Geodesics as ravity February 8, 05 It is not obvious that curvature can account for ravity. The orbitin path of a planet, for example, does not immediately seem to be the shortest path between points. Even more immediate is the way the motion of projectiles differs dependin on initial conditions. Consider the tihtly parabolic path followed by a stone thrown hih into the air, contrasted with the nearly straiht-line path of an arrow. How can these two paths, which may pass throuh nearly identical reions of space, be rearded as tracin eodesics in the same curved eometry? The answer lies in the importance of time in ivin a 4-dimensional picture. The curvature of a curve Suppose we have a curve lyin in the xy-plane, iven by y y x). At any point P x 0, y x 0 )) of the curve we place a circle. The best fit circle is the one which is tanent at P, and matches the curve as nearly as possible. To find this circle, choose coordinates with oriin at P and the x axis tanent at P. Then dy y x 0 ) y 0) 0, and the slope of the curve vanishes, 0. Now expand y x)near P. By our dx 0 choice of coordinates, the first two terms in the series vanish, y curve x) y x 0 ) + dy dx x x 0 ) + d y! dx x x 0 ) + d y! dx x + Suppose, for concreteness, that at P the curve has d y dx > 0. Now define a circle of radius R, tanent to the curve at P. The center of the circle will be a distance R up the y axis and consist of point satisfyin Solvin this for y circle x), we have x + y R) R y circle R ± R x To et points near the oriin, we require the lower sin, and expand in a power series, y circle R R x R R x R R R x ) R + x R +
The best fit circle is the circle determined by matchin these curves: y curve x) d y! dx x + y circle x R + In eneral, the hiher derivatives will not match, but we can match the ond derivative by choosin the radius of the circle: x R d y! dx x R d y dx We define the curvature at any point of the curve to be R where R is the radius of the best fit circle. The curvature of projectile motion Consider the two motions: a stone is thrown upward at a steep anle θ, landin at a distance d, while another is thrown at hih speed at a taret in the same final location. Each stone follows a parabolic motion: x v 0x t y v 0y t t so that ) x y x) v 0y ) x v 0x v 0x The distance traveled, x d, occurs when y 0: ) d 0 v 0y ) d v 0x v 0x d v 0xv 0y If the projectile is launched at an anle iven by tan θ v0y v 0x and The maximum heiht is reached at when v y 0, v 0y v 0x tan θ then d v 0x tan θ d v 0x tan θ v 0y v 0x tan θ d tan θ 0 v 0y t t v 0y
and at this time we have h v 0y d 4 tan θ We find the curvature at the top of the trajectory, P x top, h) d, d 4 tan θ). At any point, the curve may be described by x x 0 + v 0x t y y 0 + v oy t t At the top, x 0, y 0 ) d, d 4 tan θ) and the initial velocity is v 0x, v 0y ) v ox, 0). Therefore, x d + v 0xt and the curve y x) is t y d 4 tan θ t v 0x x d ) tan θ d x d ) y d 4 tan θ tan θ d d 4 tan θ tan θ d x d x d ) The best fit circle will match the ond derivative: R tan θ d and we see that the curvature is lare for steep anles and small for shallow anles. There is a dramatic difference in the curvature for different initial conditions. Suppose the total time of fliht, t v 0y ) d tan θ is ond for one stone and 0 onds for another over a distance d 5 meters. Then for the first 50 0 tan θ tan θ 5 the curvature at the top is tan θ R d 5.06 3
while for the ond, and the curvature is 0 50 0 tan θ tan θ 0 40 R 0 5.6 The curvatures differ by a factor of 00 for ordinary trajectories. This makes is impossible to model the motions of the stones by lettin them follow eodesics in a curved 3-dimensional eometry. To build a eometric model, we need the curvature to be very nearly the same for the two stones, so they both move alon comparably shaped paths. 3 The curvature in spacetime Now place the motion in spacetime. The non-relativistic) projectile follows a curve parameterized by time: ct, x, y) ct, v 0x t, ) t The motion lies in a plane rotated in the xt plane at an anle with tan ϕ v0x c proper distance proper time!) alon this direction. Then cτ c t v0x t ct tan ϕ τ t. Let τ c t x be Then we may parameterize the curve in this plane by cτ, y) cτ, τ ) ) λ, λ c where we parameterize with λ cτ. The best-fit circle to this curve ives c τ R c τ c R c and this is independent of initial conditions. This means that the hih arc of a slow stone, and the flat trajectory of a fast stone have the same curvature in spacetime! Both trajectories could lie on a spherically curved surface where the sphere has a hue radius, c, so it is only ently curved. In sharp contrast to the Euclidean case, where a factor of 0 difference in time of fliht requires a factor of 00 difference in radius of curvature, even a difference of a factor of 0 4 in time of fliht makes completely neliible difference to the curvature required in spacetime. 4
Exerise: Find the difference in curvatures m R between initial velocities of 3 Answer: If we don t nelect terms of order v c, then t and τ are related by cτ c t v0x t so ct τ t v 0x c v 0x c τ t v 0x c Now, when we parameterize the curve by cτ, we have ) cτ, y) cτ, t cτ, cτ c v 0x c Therefore, the curvature is iven exactly by c τ R c τ c v 0x c R c v 0x c Now consider stones thrown with velocities of 3 m v 0x c and 3 04 m m and 3 04. In the first case, 9 9 0 6 0 6 + 0 6 so the curvature is R c v 0x c 9.8 9 0 6 + 0 6). 0 6 +. 0 3 For the ond stone, v 0x c 9 08 9 0 6 0 8 + 0 8 5
and the curvature is R c v 0x c 9.8 9 0 6 + 0 8). 0 6 +. 0 4 The difference between these neliible. 6