A simple analytical approach that allows us to recover Newtonian physics from Schwarzschild's metric is proposed. The method elucidates why the tiny modification of the metric due to gravity entails a significant change of motion of a freely falling body.

In a recently published paper, Gould addresses a problem of Newton's and Einstein's explanations for why a ball falls when released.1 The main effort is made to visualize Einstein's model of gravity by means of ordinary wall maps having a geometry similar to the geometry of spacetime near Earth's surface. The varying scale of distance on a world map is used to represent a distortion of spacetime. This analogy enables the author to explain why geodesics (“straight” lines) are curved lines on spacetime diagrams, i.e., why the ball accelerates during its free fall near Earth. The problem is that the spacetime distortion is extremely small, so how could it so dramatically influence the ball's trajectory? To answer this question Gould refers to Schwarzschild's metric describing a weak gravitational field and finds the ball geodesic numerically (Fig. 6 in Ref. 1). It is suggested that the obtained curve is the parabola $x=12at2$, as expected on the basis of Newtonian physics, but this fact is not proved. As the Gould paper is intended to recover Newton's results from the geometrical treatment, we believe that it would be advantageous to show directly that the geodesic derived from Schwarzschild's metric is the spacetime trajectory known from Newton's theory. In this Comment, we want to demonstrate a straightforward method, at the level of an introductory course on general relativity, that allows us to obtain an analytical formula for the ball's trajectory from Schwarzschild's metric and compare it to the Newtonian result. In addition, the derived equation for x(t) shows precisely why the minuscule alteration of the time scale factor introduced to the metric by gravity1 so significantly modifies the ball's motion. Besides the notion of the metric introduced by Gould, the only other thing the reader needs to know to grasp the point discussed here is the definition of the scalar product of four-vectors $aμbμ=gμνaμbν$, where $gμν$ are the coefficients defining the metric.

The equation of motion determining the development of a body's four-velocity $uα$ (a vector tangent to the geodesic) is

$duαdτ=−Γβγαuβuγ,$
(1)

where $Γβγα$ are Christoffel symbols defined by

$Γβγα=12gαδ(∂gδβ∂xγ+∂gδγ∂xβ−∂gβγ∂xδ).$
(2)

Equations (1) and (2) are too difficult to be used at the introductory level and this is why all the calculations leading to specific curves in Ref. 1 are performed numerically. To obtain the body trajectory in a simple way, we propose employing the Killing vectors.2–4 A heuristic and not quite formal introduction to Killing vectors may proceed as follows. We know that the metric $ds2=gμνdxμdxν$ determines the free motion of a particle in spacetime. If the metric coefficients $gμν$ do not depend on some coordinate, say xk, then the particle four-velocity along this coordinate must remain constant. In other words, the projection (scalar product) of the four-velocity on a vector $ξμ=δkμ$ oriented along the coordinate xk is constant

$ξμdxμdτ=const.$
(3)

The vector $ξμ$ is called the Killing vector and it is associated with the symmetry of the metric.

The Schwarzschild metric for a weak gravitational field, after some approximations, assumes the form1

$ds2=dx2+(2ax−c2)dt2,$
(4)

where $a=GM/R2$ (the gravitational acceleration at Earth's surface), R is a distance from the initial position of the ball to Earth's center, and x is the distance the ball falls below the starting point. The metric in Eq. (4) does not depend on time t so we have the Killing vector $ξμ=(1,0,0,0)$ and the relation

$−ξμdxμdτ=−gμνξμdxνdτ=(c2−2ax)1cdtdτ=e,$
(5)

where e is a constant. In addition, dividing Eq. (4) by $dτ2$ gives

$−c2=(dxdτ)2+(2ax−c2)(dtdτ)2,$
(6)

where we have made use of the relation $ds2=−c2dτ2$, with τ the proper time of the falling ball. Calculating $dt/dτ$ from Eq. (5) and inserting the result into Eq. (6) we obtain

$(dxdτ)2=e21−2ax/c2−c2.$
(7)

Assuming that the initial velocity (for x = 0) is zero, we establish from Eq. (7) that $e2=c2$. For the non-relativistic motion considered in Ref. 1, the velocity satisfies $dx/dτ≈dx/dt$, and because $2ax/c2≪1$, one can approximate $(1−2ax/c2)−1≈1+2ax/c2$, which changes Eq. (7) into

$dxdt=2ax.$
(8)

This differential equation can readily be solved for x(t), or we can simply recognize that it describes the well known motion along a parabola $x=12at2$ with constant acceleration a (see Fig. 1). Using this simple method and omitting the problem of finding geodesics by means of Christoffel symbols, we are still able to find the ball geodesic analytically.

Fig. 1.

The solution of Eq. (8) obtained on the basis of general relativity represents free fall with acceleration a, i.e., $x=12at2$.

Fig. 1.

The solution of Eq. (8) obtained on the basis of general relativity represents free fall with acceleration a, i.e., $x=12at2$.

Close modal

To discuss the shape of the ball's trajectory it is convenient to trace how the tangent to the trajectory—the velocity—is modified by the metric. In a flat spacetime we have a = 0 so the metric coefficient for dt2 is equal to $−c2$ [see Eq. (4)]. On a spacetime diagram the trajectory would then be a straight line (motion with a constant velocity). In a curved spacetime, using Eqs. (4) and (8), we can see that the fact that the ball's velocity changes (movement along a curve) is determined not by the full scale factor $(−c2+2ax)$ but only by the term 2ax expressing a deviation of the factor from the flat-space value $−c2$. Compared to $−c2$, the modification 2ax is minuscule and, in the language of scale proposed by Gould, may seem to be unimportant. But in the light of Eq. (8), it becomes evident that the term 2ax really matters. The fact that 2ax is of the order of the metric coefficient for dx2 (equal to one) appears to be decisive and enough to influence the ball's velocity. The most important point, however, is that on the basis of Eq. (8) we obtain the explicit formula $x(t)=12at2$ for the shape of the geodesic, demonstrating that Newtonian physics is actually recovered, which would be impossible (or at least much more difficult) for numerical solutions.

Gould's idea to compare the effects predicted by general relativity with results obtained while using maps with varying scales seems to be a good introduction to understanding the unexpected shape of straight lines (geodesics) realized by moving objects in the presence of gravity. We believe that the analytical approach proposed in this Comment provides some valuable details explaining the problem of free fall in general relativity and, being an addendum to the presentation given in Ref. 1, allows the reader to fully appreciate the content of Gould's work.

The author wishes to thank P. Prawda for his continued assistance in the realization of this work.

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