A demonstration of Einstein s equivalence of gravity and acceleration

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1 IOP PUBLISHING Eur. J. Phys. 29 (2008) EUROPEAN JOURNAL OF PHYSICS doi: / /29/2/004 A demonstration of Einstein s equivalence of gravity and acceleration Ronald Newburgh Extension School, Harvard University, Cambridge, MA 02138, USA rgnew@verizon.net Received 13 August 2007, in final form 28 November 2007 Published 8 January 2008 Online at stacks.iop.org/ejp/29/209 Abstract In 1907, Einstein described a Gedankenexperiment in which he showed that free fall in a gravitational field is indistinguishable from a body at rest in an elevator accelerated upwards in zero gravity. This paper describes an apparatus, which is simple to make and simple to operate, that acts as an observable footnote to Einstein s example. It also includes an examination of acceleration-dependent forces as well as a discussion of those frames of reference in which Newton s laws of classical mechanics are not valid. Introduction The starting point of this paper is Einstein s well-known Gedankenexperiment [1] that led to the principle of equivalence. The experiment depends on understanding frames of reference, essential to the study of mechanics from the time of Newton. Nearly all books that deal with mechanics discuss frames of reference. As Sommerfeld [2] states, the laws of mechanics presuppose the existence of an inertial frame, i.e. an imaginary structure whose axes are trajectories of bodies moving purely under inertia. An equivalent statement is that Galileo s law of inertia (Newton s First Law) is valid in such a frame for a body on which the net force acting is zero. A second form of this statement is that Newton s Second Law does not hold in a non-inertial (i.e. accelerated) frame. An observer at rest in an inertial frame is called an inertial observer (I-O). Beginning students often have difficulties with this limitation on the validity of Newton s laws, laws that are presented implicitly as being universal. The problem is compounded when the students meet fictitious forces in the analysis of rotating systems, systems in which Newton s First and Second Laws do not hold. Consider a book at rest on a turntable rotating with a uniform angular velocity ω. To an I-O at rest outside the turntable there is but one horizontal force acting on the book, the friction force from the table. This force creates an acceleration towards the centre that, multiplied by the book s mass m, equals the net force, mω 2 r. The force is negative because it is in the negative radial direction; r is the radial /08/ $30.00 c 2008 IOP Publishing Ltd Printed in the UK 209

2 210 R Newburgh distance of the book from the axis of rotation. This force, unfortunately, carries the name centripetal force. The name often confuses the student who believes that it is a special force acting on the body rather than being another name for the net force. This confusion is increased when the student attempts to look at the problem from the viewpoint of the non-inertial observer (NI-O) at rest on the rotating table. Since the NI-O sees the book is at rest, he infers a zero net force. Yet he, too, can detect but one real force, the friction on the book, which implies a non-zero net force. To resolve his dilemma he then infers a second force acting radially outwards. This he calls the centrifugal force which equals +mω 2 r. However, this does not eliminate his problem, for he cannot ascribe a physical cause, the action of a second body, to this force. In classical mechanics all forces occur in pairs. The Third Law states that if body A exerts a force on body B, body B must exert a force equal in magnitude and opposite in direction on body A. In the case of the centrifugal force there is no body B. The centrifugal force is exerted on the book but is not an action arising from a second body. Some students will say that the centripetal and centrifugal forces form a Third Law pair. That this statement is false is immediately apparent for they both act on the same body, the book. We therefore conclude that the centrifugal force is a fictitious force, an example of forces that arise in non-inertial frames. Feynman [3] devoted a section of his lectures to such forces. The Berg demonstration The problem of describing and explaining fictitious forces is a common one for any teacher. Therefore, it was a great pleasure to discover a demonstration/experiment that allows the student to be simultaneously both an I-O and NI-O. We owe the demonstration to Berg [4] who has developed a striking demonstration to illustrate the effect of acceleration on the buoyant force acting on a floating body (see figure 1). In the demonstration as we have done it, a weighted test tube floats upright in a beaker of water. The beaker is fixed to a large board fixed to a spring attached in turn to a stationary support. By pulling on the board one can set it accelerating. A video camera attached to the board projects the image of the tube and beaker onto a screen. In analysing the motion, we must remember that there are two frames of reference in which we can describe the behaviour. There is the inertial or laboratory frame, the one in which the support is at rest. There is also the non-inertial or accelerated frame in which the board, beaker, test tube and camera are at rest. In the laboratory frame, the tube and beaker are both accelerating. Hence, we conclude from Newton s Second Law that a non-zero net force is acting. However, in the accelerated frame we see clearly that the tube is at rest. Therefore, an observer also at rest in that frame would infer a zero net force. How do we reconcile these two contradictory conclusions? There is another observation that is quite counterintuitive. The image of the tube and beaker as shown by the camera at rest with respect to them is of a tube completely stationary with respect to the water in the beaker. In spite of the violent motion of the board, up and down with non-constant acceleration, there is absolutely no sloshing of the water in the beaker, nor the slightest relative motion between tube and beaker. The demonstration can be seen on a University of Maryland web site ([4], loc. cit.). According to Berg, the idea was proposed in the NASA program as a possible way to reduce injury in the event of a crash of a re-entering space vehicle. However its exact origin is not certain. To see both tube and beaker accelerating up and down while remaining absolutely still with respect to each other is indeed surprising. Analysis dispels the surprise. The test tube is

A demonstration of Einstein s equivalence of gravity and acceleration 211 Figure 1. The Berg experiment the platform is attached by a harness to a spring.

3 A demonstration of Einstein s equivalence of gravity and acceleration 211 Figure 1. The Berg experiment the platform is attached by a harness to a spring. The spring in turn is attached to a fixed support, the rod to the right of the spring. The test tube floats in water in the left cylinder. A mass on a spring is in the right cylinder. The video camera is fixed to the platform. (This figure is in colour only in the electronic version) subject to two forces, its weight, mass m times the acceleration of gravity g, and the buoyant force B. Applying the Second Law, we write B + mg = ma. (1) The buoyant force acts in the upward direction and the weight acts in the downward direction. The net force, ma, acts either up or down, depending on the direction of the acceleration. From this, we conclude immediately that the buoyant force does not equal the weight. It is, in fact, a function of the acceleration so that B = mg + ma. (2) For example, if the system were in free fall, the buoyant force would be zero. The falling tube would then be weightless (note the quotation marks) as would the displaced fluid that creates the buoyant force. Weightless does not mean that gravity has ceased to act. However, a scale in free fall with a body standing on it would read zero. One might wonder in light of the experiment, how it is possible to agitate a fluid by shaking a container. The answer lies, I believe, in the fact that the simple harmonic driving force is well ordered and smooth, though not constant. As a consequence, the friction between the fluid and the beaker controls the relative motion between the two. Were the force too violent, the static friction could not balance the agitation effects. The immobility of the test tube with respect to the fluid depends on the very nature of the buoyant force, as the analysis shows. At this point we could leave the demonstration and any further discussion of it. It has forced the students to examine their preconceptions and has led to a deeper understanding

4 212 R Newburgh of buoyancy. However, the demonstration is also a useful tool for the study of the physics of accelerated frames of reference. Many books on classical mechanics state that Newton s laws are valid in inertial frames only. Joos [5] stated this rather nicely. We start with the experimental fact that there exist reference frames for which computations based on Newton s Second Law are in complete accord with experiment. Note that this is exactly the case for our analysis of the demonstration in the laboratory frame. However, this statement implies that there are frames in which Newton s Second Law does not hold. In such frames fictitious forces appear. They are called fictitious forces for they are not part of a Third Law pair, nor can we attribute their source to another body. With these fictitious forces, though, we can force the Second Law onto a Procrustean bed of conformity. Most texts discuss the appearance of such forces in relation to rotating frames. The most common fictitious forces are those associated with rotating frames, the centrifugal and Coriolis forces. The student s problem with rotating frames is the explanation of his observations. Though the behaviour of the observed bodies does not depend on the observer, the explanations do. He feels the centrifugal force to be a real force but has difficulty explaining why it appears in the non-inertial frame only. One of the beauties of Berg s demonstration is that it deals with linearly accelerated frames and allows the student to be an I-O and NI-O simultaneously. Let us now rewrite equation (1) B + mg ma = 0. (3) The introduction of a fictitious force ma, equal in magnitude and opposite in direction to the original net force, allows us to define a new net force equal to zero. This does describe the stationary behaviour of the tube in the accelerated frame in apparent agreement with the First Law. Comparison of inertial and non-inertial observations Now Berg intended this demonstration as an illustration of the nature of buoyancy, a purpose it fulfils admirably. However, it goes beyond that. The demonstration allows the I-O to be an NI-O at the same time. While watching the tube, water and beaker accelerate in the laboratory frame, the student can also view the output from the video at rest in the accelerated frame. As an NI-O he sees the tube, water and beaker completely at rest with respect to each other. Again, following Newton s laws, he concludes that the net force in the non-inertial frame is zero. This inference leads to exactly the same problem met by the NI-O in the rotating frame. Analysing the forces in a non-inertial frame, he can find but two real forces, the weight and the buoyant force. And these two forces do not balance. He therefore infers a third force, equal in magnitude and opposite in direction to the vector sum of the weight and the buoyant force (equation (3)). As with the centrifugal force, this inferred force has no origin in another physical body, nor does it form a Third Law pair with the net force of the I-O (equation (1)) since they both act on the same body. The inferred force must be a fictitious force. It is because NI-Os must use fictitious forces in analysing their observations that they are not valid observers. Newton s laws do not hold for them. Since these fictitious forces are always proportional to the mass of the body on which they act, they also have the name inertial forces. Note, however, that both the gravitational and buoyant forces are proportional to mass. Therefore they are also inertial forces but are, in addition, real forces in the Newtonian scheme. This distinction is not often made. I should like to consider a final point about the Berg experiment. In Einstein s 1907 paper ([1], loc. cit.) he expressed for the first time the equivalence of gravitational and inertial

5 A demonstration of Einstein s equivalence of gravity and acceleration 213 mass. Looking at Berg s experiment, an I-O sees the masses in the system accelerating, so he calculates with inertial mass. The NI-O sees the masses at rest, so calculates with gravitational mass. This equivalence is a direct consequence of the fact that both weight and buoyancy are inertial forces. It is this principle of equivalence that acted as a midwife in the birth of the general theory of relativity. Sciama [6] has written a particularly lucid discussion of the principle. Discussion The analysis of the buoyant force shows a clear dependence on both acceleration and gravity. If there is no acceleration, these forces are related to gravitational pull, the weight of the displaced fluid. In the event of acceleration these weights are modified by an amount proportional to the acceleration. This modification leads to the idea of an effective weight. Let us return to Einstein s Gedankenexperiment ([1], loc. cit.). A man is at rest in an elevator in a region of zero gravity. The cable is suddenly pulled upwards so that the elevator accelerates in an inertial frame. The passenger still considers himself at rest in the elevator. He releases several objects, all of which accelerate downwards, for the force exerted by the cable does not act on these objects. The man observes the objects falling, exactly as they would in a gravitational field. With this experiment Einstein established the equivalence between gravitational and inertial forces. One must be careful not to confuse Einstein s experiment with the films showing orbiting astronauts releasing objects. These remain floating in the satellite beside the astronaut and do not fall. The satellite is not in a zero gravity environment. The universal law of gravitation applies. An I-O uses this law to explain the acceleration v 2 /R of the satellite (v is the orbital velocity and R is the distance from the centre of the earth to the satellite). The astronaut who is an NI-O considers himself at rest and posits the fictitious centrifugal force to explain his perceived zero acceleration. Clearly the Berg experiment is more closely related to the orbiting astronaut than to the Einstein experiment. However, unlike the astronaut s experience, it deals with a real inertial force, buoyancy, not a fictitious inertial force. Moreover, in the Berg experiment, as equation (2) shows, it is a function of both gravity and acceleration. The student can observe directly the effect of acceleration on a real inertial force in a gravitational field. While Einstein discussed the equivalence of acceleration in a gravity-free environment with free fall in a gravitational field, he did not consider the related but different case of acceleration in a gravitational field, the subject of this paper. Yet these acceleration effects that appear when we use the idea of effective weight lead to the inevitable conclusion that there is an intimate connection between acceleration and gravity. Note, too, that we have also distinguished between real fictitious forces such as gravity and the buoyant force, and fictitious inertial forces, such as the centrifugal and Coriolis forces. However, there is also an important distinction between gravity and the buoyant force. The force of gravity does not depend on acceleration. It is the same, whether measured in an inertial or non-inertial frame. The pull of the earth on the orbiting astronaut is the same for the I-O or NI-O. However, the buoyant force is very much a function of the acceleration of the system. It is because of these effects that I offer Berg s experiment as a directly observable justification of Einstein s view of the equivalence of gravity and acceleration, even though we do not do the experiment in a gravity-free environment. I think it is also significant that one can find such justification in the seemingly simplest topics of physics, even in physics without strings attached! I should also point out that this paper does not discuss the source of inertial forces. Mach s principle locates their origin in the fixed stars, an idea first adopted and later rejected

6 214 R Newburgh by Einstein. In rejecting it Einstein found the source of these forces in the geometry of spacetime. A recent paper [8] goes into these topics in considerable detail. Acknowledgments I owe much in this paper to Thomas E Phipps of Urbana, IL and Robert S Cohen of Boston University. I cannot overemphasize the importance of discussion, critical reading and suggestion in taking an originally inchoate idea and turning it into an organized whole. These functions they served admirably. References [1] Einstein A 1907 Relativitaetsprinzip und die aus demselben gezogenen Folgerungen Jahrbuch der Radioaktivitaet [2] Sommerfeld A 1952 Mechanics (New York: Academic) p 10 [3] Feynman R, Leighton R and Sands M 1989 The Feynman Lectures on Physics vol I (Reading, MA: Addison- Wesley) pp [4] Berg R Demonstration F2 22 Buoyancy Paradox Accelerated Frame 22.htm [5] Joos G 1934 Theoretical Physics (London: Blackie) p 217 [6] Sciama D 1969 The Physical Foundations of General Relativity (New York: Doubleday) pp chapter 4 [7] Newburgh R 2007 Inertial forces, absolute space, and Mach s principle: the genesis of relativity Am. J. Phys

When equal masses don t balance

FEATURES www.iop.org/journals/physed When equal masses don t balance Ronald Newburgh 1, Joseph Peidle 2 and Wolfgang Rueckner 2 1 Harvard University Extension School, Cambridge, MA 02138, USA 2 Science