Introducing General Relativity in High Schools: a proposal for a Teaching-Learning Module

A teaching-learning module, aimed at introducing basic concepts of general relativity at the high school level, is proposed. Emphasis is on conceptual rather than technical aspects, and only familiarity with simple calculus is required on the mathematical side. The starting point is a critical overview of the principles of Newtonian mechanics, in particular the role of fictitious forces, as well as of the limits of special relativity. Part of the module is devoted to the discussion and the reproduction of key thought or real experiments, for example experiments involving non-inertial frames, or the Einstein elevator.


Introduction: teaching general relativity in high schools
The two main scientific revolutions of the 20th century, relativity and quantum mechanics, which marked the transition from "classical" to "modern" physics, introduced a deep change in the world view of humankind, and triggered major advances in science and technology.This stimulated large changes in the educational policies which today, in many countries, require students attending the last year of high school to become familiar with the main concepts of modern physics.Undoubtedly, this is a challenge for both students and teachers.Of course, students face significant difficulties in grasping phenomena which are abstract, counterintuitive and cannot be directly experienced.Indeed, modern physics implies a tremendous conceptual change [1], hence students are required to revise the basic pillars of their knowledge: this often leads to misconceptions [2].In addition, most of the mathematical framework of modern physics is quite sophisticated and well beyond high school curricula.Therefore, a guiding principle in building up a teaching-learning sequence should be to focus on conceptual issues, starting from students' prior knowledge and to guide them so that they become aware of the limits of classical physics, the key issue being how to give a novel interpretation of these limits.From the teachers' point of view, the challenge is that they often lack sufficient knowledge of topics in modern physics, hence require pre-service and in-service formation.Thus, suitable material has to be developed with this task in mind as well.
While a large number of studies has been carried out in the last decade focusing on how to bring quantum mechanics and special relativity in high schools (for detailed accounts, see respectively [3] and [4] and references therein), the amount of work focusing on general relativity (GR) -which is included in the curricula in many countries as well -is still quite small, often concentrating on the undergraduate level, although in recent years much interest has arisen towards this problem, at the moment mostly with exploratory aims (see e.g.[5,6] and references therein).In fact, although not at the heart of many technological applications (the GPS being an exception), GR is currently enjoying a new golden age, with many important discoveries and developments that often put it in the headlines.Hence, many students are naturally quite curious about it.On the other hand, GR is commonly perceived as being the archetypical difficult and heavily mathematical theory, so students may get the impression that trying to understanding it is a hopeless task.
At least in Italy, school textbooks that include GR usually present it in a way that is quite akin to popular accounts, stressing very early on the concept of curved space-time as the characteristic feature of GR.This concept is then often explained by means of the famous rubber sheet analogy [7], according to which space-time behavior is visualized by deformations of a rubber sheet induced by masses placed on it.Besides the fact that behind the introduction of the idea of curved space-time rests quite a long series of conceptual steps, the rubber sheet model shows some critical issues.In particular, it describes gravity as an effect of space curvature alone, neglecting the crucial role played by time and the deep consequences of special relativity in general (such as mass-energy equivalence), which give rise to several novel physical phenomena when gravitation is included.This may give the wrong impression that GR is something like ordinary mechanics on a curved surface, or at best a kind of "geometrization" of Newton's theory, while it is in fact much more than that.
In this contribution, we outline our own exploratory proposal for teaching GR in high school, in the form of a teaching-learning module aimed both at last year high school students and their teachers, in which emphasis is put on conceptual aspects rather than on technical or geometrical ones.An analogous point of view has been sometimes advocated in the literature [8].Besides the above considerations, the red thread of the proposal is the recognition that, from the conceptual point of view, GR is much less revolutionary than quantum mechanics, and it is in a sense a natural extension of classical mechanics and special relativity [8].Furthermore, its physical basis can be clearly stated without resorting to the allegedly fearsome mathematics necessary for the full formulation (which is of course not our aim), i.e. pseudo-Riemannian geometry.Important inspiration to us came from the rich literature on the history and philosophy of GR.In fact, the inclusion of historical, philosophical and Nature of Science (NoS) aspects in physics education has been repeatedly advocated by researchers, also in the context of GR [5,6].Our track partially retraces the early conceptual development of GR, which took place from 1907 on [9,10,11], while keeping in mind that the basic concepts and principles of GR can be stated without resorting to the geometric point of view (interestingly, the views of Einstein himself were much less geometrical than commonly thought, as argued in [12]).This point of view is also shared by several modern approaches to GR, as e.g.[13].Finally, we drew on our experience in teaching some relevant topics in classical physics and in special relativity.
Our design strategy and prerequisites for a fruitful understanding of the material contained in the module are presented in Section 2, while the core of our module is outlined in the following Sections 3, 4 and 5.This involves a critical discussion of some aspects of classical mechanics and of special relativity, which is the basis for the discussion of Einstein's equivalence principle and its consequences.The module then includes a qualitative discussion of the main classic tests of GR, discussed in Section 6.In Section 7 we list a few additional topics which we plan to add in the future, some of which are characterized by the possibility of performing simple, but quantitative calculations.Finally, our conclusions end the paper.

Foundations and prerequisites
As a starting point for the development of our proposal, let us clearly state our main research questions: first, to select those concepts in GR that can be reasonably well understood by high school students; second, to identify physical effects of GR that can be satisfactorily addressed; third, to find simple computations that can be performed without resorting to complex maths and physics tools.
Typically, when students undertake the transition from classical physics to modern physics, they still retain some basic difficulties with the former.These include the lack of a proper understanding of the crucial role played by observers in different reference frames, of the concepts of inertial and non-inertial frames [14,15], as well as those of inertial and gravitational mass.These ideas are crucial for a correct understanding of the equivalence principle and its main consequences.We therefore start with a careful critical review of some aspects of the Newtonian laws, of special relativity, and of their limits.GR is then developed as a completion of these theories [8].
The core of our track is a broad discussion of the Einstein equivalence principle and of its consequences, e.g the gravitational mass-energy equivalence, the gravitational redshift and time dilation.To develop some intuition about the workings of GR, instead of employing the curved space picture, we build on the following fundamental physical features of GR, for which plausibility arguments come from the attempt of reconciling gravity with the principles of special relativity: the effects of gravity on time; the existence of gravitational analogues of magnetic forces (called "gravitomagnetic" forces) and of electromagnetic waves (i.e.gravitational waves); gravity of energy and gravity of gravity, i.e. the fact that a gravitational field is a source for itself.Then we employ these features to give a qualitative discussion of the other classic tests of GR, i.e., the bending of light rays and the perihelion shift.
Students tackling the module should of course have completed the school curriculum in classical physics.In particular, a previous exposure to Newtonian gravity, to the concept of the energy of the electromagnetic field and of an electromagnetic wave and the Doppler effect.Special relativity is of course a prerequisite, including a discussion of the magnetic field as a consequence of relativity, of the magnetic field generated by a uniformly moving point charge and its relation with the Coulomb field, and of the classical radius of the electron.If the lack of a proper understanding of any of these topics is detected, it is certainly possible to integrate them in the module where appropriate.

Part 1: critical discussion of classical mechanics
As stated, our first step is a critical discussion of the principles of Newtonian dynamics, in particular the important role of inertial frames, of Galilean relativity and of inertial forces in non-inertial frames (the cases of uniform acceleration and uniform rotation are sufficient).Ideally, part of this discussion should have already been undertaken when discussing special relativity.This part can be enriched by many experiments and simulations, many of which have been proposed in the physics education literature (see e.g.[16]).

Non-inertial frames and inertial forces
The basic test to understand whether we are in a non-inertial reference frame is to check the validity of Newton's laws.A classic example involves an observer sitting on an accelerating (or braking, or turning) bus, who throws a ball up in the air, and notices that its trajectory is not a parabola with vertical axis as expected.From the point of view of an observer standing on the street, instead, the ball follows a parabolic trajectory neatly satisfying Newton's law.
As another example, we observe that, from our point of view, the Sun is moving around the Earth.We all take for granted that in fact it is the Earth that orbits the Sun.But it is perfectly legitimate to describe the motion of the Sun by using a reference frame which is rigidly bound to Earth.Such a reference frame is of course non-inertial.First of all, we know that the Sun gravitationally attracts the Earth, but we do not see the Earth accelerating (we would see it in the reference frame of the Sun, where the Earth is seen orbiting).On the other hand, the Sun apparently moves on an orbit which cannot be explained on the basis of the gravitational attraction of the Earth.This means that Newton's laws are not valid in this reference frame.
In a non-inertial frame, it is possible to restore the validity of Newton's laws by postulating the existence of additional forces, such as the one that deviates the ball on the bus, or the one that keeps the Sun "in orbit" around the Earth.In other words, it is as if there is some extra force acting.To see this, we need an apparently trivial mathematical step, namely we rewrite Newton's law F = ma as F − ma = 0. To a mathematician, this is just elementary algebra.But to a physicist, this is a very deep conceptual change.It means that we take F f = −ma as an additional force acting on the body.Such a force does not have any physical origin, and it does not obey Newton's third law, hence it is called a fictitious, or inertial, force.This is most readily understood if we consider again our bus, with a ball sitting on the floor.If the bus undergoes any acceleration, the passengers will see the ball accelerating, while for an external observer the ball tends to moving according to Newton's first law.The origin of the extra force needed by the passengers of the bus to explain the motion of the ball is just the inertia of the ball, i.e. its tendency to maintain its state of motion.
At this point, what is for us the most interesting property of inertial forces must be pointed out: they are always proportional to the inertial mass of the object.This means that the latter cancels out in the equations, and thus any object in the same situation undergoes the same acceleration.

Galilei's equivalence principle
There are many different versions of the equivalence principle [17], but for our purposes a simplified classification is sufficient.We consider a "Galilei version" (universality of free fall), a "Newtonian version" (the universality of free fall is a consequence of the universal proportionality of inertial and gravitational mass), an "Einstein version" (extension to all non-gravitational physics taking special relativity into account).The aim of this part of the track is to critically analyze the first two versions, which are equivalent in the framework of classical mechanics, where they are sometimes dubbed "weak equivalence principle", and to emphasize a key physical consequence, i.e., that the gravitational forces, like the inertial ones, produce motions that are independent of the mass of the object, and in particular a uniform gravitational field is indistinguishable from a uniform acceleration of the reference frame.Also, if a laboratory which is freely falling in a gravitational field is considered, no mechanical experiment can detect the gravitational field (this is nicely shown by the astronauts in orbit).The crucial observation is the mentioned fact that in classical mechanics inertial and gravitational mass coincide.It was already known to Newton that the quantity called quantitas materiae was involved both in his second law of motion, where it is a measure of the inertia of an object, and in his law of universal gravitation, where it measures the force exerted by an object on other bodies and how an object reacts to the gravitational force of other bodies.The latter in principle express different properties, and can be called active and passive gravitational mass, respectively, but it is immediate to see that, as a consequence of Newton's third law, they have to coincide, hence one simply speaks of gravitational mass.This coincidence underlies Galilei's universality of free fall, which holds in the absence of air resistance (indeed, Galilei's understanding of the role of friction was crucial for his findings).There is a simple explanation of the phenomenon, according to Newton: a heavier body is subjected to a higher force, which increases its acceleration, but it has also a higher inertia, which decreases acceleration.These two effects are tuned so that they cancel each other out.In formulas, when we consider a body subject to the gravity acceleration of the Earth, we have where m i and m g are the inertial and the gravitational mass of the object, respectively.If m i = m g , as it appears to be the case for any object, one gets a = g.This result has many consequences.For instance, in the absence of friction an object sliding down a ramp has an acceleration which depends only on the inclination angle (e.g. it does not depend on the mass of the sliding object).Likewise, the period of a simple gravity pendulum depends only on the length of the rod but is completely independent of the mass.In such situations masses always cancel, but it is important that students appreciate that on one side there is the inertial mass while on the other side the gravitational mass appears, hence this cancellation is not merely an algebraic operation, but the consequence of a very deep fact of Nature.
A useful complement, at this stage, can be the explanation of the famous Eőtvős experiment, first performed in 1885, which attempts to measure the difference between the inertial and gravitational mass of a body by means of a torsion balance.

Part 2: critical remarks on special relativity
Immediately after Einstein had published his first papers on special relativity in 1905, some shortcomings became evident.In fact, they refer to problems which had been acknowledged even before 1905, but were not addressed by special relativity, which were: the incompatibility of Newton's theory of gravity with special relativity; the extension of the principle of relativity from inertial to at least some class of accelerated motion; finally, the breakdown of the universality of free fall, since in general, in special relativity, the vertical acceleration of an object depends on its horizontal velocity.The latter feature can be appreciated by the following thought experiment [9].An observer on a moving train, and a second observer on the platform, simultaneously drop a stone, the relevant question being whether the two stones hit the ground simultaneously.According to special relativity the answer is frame-dependent, hence in general negative, since simultaneity is a relative concept.A critical discussion of the above conceptual issues is in order for a better understanding of GR.Once more our strategy is to build on students' previous knowledge.
At the time, the most concerning issue to physicists was the fact that Newtonian gravitation assumes that gravitational influence is instantaneous, which is obviously in contradiction with special relativity.In fact, this was considered an absurdity already by Newton himself, and by many later scientists.In the 19th century, the possibility that Newton's law is in fact only a static limit of a more general theory of gravity (as Coulomb's law is the static limit of Maxwell's theory) was investigated by many physicists.After special relativity, the need for such a theory became of course much more compelling, and several physicists struggled to find a theory of gravity fully consistent with special relativity ( [11], voll.[3][4].For what concerns us, these ideas make plausible the existence of magnetic-like gravitational forces and of a gravitational analogue of electromagnetic waves.Indeed, we are talking about two important features of full GR, which as a matter of fact bears such important qualitative analogies with electromagnetism.One may indeed argue that, as in electromagnetism the magnetic force can be seen as an effect of special relativity [18], something similar has to happen if gravity has to be made relativistic, e.g.moving masses must interact by a kind of "gravito-magnetic" force.While the analogy is not perfect (for example because the fact that mass, unlike the electric charge, is not Lorentz-invariant), it is still very valuable for getting qualitative and quantitative insight in GR, especially in situations where the gravitational field is not very strong, as is typical in the solar system [19].
A special relativistic theory of gravity, however, would not address the remaining issues.Such a theory indeed would not extend the relativity principle, nor predict universality of free fall.But Einstein's ideas were very different.He wanted to retain the universality of free fall, and he put Galilei's equivalence principle at the basis of his attempts to extend Newton's theory.He tackled all three conceptual problems at once, and the result of his efforts was a new, extended equivalence principle, which was the basis of a theory of gravity which reduces to special relativity only locally, and was characterized by a principle of relativity under a large class of motion.The last feature is the origin of the name of the theory, even if it is not correct to state, as is usually done, that this class includes every possible motion [10].

Part 3: Einstein's equivalence principle and its consequences
Rather than discarding Galilei's equivalence principle, Einstein made it the pivotal idea to extend the principle of relativity.As he did with Galilei's relativity principle in SR, Einstein promoted the equivalence principle from being a (perhaps accidental) feature of mechanics to a fundamental principle underlying all physical phenomena [20].This allowed to extend the relativity principle at least to motions with constant acceleration.Also, he had to incorporate the basic teachings of SR.Einstein's reasoning can be described with the famous elevator Gedankenexperiment, and his equivalence principle can in fact be stated as the requirement that the non-gravitational laws of physics reduce to those of special relativity in freely falling systems of reference (provided we are in a region of space small enough that inhomogeneities of the gravitational field are negligible).In particular, the equivalence of inertial mass and energy, combined with the equivalence of inertial and gravitational mass, implies the equivalence of energy with gravitational mass or, in other words, that also energy must have a gravitational mass.This can be seen with a simple and beautiful quantitative thought experiment, discussed below.The latter fact has the deep consequence that any physical entity which has an energy is a source of gravity and feels its effects.This is true even for objects which have zero rest mass (e.g.light), and also for the gravitational field itself.In particular, light must feel the effects of gravity.Also, gravity itself is a source of gravity (i.e.gravity "weights"), hence a gravitational field is stronger than what may be naively inferred by its material source.This is an aspect in which gravity is crucially different from electromagnetism: in fact an electromagnetic field carries no electric charge in itself, so it does not generate an additional field.

The gravitational mass of energy
The fact that energy must have a gravitational mass, equal to its inertial mass, as a consequence of the Einstein equivalence principle, can be seen through the following Gedankenexperiment, first proposed by Einstein in 1911 [21].Consider two reference systems K and K ′ , one of which, K, is at rest in a uniform gravitational field, with gravity acceleration g oriented towards the negative z direction, while the other, K ′ , is far from any source of gravity, and accelerating towards the positive z ′ axis with acceleration g.According to the weak equivalence principle, these systems are equivalent if one performs only mechanical experiments.According to Einstein, they must be equivalent for all physical phenomena, including in particular electromagnetism.Suppose now that we have two points O 1 and O 2 , which are situated along the z−axis of K at a distance h from each other.At O 2 , the gravitational potential will be higher by gh with respect to O 1 .We may set the potential at O 1 to zero, so that the potential at the position of O 2 is φ = gh.Then, O 2 emits some amount of energy E 2 , e.g. in the form of an electromagnetic wave, towards O 1 .The same situation can be considered in the system K ′ , between the points O ′ 1 and O ′ 2 .We consider also a third system K 0 free of acceleration (hence, inertial) and at rest with respect to K ′ at the time where the radiation is emitted.This third system is the one from which we make our observations.From the set-up, we see that the radiation arrives at O 1 after a time h/c.At that time, O ′ 1 will be moving with a velocity v = gh/c with respect to K 0 , hence by the laws of special relativity the radiation arriving at it will have energy By the Einstein equivalence principle, we expect that the same will be true in the system K, where now gh may be substituted by the gravitational potential φ at O 2 .Then we can write: This equation tells us that the radiation arriving at O 1 will have an energy which is higher by the potential energy that a mass E 2 /c 2 situated in O 2 has in the gravitational field.This means that an energy E must itself have a potential energy due to gravity, corresponding to a gravitational mass E/c 2 , equal to the inertial mass associated with it.

Gravitational redshift and gravitational time dilation
Let us now consider a light beam which is emitted by a source on the floor of our lab, and moves towards the ceiling (the height h of the lab is assumed to be not too large, so we can consider the gravitational field as uniform in all the lab).By the equivalence principle, the effect of this uniform gravitational field is equivalent to that of a uniform acceleration of the same magnitude in the opposite direction.In that case, as the light is emitted, the floor and the ceiling are moving with an accelerated motion in the same direction as the light, hence the frequency of the light will be Doppler-shifted by the following amount (we assume that the lab always moves with a speed much less than the speed of light, so we do not need the relativistic Doppler formula): where in the second equality we have used the fact that when the light ray reaches the ceiling, the floor has been moving with an acceleration g for a time h/c.If we again view the situation as occurring in a lab sitting in a uniform gravitational field, the quantity gh is nothing but the difference δφ in gravitational potential between the ceiling and the ground, hence Thus, by the equivalence principle, we conclude that if light is emitted from the floor towards the ceiling in a uniform gravitational field, it will be red-shifted by an amount which is directly proportional to the variation in gravitational potential between the source and the receiver.This effect is very small and has been observed for the first time only in 1960 by Pound and Rebka [22], and constitutes one of the so-called three classic tests of GR.The above effect is essentially the same as the gravitational time dilation.To see this, we may consider the following situation [22].A light source emits flashes of light towards the ceiling, say one per second.Since the ceiling is meanwhile accelerating, the distance to be covered by the light pulses increases, hence they will be observed with a lower rate by the receiver.An observer standing by the receiver can thus argue that a clock standing by the source ticks slower than her clock.By the Einstein equivalence principle, a clock at a lower gravitational potential ticks more slowly.This effect can be measured directly with two initially synchronized atomic clocks, put at different heights, i.e. at different values of the gravitational potentials [22].The effect is crucial in the working of the GPS, being in fact larger than the corresponding special relativistic time dilation due to the motion of the satellite.A discussion of the GPS, which is the most widespread technological application of GR, can be included at this point.

Gravitational light bending and Mercury's perihelion shift
Once the core of our track has been developed, a qualitative discussion of the other two classic tests of GR, i.e., light ray bending by gravitational fields and the perihelion shift of orbits, can be given.Let us start with light ray bending.Consider a light source emitting light in a direction initially parallel to the ground of an elevator.When the elevator accelerates, an observer in it will see the light ray as curved (parabolic).Hence, we may say that inertial forces can curve light rays.By the equivalence principle (and by the above considerations on the gravitational mass of light) we expect that the same can be done by a gravitational field.In the elevator case, if the latter is put in a gravitational field, light rays fall like the flow of water leaking out of a bucket.This effect is at the basis of another classic test of Einstein's theory, namely the deflection of light rays by the Sun, which was first predicted in [21] and then verified by Eddington in 1919 [22] by observing the positions of stars close to the Sun during a total solar eclipse.The effect is known in general as gravitational lensing (discussed for the first time in general again by Einstein [23]), and is now plainly evident in the majestic pictures of far galaxy clusters taken by space telescopes.It is interesting to notice that, if one makes the hypothesis that light is made up by massive point particles (this is just Newton's corpuscular theory of light), this effect is present in Newtonian gravity as well, and it was already considered by Cavendish in 1784 [24].It can in fact be checked [24] that if one computes the deflection angle of a light ray that grazes the Sun, the result is the same as that obtained by the equivalence principle [21].The computation is a bit too advanced for high school pupils, but it can be proposed to teachers.Notice that the effect is not present in the context of classical electromagnetism, where light is made of electromagnetic waves, which are unaffected by gravity, while it is a necessary consequence of Einstein's equivalence principle regardless of the nature of light.But there is an important point to stress.As Einstein later showed [25], when this effect is computed in full GR, the result is twice that predicted from the equivalence principle (or from Newtonian mechanics with light corpuscles).This doubling is usually explained as a being a consequence of the curvature of space [22], which is relevant in the case of the non-uniform gravitational field produced by a localized body such as the Sun.However, there is an alternative qualitative explanation involving concepts which have been discussed earlier.As argued by Feynman [13], indeed, the doubling can be seen as due to magnetic-like gravitational forces.In fact, it can be easily shown in electromagnetism that the magnetic force exerted by a charge moving with speed v is v 2 /c 2 times the electrostatic force exerted by the same charge hence, in the limit where v = c, the two forces have equal magnitude.Analogously, gravitomagnetic forces act on something moving with the speed of light, such as light itself, with the same intensity as electriclike gravitational forces.Hence, when passing close to the Sun, a light ray will feel a force that is twice that predicted by Newtonian gravity alone, which explains the doubled deflection angle.
The remaining classic test is the precession of the perihelion of the orbit of Mercury (which was observed already in the 19th century by Le Verrier and Newcomb [22]).In fact, this was the only observation which, at the beginning of the 20th century, could not be explained by Newtonian theory.Its explanation was the first triumph of Einstein's GR [25].Also in this case we may use the previous concepts to give a qualitative explanation.Although the gravitational fields in the Solar System are very weak, the modifications induced by GR on the Newtonian theory have small but in principle measurable effects on the planetary orbits.These effects can be computed by successive approximations.Such computations are obviously complex and outside our scope, but a flavor of the physical explanation can be given anyway.As emphasized, GR modifies Newtonian theory because a gravitational field has a weight itself.In fact, there are also other effects, the first due to the special relativistic modifications of planetary dynamics [26], the second due to the magnetic-like forces (both are very small due to the very small orbital velocities in the solar system, compared with the speed of light).Other effects include the not perfect sphericity of the Sun.However, the first effect is by far the largest one.Such effects are largest for the case of Mercury, because of its closeness to the Sun and the speed with which it orbits it, hence it is for the case of Mercury that they were first discovered.
As we have seen, a gravitational field, being something which has an energy, must weight, i.e. it must generate a gravitational field of its own.Of course also this "secondary" gravitational field will generate a "tertiary" gravitational field and so on.For ordinary celestial bodies like the Earth, but also the Sun, the energy contained in the "primary" gravitational field is much smaller than the energy contained in the celestial body itself, hence the "secondary" field will just be a small correction to the "primary" field, and so on.Like in Zeno's paradox [27], the result of the addition of all these smaller and smaller contributions is finite but in practice, in such cases, only the "secondary" field will be in some way detectable.This additional field is then the main cause of the precession of the perihelion of the orbits of planets.We know in fact that according to Kepler's laws, the orbits of planets are elliptical, with the Sun in one of the foci of the ellipse.In particular, the orbits of the planets are closed curves.It can be shown that this is happens precisely because the gravitational fields decreases with the square of the distance from the Sun.Instead, when the above mentioned correction is taken into account (even if only the "secondary field"), the corrected gravitational field does not decrease any more like the square of the distance.If the orbit of the planet is circular, the additional field only shows up in a slightly larger attraction, hence a Newtonian physicist would simply state that the Sun has a mass which is slightly larger than the real one.But if the orbit is not circular, and the planet varies its distance from the Sun, the deviation from the inverse square behavior will show up in making the orbit not closed.In fact, the orbit is still approximately elliptic, but its axis (in particular its perihelion) rotates itself around the Sun.The result is that, for each orbit, the perihelion of Mercury is shifted by 43 seconds of arc, in full agreement with observations.More recently, the effect was measured also for farther planets [22].In more modern times, this effect has been measured for much stronger gravitational fields, where it is much larger, such as binary neutron stars [22].

Further developments
What we considered up to now gives, in our view, already a satisfactory, if qualitative, idea of the workings of Einstein's equivalence principle and of GR, which is sufficient for considering the most important technological application of the theory, the GPS, and also to understand something about the marvellous recent astrophysical discoveries involving neutron stars, black holes, gravitational waves and gravitational lensing.Nevertheless, if time permits, there is the possibility of adding more topics to the track.As stated above, this part is currently in a very preliminary state, hence we will limit ourselves to a very quick overview, referring to future work for details and results.First, although our track does not emphasize the curved space picture of GR, a connection with it is certainly desirable, also in view of the nice possibility of making contact with the mathematical topic of non Euclidean geometries.A useful starting point to discuss this may be Einstein's famous rotating disc analogy [10], which is used to argue that in a gravitational field, space has to be curved.In fact, these effects can be combined with the gravitational red-shift in the (quite counter-intuitive) statement that it is space-time, not space alone, that is curved.A further step is a qualitative discussion of the Einstein equations and how they relate the space-time curvature to the matter and energy present in space-time.A nice semi-quantitative discussion can be found in [28], and it would be nice to find a way of making it accessible to high school students (while it is accessible to teachers of course).
Finally, we have identified in the literature a few relevant physical applications of GR where some quantitative calculations, within grasp of students, can be performed.An example is the computation of the power radiated by gravitational waves by dimensional analysis [29].Another is a discussion of black holes starting with the standard calculation of the Schwarzschild radius using Newtonian gravity and the concept of escape velocity.This should be supplemented by a discussion of black holes as objects made up only by gravitational binding energy (thus taking to the extreme the idea of the weight of gravity), which parallels the classic idea of the classic radius of the electron.Some more discussion of black holes can involve the Paczyński-Wiita potential [30], which is commonly used in astrophysical applications.A further nice topic is the statement of the Hawking area theorem, which can be applied to the computation of the largest possible radiated energy in a black hole merger [31].We are currently aiming at enlarging the list, also by original ideas.

Conclusions
We have presented a proposal for a teaching-learning sequence for introducing the basics of GR to last year high school students, which can also be used for pre-service and in-service teacher formation.The module strongly relies on conceptual aspects over geometric ones, and builds on the principles of classical dynamics and special relativity and their limits, viewed as the seeds to construct a new, more general theory.It includes some very simple calculations and thought experiments, but it can be easily extended to include real experiments (this is of course possible only in part 1) and simulations.At the moment, only part of the first half of the sequence, which in fact mostly concerns Einstein's equivalence principle, has been tested in class, albeit with encouraging partial qualitative results.We plan to start a full trial with quantitative tests soon, and to use the results to revise the teaching-learning sequence according to standard guidelines [5,32].Also, we are currently working for extending the module as briefly discussed in Section 7.