any other coordinate system and the result will be the same.

The highly directional light from a synchrotron in the lab frame is omnidirectional in the rest frame of the electron that emits it. So I am not sure why the highly directional result would be the same in any other coordinate system.

AM

 

[nrqed]

No, this is again just plain wrong. What is observed is that, e.g., muons from the upper atmosphere reach the ground. The reason for this is that their proper time between the creation and hit the ground events is short enough for them not to decay. This will be described differently in different coordinate systems and you may refer to it using whatever coordinates you select. But "time dilation" is the effect of the proper time of an observer being shorter than the difference in time coordinates. Obviously this is a coordinate dependent statement. Just the fact that the wording of the description in different frames is different should tell you that the description is coordinate dependent. The physical observable is the proper time and its value depends only on the world line and the geometry of space-time..

You are saying that the lifetime of the muon in a frame where it is not at rest is not a physical observable?? How do you define "physical observable"? I define it as something that can be physically measured. I can certainly measured the time between the creation of a muon and its disintegration in a frame where it is not at rest. Why is it not a physical observable?? You are saying that only invariant quantities are observables but this is plain wrong.

 

[George Jones]

I guess it comes down to what one means by "observable".

Suppose the muon is created at event A and dies at event B. A clock carried by the muon is the only inertial clock that experiences both of these events. If I want to measure the time, then I have to say "Well, event A is simultaneous with event A' on my worldline, event B is simultaneous with event B' on my worldline, and my inertial clock (assuming that I am am inertial observer) measures the time difference between A' and B' to be $\Delta t$."

But, since A and B are not on my worldine and thus not experienced by me, I have to use a simultaneity convention to do this.