Snellius meets Schwarzschild - Refraction of brachistochrones and time-like geodesics
aa r X i v : . [ phy s i c s . pop - ph ] S e p Snellius meets Schwarzschild – Refraction of brachistochrones and time-like geodesics
Heinz-J¨urgen Schmidt Universit¨at Osnabr¨uck, Fachbereich Physik, D - 49069 Osnabr¨uck, Germany
The brachistochrone problem can be solved either by variational calculus or by a skillful applicationof the Snellius’ law of refraction. This suggests the question whether also other variational problemscan be solved by an analogue of the refraction law. In this paper we investigate the physicallyinteresting case of free fall in General Relativity that can be formulated as a variational problemw. r. t. proper time. We state and discuss the corresponding refraction law for a special class ofspacetime metrics including the Schwarzschild metric.
I. INTRODUCTION
Sometimes the same result can be derived in differentways. In physics we may benefit from considering side-by-side approaches solving the same problem since theymay have varying virtues; one approach may be moregeneral and the other may provide more physical insight.For example, the brachistochrone problem posed byJohann Bernoulli in 1696 often appears in textbooks asan exercise in the variational calculus and is solved viathe corresponding Euler-Lagrange equation. However, atthat time the variational calculus was not yet developed(the corresponding textbook of Leonhard Euler [1] wasonly published in 1744). Hence it is not surprising thatBernoulli solved the problem in a different way: He con-sidered the analogy with a beam of light in a mediumwith varying refractive index that, according to Fermat’sprinciple (stated in 1662), chooses the path between twopoints that takes the least time. At first sight this onlymeans that we have two different physical variationalproblems that have the same solution, but for the opticalproblem there exists a local law determining the path oflight: the Snellius’ law of refraction (1632), formulatedin a way suitable for a smoothly varying refractive index.Bernoulli succeeded in showing that the cycloid curvesatisfies the refraction law for the problem under consid-eration and thus solved the brachistochrone problem.Undoubtedly, the variational approach is straightfor-ward and applicable to a large class of similar problems.Given the ubiquity of variational principles in modernphysics it is in order that students have to learn how tofind and (sometimes) solve the Lagrange equations andthe brachistochrone problem could serve as an entertain-ing exercise. However, Bernoulli’s solution has a uniqueappeal in so far as it uses the transfer from other branchesof physics and solves a problem not by setting into oper-ation a heavy machinery but by a clever application of asimple law. Our intuition is not well-trained to solve suchkind of variational problems. One may concede that thenaive choice of a straight line between two given points A and B does not yield the path of least time and that it isa promising strategy to speed up early even if the path isthereby prolonged. But the detailed form of the optimalcurve is not intuitively clear. Recall that Galileo Galileialso considered the brachistochrone problem and conjec-tured its solution to be the circle. On the other hand, the Snellius’ law is intuitively clear in the light of Fermat’sprinciple, at least qualitatively. Hence, if we divide thespace between A and B into a large number of layers withdifferent but constant refractive indices and imagine therefraction of light at each boundary between the layers,the form of the optimal curve will become plausible, in-cluding the possible minimum of the brachistochrone dueto total internal reflection of the light beam.In this paper we will try to transfer Bernoulli’s idea toanother problem in physics that can be cast into the formof a variational problem: The motion of a freely fallingpoint particle according to General Relativity. Given twopoints A and B in spacetime the actual world-line takenby the particle is the time-like geodesic that assumes themaximal proper time τ among all time-like curves join-ing A and B . Variational calculus yields the differentialequation governing the freely falling particle, namely thegeodesic equation d x λ dτ + Γ λµν dx µ dτ dx ν dτ = 0 , (1)see, e. g. , [2] (4.4.18). In the limit of low velocities(compared with the speed of light) and linearly varyinggravitational potential the solutions of (1) would includethe parabolic trajectories well-known from high schoolphysics. However, due to the complicated way to de-rive these elementary results one could hardly justify theclaim that the parabolic trajectory is intuitively under-stood in terms of General Relativity. In the light ofBernoulli’s problem one would have to look for an analogyof the Snellius’ law for time-like geodesics. This wouldrequire a situation where the spacetime metric g µν is con-stant within two regions A and B separated by a space-like surface. If the metric in A and B would be the samewe have essentially a situation described by Special Rel-ativity. Then the straight time-like line joining two givenpoints A ∈ A and B ∈ B would realize the maximalproper time among all time-like curves joining A and B .But if the spacetime metric in A and B is different thenthe maximal proper time is realized by a world line com-posed of two straight lines within A and B with differentvelocities. The equation relating these velocities to thechange of the metric would be the ”Snellius’ law for time-like geodesics” we are looking for. Intuitively, the particlewants to spend more time in the region with higher grav-itational potential, say, in B , since there its internal clockruns faster and it can gain more proper time than in theregion with lower potential, say, A . But it would not be agood idea to rush too quickly into B since this would slowdown the internal clock according the time dilation effectalready known from Special Relativity. This qualitativeexplanation of free fall according to General Relativityhas also be given in [8].Admittedly, the situation of two regions A and B witha jump of the metric at its boundary is un-physical andonly a fictitious situation in which we can argue in a sim-plified way. This is so because a physical metric has tosatisfy Einstein’s field equations and the described jumpcould only be realized by a double layer of positive andnegative mass density and therefore has to be ruled out.This is in contrast to the situation in geometrical opticswhere two regions with different but constant refractiveindices can be easily realized. Nevertheless, the sequenceof different finer and fine unphysical layers has a phys-ical limit with a smoothly varying metric that can bedescribed by the continuum version of the refractive law.The paper is organized as follows. In section II A werecapitulate the classical brachistochrone problem. Al-though this material can be found at various places it willbe convenient for the reader to present a concise accountsuited for the present purpose. In section II B we defineda non-Euclidean metric of the half-plane such that thebrachistochrones are exactly the geodesics of this metric.The classical brachistochrone problem can be generalizedto a situation with a two-dimensional potential havinga one-parameter family of symmetries, see section II C.Here the Snellius’ refraction law acquires an extra fac-tor that compensates the change of the normal directionneeded to define the angle of refraction θ . As an exampleof this generalization we calculate the brachistochronesfor the harmonic oscillator potential in section II D.Section III contains the derivation of the Snellius’ lawfor time-like geodesics for the special case where the met-ric has the form of a Schwarzschild (or slightly more gen-eral) metric restricted to (1 + 1) dimensions, namely dτ = ϕ ( r ) dt − ϕ ( r ) − dr c , (2) c denoting the velocity of light in vacuo. Note that theReissner-Nordstr¨om metric describing a charged blackhole, see, e. g. [2], is also of the form (2). We will showthat the Snellius’ law is equivalent to the energy conserva-tion law for one-dimensional motion in the Schwarzschildmetric, see [2], (6.3.12). Hence it does not represent anew result about general-relativistic free fall motion butrather a new interpretation of a well-known law. A Bxy dy dxds θ FIG. 1: A curve C ( A, B ) joining the points A and B andfurther notations explained in the text. II. THE BRACHISTOCHRONE PROBLEMREVISITEDA. The classical brachistochrone problem
Despite its age of over 300 years the brachistochroneproblem has also found recent attention [3], [4], especiallyin connection with its generalization including friction [5]– [7] that was already considered in [1]. For the purposeof this paper we will shortly recapitulate its formulationand solution.Let C ( A, B ) be a plane curve between the points A and B and consider the constrained motion of a point parti-clealong C ( A, B ) under the influence of a constant (grav-itational) acceleration of absolute value g . The initialvelocity at the point A is assumed to vanish. Upon in-troducing coordinates x, y ≥ C ( A, B )will, at least locally, be described by a smooth function x y ( x ) with derivative y ′ ( x ) = dydx . Due to the energyconservation m v = m g y , (3)where m is the (irrelevant) mass of the particle, the ab-solute velocity v of the particle is given by v = p g y . (4)The time dt needed to pass an infinitesimal part of thecurve with length ds can be written as dt = dsv = p dx + dy v ( ) = s y ′ g y dx . (5)Hence the total time T to pass the curve C ( A, B ) will begiven by T = Z x B x A s y ′ ( x ) g y ( x ) dx , (6)where ( x A , y A = 0) and ( x B , y B ) are the coordinates ofthe points A and B , resp. .The brachistochrone problem consists in finding thecurve C ( A, B ) that makes T minimal for given A and B . We will first recapitulate the approach due to thevariational calculus. To this end we re-write (6) in theform T = Z x B x A L ( y ( x ) , y ′ ( x )) dx , (7)introducing the Lagrangian L ( y, y ′ ) ≡ s y ′ g y . (8)Then the solution of the above variational problem isgiven by a solution of the Euler-Lagrangian equation0 = ddx ∂ L ∂y ′ − ∂ L ∂y . (9)Conversely, each solution of the Euler-Lagrange equationyields a solution where the quantity (7) has locally a sta-tionary value. We will not consider this equation directlybut use the fact that x is a cyclic coordinate of (8) andhence the “Hamiltonian” H ≡ y ′ ∂ L ∂y ′ − L (10)is a constant of motion, invoking Noether’s theorem. Af-ter evaluating (10) we thus obtain − H = 1 p g y (1 + y ′ ) = sin θv = const. , (11)where the angle θ is introduced according to Figure 1 andsatisfiessin θ = dxds = dx p dx + dy = 1 p y ′ . (12)Formulated in this way the conservation law (11) assumesthe form of Snellius’ law of refraction. Moreover, thelatter is equivalent to2 R ≡ y (cid:0) y ′ (cid:1) = const. , (13)where R ≥ H = 0 that correspondsto the limit R → ∞ is treated separately. By contrast,the case of R = 0 has to be excluded since it leads to y ( x ) ≡ x and g R x y ϕ FIG. 2: The circle in the ˙ x − ˙ y − plane with parameter repre-sentation (21),(22) . y , denoted by a dot, where the time dependence of theinvolved functions is usually suppressed:˙ x ( ) = r gy y ′ ) = r gR y, (14)˙ y = y ′ ˙ x ( ) = r gR y y ′ ( ) = r gR p y (2 R − y ) . (15)We will shortly comment on the sign ambiguity intro-duced by the square root in (15). The positive sign cho-sen in (15) holds for the descending part of the brachis-tochrone. It will be tacitly understood in what followsthat there exists also an ascending part where a negativesign would have to be inserted into (15).Since both derivatives ˙ x and ˙ y only depend on y thiscan be viewed as a parameter representation of a curvein the ˙ x − ˙ y − plane. The form of the curve follows fromthe following calculation:˙ y ) = gR y (2 R − y ) (16)= gR (cid:0) R − ( y − R ) ) (cid:1) (17) ( ) = gR R − s Rg ˙ x − R ! (18)= gR (cid:18) R − Rg (cid:16) ˙ x − p g R (cid:17) (cid:19) (19)= g R − (cid:16) ˙ x − p g R (cid:17) . (20)It is a circle with center ( √ g R,
0) and radius √ g R , seeFigure 2. Another parameter representation of this circleuses the angle φ , see Figure 2,˙ x = p g R (1 − cos φ ) , (21)˙ y = p g R sin φ . (22)Differentiating (21) w. r. t. time yields¨ x = p g R sin φ ˙ φ . (23)On the other hand, we may differentiate (14) and obtain¨ x = r gR ˙ y ( ) = r gR p g R sin φ . (24)Comparison of (23) and (24) shows that the angular ve-locity ˙ φ assumes the constant value˙ φ = r gR . (25)Since both t = 0 and φ = 0 at the point A we may furtherconclude φ ( t ) = r gR t . (26)This enables the t -integration of (21) and (22) in astraightforward manner with the result x ( t ) = x A + p gR t − p gR s Rg sin r gR t (27)= x A + R ( φ ( t ) − sin φ ( t )) , (28) y ( t ) ( ) = s Rg ˙ x (29) ( ) = R (1 − cos φ ( t )) . (30)This is obviously the parameter representation of a cy-cloid, the curve traced by a point on the rim of a circularwheel with radius R as the wheel rolls along a straightline without slipping. This completes the solution of theclassical brachistochrone problem. FIG. 3: A family of cycloids, geodesics of the metric (31),passing through a point of the half-plane H . The limit case R → ∞ is indicated by a vertical dashed line. B. The classical brachistochrone metric
We may reformulate the brachistochrone problem in aslightly different language by writing the square of (5) as dt = dx + dy g y , (31)and viewing this equation as the definition of a Rieman-nian metric in the open half-plane H given by H = { ( x, y ) ∈ R | y > } . (32)This approach is well-known from geometrical optics, see,e. g. [4]. The length of a curve C ( A, B ) in H w. r. t. thismetric is the time that a point particle constrained tomove on C ( A, B ) needs to run through the curve, simi-larly as described above. The only difference is that thestarting point A of the curve can be chosen arbitrarily in H and hence the particle has a non-vanishing initial veloc-ity v A = √ g y A according to (4). Then the precedingconsiderations show that the family of cycloids definedby (28) and (30), where the two parameters x A ∈ R and R > R → ∞ . Then it follows that for each point A ∈ H andeach direction t there exists a cycloid (in the extendedsense) that passes through A and is tangent to t , seeFigure 3. It remains an open problem to further analyzethe metric (31) and to decide whether it is isometric to aknown structure. As a first result into this direction wemention that the scalar curvature R of the metric (31) isgiven by R = − gy , (33)and hence negative and not constant. C. The generalized brachistochrone problem
We may generalize the classical brachistochrone prob-lem by considering a more general two-dimensional po-tential and (local) orthogonal coordinates x, y such thatthe analogue of (3) leads to a velocity field v ( y ) and theEuclidean metric has the form ds = g ( y ) dx + g ( y ) dy . (34)It is thus invariant under the one-parameter group oftranslations into x − direction. Then the analogue varia-tional problem leads to a Lagrangian L ( y, y ′ ) = p g ( y ) + g ( y ) y ′ v ( y ) . (35)By assumption, x is still a cyclic coordinate and hencethe “Hamiltonian” H = y ′ ∂ L ∂y ′ − L = − g v p g + g y ′ (36)will be a constant of motion again invoking Noether’stheorem. The angle θ between the brachistochrone andthe local y -direction now satisfiessin θ = r g g + g y ′ (37)and hence the Snellius law of refraction assumes the form − H = √ g sin θv = C = const. . (38)After a straightforward calculation the analogue of (14)and (15) is obtained as˙ x = v g C, (39)˙ y = − v √ g g p g − v C . (40)This describes again a curve in the ˙ x − ˙ y − plane but itsform depends on the potential and an explicit solutionanalogous to the classical brachistochrone problem is notpossible in general. θ ϑ θ P Qx x + dx y y + dy g ( y ) dx g ( y + dy ) dx v v R S
FIG. 4: The geometry of the refraction law for the general-ized brachistochrone problem. Since the direction normal tothe equipotential lines y = const. changes from P to Q thecorrection factor √ g occurs in (38) and (44). It remains to make the factor √ g in the refraction law(38) plausible. To this end we consider an infinitesimalquadrangle P, R, Q, S , see Figure 4, and the light-rayfrom P to Q . The Snellius’ law only yields a relationbetween θ and the alternate angle ϑ , namelysin θ v = sin ϑ v . (41)We cannot assume that ϑ = θ holds in general since thenormal to the equipotential lines y = const. has differentdirections at P and Q . This difference is of first order in the distance P Q and hence of the same order as thechange of the velocity v . More precisely, in the first orderwe obtain sin ϑ = R QP Q = p g ( y ) dxP Q , (42)sin θ = P SP Q = p g ( y + dy ) dxP Q . (43)Together with (41) it follows that p g ( y + dy ) sin θ v = p g ( y ) sin θ v , (44)and hence (38) holds. D. Example: The harmonic oscillatorbrachistochrone problem
As an example of the generalized brachistochrone prob-lem we consider the two-dimensional harmonic oscillatorpotential V ( ρ ) = m ω ρ . (45)where ρ, φ are polar coordinates such that the Euclideanmetric assumes the form ds = ρ dφ + dρ . (46)We choose the coordinates x = φ and y = σ ≡ ρ suchthat ds = g ( σ ) dφ + g ( σ ) dσ = σ dφ + 14 σ dσ . (47)Let the starting point A of the brachistochrone have thecoordinate ρ A = R such that the velocity field can bewritten as v ( σ ) = ω p R − σ , (48)and the refraction law reads √ σ sin θv = C = const. . (49)Then (39) and (40) assume the form˙ φ = ω (cid:0) R − σ (cid:1) σ C, (50)˙ σ = − ω p C ω p ( R − σ ) ( σ − r ) , (51)where r ≡ CωR √ C ω (52)is the minimal radius of the brachistochrone, see Figure5.We may solve the differential equation (51) by sepa-ration of variables, insert the result into (50) and finallyintegrate over t . The result allowing for the initial con-dition σ (0) = R reads σ ( t ) = R (cid:0) cos (cid:0) ωt √ C ω (cid:1) + 2 C ω + 1 (cid:1) C ω ) , (53)and φ ( t ) = φ − ω t + 1 C arctan Cω tan (cid:0) ω t √ C ω (cid:1) √ C ω ! . (54)It follows that the minimal radius r is reached after thetime T = π ω √ C ω . (55)An example of the resulting brachistochrone is shownin Figure 5. Here we see that the brachistochrone of theharmonic oscillator potential will have a point S of self-intersection. The minimal time to pass from S to S is,of course, T = 0, whereas one would need a finite time T to follow the brachistochrone from S to S by windingonce around the origin O . This is no contradiction sincethe brachistochrone only represents a local minimum oftime, not a global one, as it is illustrated by numericalexamples in Figure 5. III. REFRACTION OF TIME-LIKE GEODESICS
As pointed out in the Introduction we will look for ananalogue of Snellius’ law in the theory of freely fallingparticles according to General Relativity (GR). It will beadvisable to stress the differences to the classical brachis-tochrone problem (BP) in order to avoid misunderstand-ings. Both problems have to do with the motion of pointparticles in a gravitational field but the GR case dealswith free fall in contrast to the motion constrained to acurve C ( A, B ) in the BP. Here A and B are two pointsin the (closure of the) half-space H defined in SectionII and the brachistochrone realizes the minimal (non-relativistic) time. The world-line of a freely falling parti-cle in GR realizes the maximal proper time among alltime-like curves that connect two points A and B in spacetime . Light rays or paths only occur in the BP bymeans of analogy. They do not play any role in the con-sidered GR case although Fermat’s principle also holds inGR, see [9], theorem 7.3.1. But despite these differencesboth solution curves are geodesics of a (pseudo) Rieman-nian metric and hence solutions of an Euler-Lagrangeequation. This common ground means that they can beinvestigated by similar techniques.It will be instructive to repeat the well-known deriva-tion of the Snellius’ law by using Fermat’s principle. Forthe illustration we will use the same Figure as for thederivation of the refraction law for time-like geodesics R r SO FIG. 5: The brachistochrone resulting from a two-dimensionalharmonic oscillator potential, consisting of a descending part(blue curve) and an ascending part (red curve). Both partscross at the point S . The chosen values are ω = 1 s − , R =4 m and C = 0 . s . It follows from (55) that the minimalradius r is reached after the time T = 1 . s . To passthe straight (green) line of length R from the starting pointto the center O would already require the longer time T = π ω ≈ . s . To pass the closed part of the brachistochronefrom S to S requires T = 0 . s , whereas one would need1 . s to go from S on a straight line to the center O andback. and hence adopt an apparently strange notation denot-ing the vertical coordinate by t , the horizontal one by r and the time by τ . We assume that the velocity of lighthas two different but constant values v and v in two re-gions A and B separated by plane (the blue line in Figure6). Hence according to Fermat’s principle the light willchoose straight lines inside the regions A and B . Thusthe total path from A ∈ A to B ∈ B will be composedof a straight line joining A and P , and another straightline joining P and B where P is an arbitrary point atthe boundary between A and B . The total time to travelfrom A to B then amounts to τ = τ + τ = √ t + X v + p ( T − t ) + X v , (56)cf. the notation introduced in Figure 6. It will be a min-imum for0 = ∂τ∂t = tv √ t + X − T − tv p ( T − t ) + X , (57)or, equivalently, sin θ v = sin θ v , (58)where the angles θ and θ are defined according to Figure6. This is Snellius’ law of refraction; the version thatcovers also the case of a continuously varying velocityfield reads sin θv = C = const. (59)and appeared in the above solution of the classicalbrachistochrone problem in equation (11). In the con-tinuous case the light path cannot enter regions with | C v | > | C v | = 1 due to total internal reflection. X X tT - tA B rt v v Φ Φ P θ θ FIG. 6: The geometry of the Snellius’ law for two regions withdifferent velocities v > v . The same figure can be used toillustrate the spacetime geometry of the corresponding law fortwo spacetime regions with different gravitational potentials(values of the metric) Φ < Φ . Next we will perform the analogous calculation fortime-like curves in two-dimensional spacetime with themetric (2). We will assume that the function φ ( r ) thatoccurs in (2) has two different but constant values Φ > > A and B , resp., andcalculate the total proper time τ to travel from A to B via P : τ = τ + τ = s Φ t − X Φ c + s Φ ( T − t ) − X Φ c . (60)It will be a maximum for0 = ∂τ∂t (61)= Φ t q Φ t − X Φ c (62) − Φ ( T − t ) q Φ ( T − t ) − X Φ c . (63)In order to cast this equation into a similar form as (58) we introduce positive variables a, b, u , u such that p Φ t = a cosh u , (64) X √ Φ c = a sinh u , (65) p Φ ( T − t ) = b cosh u , (66) X √ Φ c = b sinh u . (67)This implies that tanh u = Xc t Φ (68)is the velocity of the particle in the region A , measuredw. r. t. the metric in this region using the constant valueΦ , analogously fortanh u = Xc ( T − t ) Φ . (69)Then the “refraction law” (61) - (63) can be conciselywritten as p Φ cosh u = p Φ cosh u . (70)The version that also covers the case of a continuouslyvarying metric reads p ϕ ( r ) cosh u = E = const. , (71)where u is implicitly defined by the analogue of (68):tanh u = drc dt ϕ . (72)In the continuous case the world-line cannot enter regionswith E √ ϕ ( r ) < E √ ϕ ( r ) = 1 due to the analogue of total internal reflec-tion.An equivalent form of the continuous refraction law isobtained by the following calculation dτ = s ϕ dt − dr ϕ c (73)= √ ϕ dt s − (cid:18) drc ϕ dt (cid:19) (74) ( ) = √ ϕ dt p − tanh u (75)= √ ϕ dt cosh u , (76)and hence ϕ dtdτ = √ ϕ cosh u = E = const. . (77)We remark that (77) is not a new equation but well-known. Consider the case of the Schwarzschild metric,i. e. , (2) with the special choice ϕ ( r ) = 1 − G Mc r , (78)where, as usual, G denotes the gravitational constant and M the mass of the gravitational center. Then equation(77) is equivalent to (6.3.12) in [2] and the latter followsfrom the fact that the Schwarzschild metric is static. Thequantity (77) hence represents some sort of energy. Thisis reminiscent to the brachistochrone problem where theSnellius law can be derived from the fact that the “Hamil-tonian” H will be constant of motion, see Section II.Finally, we will consider terrestrial free fall experimentswith a height of, say, y ≈ m and estimate the order ofmagnitude of the involved quantities. Let r = 6 . × m be the radius of the earth such that M Gr = g = 9 . ms . (79)Then we write r = r + y and expand p ϕ ( r ) according tothe Schwarzschild metric w. r. t. y in the following way: p ϕ ( r ) = s − G Mc ( r + y ) ( ) ≈ − r gc + g yc , (80) Since the velocities v involved are small compared with c we may set u ≈ vc and hencecosh u ≈ v c . (81)Then the law of refraction (71) leads to p ϕ ( r ) cosh u ( )( ) ≈ − r gc + v + g yc = const. , (82)thus recovering the non-relativistic energy conservationlaw (3). Although the relative variation of ϕ ( r ) is of theorder of 10 − this is compatible with the change of thenon-relativistic quantities v and g y of the order of afew m s since these quantities are divided by c in (82). Acknowledgment
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