Comment on “Determining the difference between local acceleration and local gravity: Applications of the equivalence principle to relativistic trajectories” [Am. J. Phys. 92(6), 444–449 (2024)]

Wang, Frank

doi:10.1119/5.0227213

In a recent publication,¹ Balbus pointed out that the common equivalence principle explanation of gravitational bending of light is quantitatively incorrect by a factor of three in the Schwarzschild geometry. Typical books assert that the vertical deflection of light in an stationary elevator on earth is $- (1 / 2) g t^{2}$ ⁠, where g is the acceleration due to gravity and t is a photon's time of flight across the elevator, but his calculation indicated that it should be $- (3 / 2) g t^{2}$ ⁠. He derived a transformation between the Schwarzschild coordinates and Riemann normal coordinates, which involved solving a system of partial differential equations and some approximations. In general relativity, the physical interpretation of coordinates is not always obvious; in this Comment, we derive a transformation between the Schwarzschild coordinates and local inertial coordinates, which have a special meaning related to the equivalence principle.² Local inertial coordinates measure proper time and length, and our procedure of finding them is exact and generalizable. The method presented below requires solving algebraic equations only, and the solution is guaranteed. Our calculation reveals that g needs to be further modified by a relativistic factor.

With the commonly used symbols G for the gravitational constant and c for the speed of light, the Schwarzschild metric for a mass M is

d s^{2} = - (1 - 2 r_{g} / r) c^{2} d t^{2} + {(1 - 2 r_{g} / r)}^{- 1} d r^{2} + r^{2} (d θ^{2} + {sin}^{2} θ d ϕ^{2}),

(1)

where

r_{g} = G M / c^{2}

⁠. Our goal is to find a coordinate system

x^{' μ} = (T, X, Y, Z)

to make this metric locally flat near a point P in this form:

d s^{2} = - c^{2} d T^{2} + d X^{2} + d Y^{2} + d Z^{2} + O [{(x^{' α})}^{2}] d x^{' μ} d x^{' ν} .

(2)

A different way to state the locally flat condition is that the first derivative of the metric elements is zero:

\partial g_{μ ν} / \partial x^{' α} = 0

at P.²

Albert Einstein's “happiest thought” occurred to him in 1907: “for an observer falling freely from the roof of a house there exists—at least in this immediate surroundings—no gravitational field” [his italics].³ To implement Einstein's thought mathematically, let us imagine an observer inside a free-falling elevator. It is natural to use a rectangular X, Y, Z grid attached to the walls as spatial coordinates, and a fixed clock on the wall to measure time T. According to Einstein's insight, the metric written in the coordinate system $(T, X, Y, Z)$ should automatically satisfy Eq. (2). A free-falling observer is the most central idea in general relativity, and the Schwarzschild solution is one of the most used models. Surprisingly, the explicit formulas for transforming coordinates between these two systems are hard to locate in the literature.

To develop the formulas, it is convenient to write the Schwarzschild metric in Cartesian coordinates

x^{μ} = (t, x, y, z)

⁠; a complete derivation can be found in Ref. 4. The first step is to expand the metric elements around the Cartesian coordinates

P (0, 0, 0, r_{0})

⁠,

r_{0} > 2 r_{g}

⁠, to linear order, as

\begin{matrix} d s^{2} = - [1 - \frac{2 r_{g}}{r_{0}} + \frac{2 r_{g} (z - r_{0})}{r_{0}^{2}}] c^{2} d t^{2} + [\frac{1}{1 - \frac{2 r_{g}}{r_{0}}} - \frac{2 r_{g} (z - r_{0})}{r_{0}^{2} {(1 - \frac{2 r_{g}}{r_{0}})}^{2}}] d z^{2} \\ + d x^{2} + d y^{2} + \frac{4 r_{g} x}{r_{0}^{2} (1 - \frac{2 r_{g}}{r_{0}})} d xdz + \frac{4 r_{g} y}{r_{0}^{2} (1 - \frac{2 r_{g}}{r_{0}})} d ydz . \end{matrix}

(3)

We then rescale

t \to t / \sqrt{1 - 2 r_{g} / r_{0}}

(time dilation) and

z \to z \sqrt{1 - 2 r_{g} / r_{0}}

to make the constant term of the diagonal metric elements unity. Next we write

(t, x, y, z)

as quadratic functions of

x^{' μ} = (T, X, Y, Z)

⁠:

x^{μ} = x^{' μ} + a_{α β}^{μ} x^{' α} x^{' β}

(see Eq. for explicit expressions) and calculate dt, dx, dy, and dz and then substitute them into Eq. . (See Ref. 7 for a simplified example.) To impose the locally flat condition, we want to ensure that there are no linear terms in each metric element when we express the metric in Eq. in terms of

(T, X, Y, Z)

⁠. In theory, there are

4 +_{4} C_{2} = 10

coefficients for each variable, thus 40 coefficients for T, X, Y, and Z. For a Taylor expansion of

g_{μ ν}

⁠, there are 4 linear terms for each metric tensor element, and for 10 elements, there are 40 linear terms. We can set 40 linear equations with 40 undetermined coefficients, so the system is consistent and solvable, although it may be tedious to do so.⁵ One can utilize a computer algebra system to carry out the computation (see the supplementary material). In this highly symmetric case, the only nonvanishing terms are

t = T + b T Z, x = X, y = Y, z - r_{0} = Z + d T^{2} + e Z^{2} + f (X^{2} + Y^{2}),

(4)

and it is not too laborious to find the undetermined coefficients. After restoring

t \to t / \sqrt{1 - 2 r_{g} / r_{0}}

and

z \to z \sqrt{1 - 2 r_{g} / r_{0}}

⁠, the final result is

\begin{matrix} t = \frac{T}{\sqrt{1 - \frac{2 r_{g}}{r_{0}}}} - \frac{r_{g}}{r_{0}^{2}} \frac{1}{(1 - \frac{2 r_{g}}{r_{0}})} T Z, x = X, y = Y, \\ z - r_{0} = \sqrt{1 - \frac{2 r_{g}}{r_{0}}} Z + \frac{r_{g}}{2 r_{0}^{2}} Z^{2} - \frac{r_{g}}{r_{0}^{2}} (X^{2} + Y^{2}) - \frac{r_{g}}{2 r_{0}^{2}} c^{2} T^{2} . \end{matrix}

(5)

This is the transformation from the coordinates

(T, X, Y, Z)

in a free-falling elevator to the Schwarzschild coordinates

(t, x, y, z)

near the point

P (0, 0, 0, r_{0})

⁠.

With this transformation at hand, one can unleash the power of the equivalence principle as imagined by Einstein. In the absence of non-gravitational forces, particles in a free-falling elevator will just undergo uniform motion. Consider a particle that moves horizontally on the XZ-plane with a speed v,

X = v T, Y = Z = 0.

(6)

Geometrically, it is a geodesic in a flat spacetime, and it's a straight line. Imagine Alice in a stationary elevator on earth, using the Schwarzschild coordinates

(t, x, y, z)

centered at P, the z-component of the geodesic is varying; thus, the particle trajectory is perceived to be curved due to gravity, even for a photon. Precisely, based on the transformation Eq. ,

z - r_{0} = - \frac{r_{g}}{2 r_{0}^{2}} c^{2} T^{2} - \frac{r_{g}}{r_{0}^{2}} v^{2} T^{2} = - \frac{1}{2} g T^{2} (1 + 2 β),

(7)

where

g = r_{g} c^{2} / r_{0}^{2} = G M / r_{0}^{2}

and

β = v / c

⁠. Unlike X, Y, and Z, which represent proper distance, z is merely a coordinate. The rate of change of proper distance

ℓ

with respect to z near P is

d ℓ / d z = 1 / \sqrt{1 - 2 r_{g} / r_{0}}

(see the constant term of

g_{z z}

in Eq. ), so the physical deflection in the vertical direction is

ℓ = - \frac{1}{2} \frac{g}{\sqrt{1 - \frac{2 r_{g}}{r_{0}}}} T^{2} (1 + 2 β),

(8)

and

g_{S} = g / \sqrt{1 - 2 r_{g} / r_{0}}

(Schwarzschild acceleration) is the acceleration needed for an observer to stay stationary at point P.⁶ Although the difference between g and

g_{S}

is minute in the weak field limit

r_{g} / r_{0} ≪ 1

⁠, we have not invoked such an approximation throughout our calculation. It is

g_{S}

not g that will be detected by an accelerometer if one undertakes an experiment.

We note that when solving the equations for the transformation, the coefficient for $X^{2} + Y^{2}$ in Eq. (4) are needed to eliminate the linear expansion of the off-diagonal metric elements $g_{x z}$ and $g_{y z}$ ⁠.⁷ Moreau et al.⁴ have demonstrated that if $g_{x z}$ and $g_{y z}$ are removed from the metric, the spacetime curvature will be zero. In their words, “only [their italics] the off-diagonal terms contribute to the elements of the Riemann tensor.” Because of the $X^{2}$ term, the vertical deflection in Eq. (8) is not just $- (1 / 2) g_{S} T^{2}$ but also with an added contribution $- g_{S} T^{2} β$ ⁠, even in a small region. For a photon (⁠ $β = 1$ ⁠), the deflection is $ℓ = - (3 / 2) g_{S} T^{2}$ ⁠. We reach the same conclusion as Balbus¹ that the departure from the commonly stated value is by a factor of three. Previously, Díaz-Miguel also found such a factor of three for a light ray traveling in a nearly uniform gravitational field,⁸ but Linet disputed the interpretation of the coordinates and insisted that $- (1 / 2) g t^{2}$ was correct.⁹ Because Linet considered a homogeneous gravitational field with null spacetime curvature, based on the above discussion, we conclude that his objection was unfounded. Although Balbus and Díaz-Miguel's g and our relativistic $g_{S}$ differ little in the weak field limit, if one could perform a measurement in a strong-field environment such as near the surface of a neutron star (⁠ $r_{g} / r_{0} \sim 0.4$ ⁠), the result should match our calculation.

For readers interested in history, in 1911, Einstein made a wrong prediction of the gravitational bending of light,¹⁰ but the planned observation during the solar eclipse in 1914 was preempted by the outbreak of World War I.¹¹ If it had happened, the emerging theory would have disagreed with the data. Einstein did not know about Riemannian geometry until 1912,³ and his calculation based on the equivalence principle without considering curvature is half as large as the true value confirmed in 1919 based on the fully developed general theory of relativity published in 1915. Moreau et al.¹² have performed a detailed analysis on the sources of discrepancy.

SUPPLEMENTARY MATERIAL

See the supplementary material for the computer algebra code.

REFERENCES

1.

S. A.

Balbus

, “

Determining the difference between local acceleration and local gravity: Applications of the equivalence principle to relativistic trajectories

,”

Am. J. Phys.

92

(

6

),

444

–

449

(

2024

).

https://doi.org/10.1119/5.0162082

Google Scholar

Crossref

2.

R. H.

Price

, “

General relativity primer

,”

Am. J. Phys.

50

(

4

),

300

–

329

(

1982

).

https://doi.org/10.1119/1.12853

Google Scholar

Crossref

3.

A.

Pais

,

Subtle Is the Lord

(

Oxford University Press

,

Oxford

,

1982

), pp.

178

and 210–216.

Google Scholar

4.

W.

Moreau

,

R.

Neultz

, and

D. K.

Ross

, “

The equivalence principle in the Schwarzschild geometry

,”

Am. J. Phys.

62

(

11

),

1037

–

1040

(

1994

).

https://doi.org/10.1119/1.17703

Google Scholar

Crossref

5.

One might wonder if we can also eliminate the quadratic terms using the procedure. If we write

x^{μ}

as a cubic function of

x^{' ν}

⁠, there will be

4 + 2_{4} C_{2} +_{4} C_{3} = 20

coefficients for each variable, thus 80 coefficients all together. If we expand each

g_{μ ν}

to the second power, there will be 10 terms, and for 10 metric elements, there will be 100 terms. We cannot eliminate the quadratic terms of the Taylor expansion of

g_{μ ν}

⁠; there will be

100 - 80 = 20

values left, and 20 is precisely the number of independent Riemann tensor elements in a four-dimensional spacetime.

6.

S.

Carroll

,

Spacetime and Geometry: An Introduction to General Relativity

(

Addison Wesley

,

San Francisco

,

2004

), p.

248

.

Google Scholar

7.

To elaborate on this point, we observe that

g_{x z}

in Eq. is proportional to 4x. Focusing on X and Z variables only in Eq. , we have

d x = d X

and

d z = 2 f XdX + (1 + 2 e Z) d Z

⁠. The relevant part of metric (dropping the proportionality constant) is

d z^{2} + 4 x dxdz = (4 f^{2} + 8 f) X^{2} d X^{2} + 2 [(2 + 2 f) X + 4 e (f + 1) X Z] dXdZ + {(1 + 2 e Z)}^{2} d z^{2}

⁠. We find

f = - 1

as the solution to eliminate the linear term

2 (2 + 2 f) X dXdZ

⁠; if the coefficient f is absent, we cannot make the metric locally flat. The same argument applies for

g_{y z}

⁠.

8.

E.

Díaz-Miguel

, “

Gravity fall of light: An outline of a general relativity test

,”

Phys. Rev. D

69

(

2

),

027101

(

2004

).

https://doi.org/10.1103/PhysRevD.69.027101

Google Scholar

Crossref

9.

B.

Linet

, “

Comment on ‘Gravity fall of light: An outline of a general relativity test

,’”

Phys. Rev. D

70

,

048101

(

2004

).

https://doi.org/10.1103/PhysRevD.70.048101

Google Scholar

Crossref

10.

A.

Einstein

, “

Über den Einfluß der schwerkraft auf die ausbreitung des lichtes

,”

Ann. Phys.

340

(

10

),

898

–

908

(

1911

).

https://doi.org/10.1002/andp.19113401005

Google Scholar

Crossref

11.

L. M.

Krauss

, “

What Einstein got wrong

,”

Sci. Am.

313

(

3

),

50

–

55

(

2015

).

https://doi.org/10.1038/scientificamerican0915-50

Google Scholar

Crossref

PubMed

12.

W.

Moreau

,

C.

Purdie

, and

J.

Wood

, “

The rate of deflection of light in an accelerated frame and a gravitational field

,”

Am. J. Phys.

67

(

3

),

241

–

251

(

1999

).

https://doi.org/10.1119/1.19232

Google Scholar

Crossref

2024

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Comment on “Determining the difference between local acceleration and local gravity: Applications of the equivalence principle to relativistic trajectories” [Am. J. Phys. 92(6), 444–449 (2024)]

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