Fermat's Library | The young center of the Earth annotated/explained version.

### TL;DR In this paper, a group of scientists from Denmark revisi...

Here's the page from "Feynman's Lectures on Gravitation" in which h...

## Inside the sphere $(r \le R)$ ### Mass enclosed inside radi...

The Earth's density profile, as described in the Preliminary Refere...

\[ 5 \times 10^{-3} \,\text{years} = 0.005 \,\text{years} = 0.00...

The Sun's density profile, such as the Model S, is calculated using...

The young center of the Earth

U.I. Uggerhøj,

R.E. Mikkelsen,

and J. Faye

Department of Physics and Astronomy, Aarhus University, Denmark

Department of Media, Cognition and Communication, University of Copenhagen, Denmark

(Dated: April 20, 2016)

We treat, as an illustrative example of gravitational time dilation in relativity, the observation that

the center of the Earth is younger than the surface by an appreciable amount. Richard Feynman ﬁrst

made this insightful point and presented an estimate of the size of the eﬀect in a talk; a transcription

was later published in which the time diﬀerence is quoted as ’one or two days’. However, a back-

of-the-envelope calculation shows that the result is in fact a few years. In this paper we present

this estimate alongside a more elaborate analysis yielding a diﬀerence of two and a half years. The

aim is to provide a fairly complete solution to the relativity of the ’aging’ of an object due to

diﬀerences in the gravitational potential. This solution - accessible at the undergraduate level - can

be used for educational purposes, as an example in the classroom. Finally, we also brieﬂy discuss

why exchanging ’years’ for ’days’ - which in retrospect is a quite simple, but signiﬁcant, mistake -

has been repeated seemingly uncritically, albeit in a few cases only. The pedagogical value of this

discussion is to show students that any number or observation, no matter who brought it forward,

must be critically examined.

I. INTRODUCTION

The gravitational potential inﬂuences the rate at which time passes. This means that a hypothetical measurement

of the age of a massive object like the Sun or the Earth would yield diﬀerent results depending on whether performed

at the surface or near the center. In this connection, clearly, issues such as the initial assembly of cosmic dust to form

the protoplanet eventually leading to the Earth is not what is alluded to when considering the age. Rather, the age

is understood as e.g. the ’aging’ of radioactive elements in the Earth, i.e. that fewer radioactive decays of a particular

specimen have taken place in the Earth center than on its surface. Furthermore, arguments based on symmetry will

convince most skeptics, including those from ’the general public’, that there is no gravitational force at the Earth

center. Consequently, such an eﬀect cannot be due to the force itself, but may instead be due to the ’accumulated

action of gravity’ (a layman expression for the gravitational potential energy being the radial integral of the force).

Thus, there is also a good deal of pedagogical value in this observation.

In a series of lectures presented at Caltech in 1962-63, Feynman is reported to have shared this fascinating insight

with the audience using the formulation ”...since the center of the earth should be a day or two younger than the

surface!” [2]. This thought experiment is just one among a plethora of fascinating observations about the physical

world provided by Richard Feynman. Although this time diﬀerence has been quoted in a few papers, either the

lecturer or the transcribers had it wrong; it should have been given as ’years’ instead of ’days’.

In this paper, we ﬁrst present a simple back-of-the-envelope calculation which compares to what may have been

given in the lecture series. We then present a more elaborate analysis which brings along a number of instructive

points. We believe that this correction only makes the observation of age diﬀerence due to gravity even more intriguing.

We stress that this paper is by no means an attempt at besmearing the reputation of neither Feynman nor any of

the authors who trustingly replicated his statement (including one of the authors of the present paper, UIU). Instead

the, admittedly small, mistake is used as a pedagogical point much like the example ’the human failings of genius’

that Ohanian has used in his book about Einstein’s mistakes [1]. Realising that even geniuses make mistakes may

make the scientist more inclined towards critically examining any postulate on his/her own.

II. THE CENTER OF THE EARTH IS YOUNGER THAN ITS SURFACE

A. Homogeneous Earth

We initially suppose that the object under consideration is a sphere with radius R and mass M, homogeneously

distributed. Its gravitational potential as a function of distance r to its center is then given by

Φ = −G

r ≥ R (1)

Φ = −G

M(3R

− r

)

r ≤ R (2)

arXiv:1604.05507v1 [physics.ed-ph] 19 Apr 2016

such that the potential on its surface is Φ(R) = −GM/R and the potential in its center is Φ(0) = −3GM/2R. The

diﬀerence between the gravitational potential at the center and at the surface is then

∆Φ = Φ(R) − Φ(0) =

. (3)

A diﬀerence in gravitational potential implies a time dilation at the point with the lower potential. This is given by

the standard ’gravitational redshift’

ω = ω

(1 −

∆Φ

), (4)

which here relates the (angular) frequencies at the center, ω, and at the surface ω

. Being the inverse of the period, the

frequency is indirectly a measure of how quickly time passes. It is customary to use the symbol ω in this connection,

and we emphasise that this variable has nothing to do with the Earth rotation.

We combine equation (4) with the result for ∆Φ in equation (3) and use that ∆ω = ω − ω

∆ω = −

Gω

(5)

We note that this treatment is based on equation (4) which ”...refers only to identically constructed clocks located

at diﬀerent distances from the center of mass of a gravitating body along the lines of force. All that is required is

that the clocks obey the weak equivalence principle [...] and the special theory of relativity.” [3]. See Ref. [4] for a

recent, instructive example that can be easily performed in the undergraduate laboratory to display one aspect of the

equivalence principle.

For the case of the Earth, upon rewriting and setting the surface acceleration GM

= g = 9.82 m/s

, with R

being the Earth radius, equation (5) becomes

∆ω

= −

, (6)

such that the Earth mass M

and the gravitational constant G are not needed explicitly.

For the sake of a back-of-the-envelope calculation we may exploit that c/g ' 1 year (within 3%, although there is

no direct connection between the motion of the Earth around the Sun and c). The Earth age is T

= 4.54 · 10

years

and its average radius is R

= 6371 km so that R

/2c is approximately 10 ms. A year is approximately π · 10

Clearly, the use here of π is a mnemonic device, not an expression of precision, although it is precise to about half a

percent (one could use 3 instead of π which, however, is imprecise to 5 percent). Thus the diﬀerence between the age

of the Earth surface and its center becomes approximately 4.5 · 10

· 10

−2

/π · 10

' 1.4 years, with the center being

youngest. This is the type of ’back-of-the-envelope’ calculation that one could imagine that Feynman had in mind

when he expressed his ”...since the center of the earth should be a day [which thus should have read ’year’] or two

younger than the surface!” [2]. Where the mistake actually entered in the lecture and transcription process is unlikely

to ever be ascertained, and its exact origin is not important for the following discussion.

With tabulated values for M

, G, R

, c and T

a more precise number for the homogeneous Earth is obtained:

∆T

= T

= 1.58 years, (7)

with the center being youngest.

B. Realistic Earth

Rather than assuming a homogeneous Earth, we now turn to a more realistic density distribution. This yields a

signiﬁcantly diﬀerent result and reveals some insights to the origin of the time diﬀerence. A rather precise description,

but not the only one available, of the Earth density proﬁle is tabulated in the so-called ’Preliminary Reference Earth

Model’ (PREM) [5]. Very recently, the PREM has been applied to give a detailed description of the Earth ’gravity

tunnel’ problem [6].

We shall consider a spherically symmetric Earth with a density only dependent on radius, ρ(r), as given by the

PREM, see Figure 1. The gravitational potential caused by this sphere is then given by

0 1000 2000 3000 4000 5000 6000

2000

4000

6000

8000

10000

12000

r [km]

density [kg/m

]

PREM

Homogeneous

FIG. 1: The density of the Earth as a function of distance to the Earth center for two diﬀerent models. The blue line shows

the PREM of the Earth density and the red curve is the constant density in the approximation of a homogeneous Earth with

mass M

Φ(r) = −

∞

grav

, (8)

where

grav

/m is the mass speciﬁc force, or acceleration ~a

grav

, due to gravity, with

grav

being the gravitational

force (which is why Φ is the gravitational potential and not the gravitational potential energy). The gravitational

potential energy is equal to the work done by taking a test particle of mass m from inﬁnity to a distance r away from

the center of the Earth.

We split the expression in two parts:

Φ(r) = −

grav

−

∞

grav

, (9)

with the ﬁrst term being the work per unit mass done inside the object - in this case the Earth - and the last term the

work per unit mass done moving the test particle from inﬁnity to the Earth surface. The gravitational acceleration

at a distance, r, outside a sphere of mass M is ~a

grav

= −ˆrGM/r

, the sign showing that it is directed towards

the center. Inside the sphere, when r < R, only the mass closer than r to the center matters. We denote this by

M(r) =

4πr

ρ(r

)dr

. (10)

Now we can write the sum speciﬁcally for r < R as

Φ(r) =

M(r

)

− G

(11)

where the last term is the potential at the surface of the object. The integrand in the ﬁrst term is the gravitational

acceleration as a function of r. When evaluated at the surface, r = R

, the result is the normal gravitational

acceleration, g. This can be seen on Figure 2 where the acceleration felt at diﬀerent distances to the Earth center is

shown. Due to the mass distribution ’kink’ seen in Figure 1 at a radius of about 3500 km, the acceleration becomes

almost constantly equal to its surface value from this radius, outwards.

Using the PREM density distribution ρ(r

) in eq. (10) as an input to eq. (11), the more elaborate result for the age

diﬀerence of the Earth center and the surface is

∆T

= 2.49 years, (12)

with the center being youngest.

As a, perhaps, intriguing side-eﬀect, we show the time diﬀerence as a function of radius, see Figure 3.

As expected, the two theories predict similar time diﬀerences near the surface of the Earth. Closer to the center,

the PREM yields a larger result than the homogeneous distribution. This is because M(r)

PREM

> M(r)

Homog.

for

small r. In fact, assuming for simplicity that the object of radius R consists of a region of high density for 0 ≤ r ≤ r

0 1000 2000 3000 4000 5000 6000

r [km]

acceleration [m/s

]

PREM

Homogeneous

Analytical

FIG. 2: The size of the gravitational acceleration as a function of distance to the Earth center. It reaches the familiar value

of 9.82 m/s

at the surface. The analytical curve is given by the simple scaling g · r/R

by assuming a homogeneous mass

distribution.

and zero density for r

< r ≤ R, respecting that the total mass equals M, and with r

= k

R, 0 < k

≤ 1, the

potential diﬀerence between center and surface becomes

∆Φ = (

− 2)∆Φ

, (13)

where ∆Φ

is that of the homogeneous distribution. Thus, the factor 3/k

− 2 yields the increase in time diﬀerence

compared to the homogeneous model. So for the Earth, where this approximation is rather crude, we may set r

to be 3480 km as seen from the PREM curve in Figure 1, i.e. k

' 3480 km/6371 km ' 0.55 such that ∆Φ '

3.5∆Φ

somewhat above the factor 1.7 obtained by the numerical method, as expected from the crudeness of this

approximation.

0 2000 4000 6000 8000

r [km]

Δ T [years]

PREM

Homogeneous

FIG. 3: The ﬁgure shows the time diﬀerence between a point inside the Earth and the surface (center) of the Earth. The blue

curve is calculated with PREM and the red curve using a homogeneous mass distribution.

We end this section by showing that the time dilation due to the rotational speed of the surface of the Earth makes

a negligible contribution. The surface speed is given from the period of rotation as v

= R

2π/T

' 464 m/s where

= 86.164, 099 s is the stellar day (the Earth rotation period with respect to the ’ﬁxed’ stars). Since the time

dilation in special relativity is given from the Lorentz factor γ = 1/

1 − v

' 1 + v

/2c

as ∆T = T v

/2c

get with T

that ∆T

' 5 · 10

−3

years, which can be neglected in the present discussion.

III. THE CASE OF THE SUN

Clearly, the calculations performed in connection with the Earth can be performed for essentially any other cosmic

object with known mass and radius, at least in the limit of a homogeneous mass distribution. However, we limit the

additional cases to our cosmic neighbourhood, i.e. to that of the Sun, in order to demonstrate the applicability of eq.

(13). For the Sun, in analogy with the PREM which is based on seismic data, we choose the so-called ’Model S’ for

its density distribution, a model in good agreement with helioseismic data [7].

0 1 2 3 4 5 6 7

x 10

0.5

1.5

x 10

Sun density

r [km]

density [kg/m

]

Sun S

Homogeneous

FIG. 4: The density of the Sun as a function of distance to the Sun center for two diﬀerent models. The blue line shows the

’Model S’ of the Sun density, obtained from helioseismic data [7], and the red curve is the uniform density Sun.

In the homogeneous case, the age diﬀerence between the Sun center and surface, which can be rewritten as ∆T =

esc

/4c

with v

esc

2GM/R being the surface escape velocity, is

∆T

= 4.8 · 10

years, (14)

whereas with the ’Model S’ solar model it becomes

∆T

= 3.9 · 10

years, (15)

see Figure 5.

The factor of 8.0 diﬀerence between these two numbers is substantially larger than that between the same two

numbers for the Earth. This is a result of the Earth being relatively homogeneous while for the Sun, a signiﬁcantly

larger part of its mass is located close to its center. Using eq. (13) and approximating k

' 2.5/7 from the density

distribution, the Model S curve in Figure 4, we get a factor 8.4, a much better approximation than for the case of the

Earth.

IV. DISCUSSION AND CONCLUSION

As a ﬁnal discussion we address the question: why did famous, respectable and clever physicists publish Feynman’s

claim (although not verbatim, actually) that ”[Feynman] concluded that the center of the Earth should be ’a day

or two younger than its surface’” [8], or ”[Feynman] concludes that the center of the Earth should be by a day or

two younger than its surface” [9] and reversely ”’atoms at the surface of the earth are a couple of days older than at

its center’”, in the latter case even with the comment ”this was conﬁrmed by airplane experiments in 1970s” [10]?

And why did other, equally talented physicists not correct that particular mistake in the foreword to the transcribed

lectures, in spite of quite extensive discussions, spanning 24 pages, of among other things a few misconceptions etc.

[11]? Not to mention the transcribers - postdocs with Feynman - who, along the way, probably have corrected a few

mistakes here and there? Or the editor, who also provided introductory notes on quantum gravity [12]? Why did one

of us (UIU), repeat the same mistake in a science book for the layman [13]?

This, of course, was not because any of these physicists were unable to check the original claim, or found it

particularly laborious to do so. Instead, it seems likely that they knew that the qualitative eﬀect had to be there, and

simply trusted that Feynman and his transcribers had got the number right. This is here considered an example of

’proof by ethos’ [14].

The term ’proof by ethos’ refers to cases where a scientist’s status in the community is so high that everybody else

takes this person’s calculations or results for granted. In other words, nobody questions the validity of that scientist’s

0 2 4 6 8

x 10

r [km ]

Δ T [years]

Sun S

Homogeneous

FIG. 5: The ﬁgure shows the time diﬀerence between a point inside the Sun and the surface (center) of the Sun. The blue

curve is calculated with the ’Model S’ density model for the Sun and the red curve using a homogeneous mass distribution.

claim because of the particular ethos that is associated with that person. The result is accepted merely by trust.

Indeed, the proof by ethos is not really a proof as it does not follow logically from a set of premises. But it is a proof

in the sense that it is persuasive, and tells us something about how scientists work in practice when they accept a

calculation or an experimental result. Scientists must to a large extent rely on the validation of other fellow’s work,

and it happens to be a psychological default condition among many (scientists), that if a famous peer has publicly

announced a result, it is accepted at face value. This seems also to be the situation in the case of the ﬂawed estimate

of the relativistic age of the Earth’s core.

In science, one route to becoming famous is being right on some important topics. However, just because

someone has become famous, this person is evidently not necessarily right on all matters. Feynman himself would

most likely have agreed with this and he would probably not have fallen for his own miscalculation: For a long

time, his own theory of beta decay was at odds with the then prevalent, but false, understanding of existing

experimental results. Upon ﬁnally realizing and correcting this community-wide misunderstanding Feynman wrote:

”Since then I never pay any attention to ”experts”. I calculate everything myself.” [15]. And when faced with

a mistake of his own, he put it even more bluntly: ”What it says in the book [I have written] is absolutely wrong!” [16].

In spite of the small numerical mistake, Feynman’s observation that the center of the Earth is younger than its

surface is a fascinating demonstration of time dilation in relativity, and as such a very illustrative example for use in

the classroom.

[1] H.C. Ohanian, Einstein’s mistakes, the human failings of genius, (W.W. Norton 2008)

[2] R.P. Feynman, F.B. Morinigo og W.G. Wagner, Feynman Lectures on Gravitation, edited by B. Hatﬁeld, (Westview Press

Advanced Book Program, 2003).

[3] A.M. Nobili et al., ”On the universality of free fall, the equivalence principle, and the gravitational redshift,” Am. J. Phys.

81, 527-536 (2013).

[4] R.S. Christensen et al., ”Laboratory test of the Galilean universality of the free fall experiment” Phys. Ed. 49, 201 (2014).

[5] Adam M. Dziewonski and Don L. Anderson, ”Preliminary reference Earth model,” Physics of the Earth and Planetary

Interiors 25, 297-356 (1981).

[6] A.R. Klotz, ”The gravity tunnel in a non-uniform Earth,” Am. J. Phys. 83, 231-237 (2015).

[7] J. Christensen-Dalsgaard, et al., ”The Current State of Solar Modeling,” Science 272, 1286 - 1292 (1996); available at

http://astro.phys.au.dk/∼jcd/solar models/.

[8] L.B. Okun, K.G. Selivanov and V.L. Telegdi, ”Gravitation, photons, clocks,” Phys. Usp. 42, 1045-1050 (1999).

[9] L.B. Okun, K.G. Selivanov and V.L. Telegdi, ”On the interpretation of the redshift in a static gravitational ﬁeld,” Am. J.

Phys. 68, 115-119 (1999).

[10] L.B. Okun, ”Photons, clocks, gravity and the concept of mass,” Nucl. Phys. B (Proc. Suppl.) 110, 151-155 (2002).

[11] J. Preskill and K.S. Thorne, ”Foreword,” foreword to [2].

[12] B. Hatﬁeld, ”Quantum gravity,” Introductory pages to [2].

[13] U.I. Uggerhøj, Tid, Tænkepauser 17 (in danish), (Aarhus University Press 2014).

[14] J. Faye, The nature of scientiﬁc thinking. On interpretation, explanation and understanding, (Palgrave Macmillan 2014).

[15] R. Feynman, R. Leighton and E. Hutchings, Surely You’re Joking Mr. Feynman!, (W W Norton 1985).

[16] T. Hey, ”Richard Feynman and computation,” Contemp. Phys. 40, 257-265 (1999).

Discussion

The Earth's density profile, as described in the Preliminary Reference Earth Model (PREM), is a detailed, 1D model that provides the radial distribution of density, seismic velocities (P-wave and S-wave speeds), and attenuation throughout Earth's interior. PREM was developed by A. Dziewonski and D. Anderson in 1981 and serves as a standard for geophysical studies. To create PREM, researchers use seismic data, particularly travel times of P- and S-waves from earthquakes, which vary depending on the material properties of the Earth's layers. These variations are analyzed to infer the density and velocity of seismic waves at different depths. PREM assumes radial symmetry, averages over lateral variations, and accounts for factors like phase transitions, anisotropy, and temperature effects to ensure the model is as accurate as possible. ### TL;DR In this paper, a group of scientists from Denmark revisited a claim by Richard Feynman, calculating that Earth’s core is approximately 2.5 years younger than its surface due to gravitational time dilation. In the paper U.I. Uggerhøj, R.E. Mikkelsen and J. Faye describe the methods used to arrive at this result. Feynman famously suggested during a 1960s Caltech lecture that time dilation, an effect of general relativity, would make the core younger than the crust by "a day or two". While the general concept was accepted by physicists, his estimate went largely unchecked until recently. Gravitational time dilation occurs because massive objects like planets warp spacetime, causing time to move more slowly in regions of stronger gravitational pull. ## Inside the sphere $(r \le R)$ ### Mass enclosed inside radius $r$ For a sphere of total radius $R$ and **uniform mass density** $\rho$, the total mass is \[ M = \rho \,\frac{4\pi R^{3}}{3}. \] If we consider a smaller sphere of radius $r < R$, the enclosed mass is \[ M(r) = \rho \,\frac{4\pi r^{3}}{3}. \] Since $\rho$ is the same everywhere, we can relate $M(r)$ to the total mass $M$ by \[ M(r) = M \,\frac{r^{3}}{R^{3}}. \] ### Gravitational field inside the sphere By **Newton’s shell theorem**, only the mass enclosed within radius $r$ contributes to the gravitational field at $r$. For $r < R$, the radial component of the gravitational field $g(r)$ is \[ g(r) = \frac{G\,M(r)}{r^{2}} = \frac{G}{r^{2}} \left(M \,\frac{r^{3}}{R^{3}}\right) = G\,M \,\frac{r}{R^{3}}. \] This $g(r)$ points inward. ### Relation between potential $\Phi$ and field $g$ The gravitational field $g(r)$ is related to the potential $\Phi(r)$ by \[ g(r) = -\frac{d\Phi}{dr}. \] Hence, \[ -\frac{d\Phi}{dr} = G\,M \,\frac{r}{R^{3}} \quad\Longrightarrow\quad \frac{d\Phi}{dr} = -\,G\,M \,\frac{r}{R^{3}}. \] ### 2.4 Integrate to find $\Phi(r)$ To find $\Phi(r)$ for $r \le R$, integrate from $r$ to $R$ (using that $\Phi(R) = -\tfrac{G\,M}{R}$ must match the external solution at $r=R$): \[ \Phi(r) - \Phi(R) = \int_{R}^{r} \frac{d\Phi}{dr'} \, dr' = \int_{R}^{r} \left[-\,G\,M \,\frac{r'}{R^{3}}\right] dr'. \] Be mindful of the limits; it is often easier to reverse them and adjust the sign: \[ \Phi(r) = \Phi(R) - \int_{r}^{R} G\,M \,\frac{r'}{R^{3}} \, dr'. \] 1. **At $r = R$:** \[ \Phi(R) = -\,\frac{G\,M}{R}. \] 2. **Perform the integral:** \[ \int_{r}^{R} \frac{r'}{R^{3}} \, dr' = \frac{1}{R^{3}} \int_{r}^{R} r' \, dr' = \frac{1}{R^{3}} \left[ \frac{r'^{2}}{2} \right]_{r}^{R} = \frac{1}{R^{3}} \,\frac{R^{2} - r^{2}}{2} = \frac{1}{2\,R} \left(1 - \frac{r^{2}}{R^{2}}\right). \] Putting it all together: \[ \Phi(r) = -\frac{G\,M}{R} - G\,M \left[ \frac{1}{2\,R} \left(1 - \frac{r^{2}}{R^{2}}\right) \right] = -\frac{G\,M}{R} - \frac{G\,M}{2\,R} \left(1 - \frac{r^{2}}{R^{2}}\right). \] Simplify: \[ \Phi(r) = -\frac{G\,M}{2\,R} \left( 3 - \frac{r^{2}}{R^{2}} \right) \quad\text{for}\; r \le R. \] Alternatively: \[ \Phi(r) = -\frac{G\,M}{R} \left(\frac{3}{2} - \frac{r^{2}}{2\,R^{2}} \right). \] \[ 5 \times 10^{-3} \,\text{years} = 0.005 \,\text{years} = 0.005 \times 365 \,\frac{\text{days}}{\text{year}} = 1.825 \,\text{days}. \] It's interesting to note that the time dilation due to the rotation of Earth's surface alone is very close to the initial time difference suggested by Feynman. Here's the page from "Feynman's Lectures on Gravitation" in which he mentions it. ![](https://i.imgur.com/WeCBMn4.png) The Sun's density profile, such as the Model S, is calculated using helioseismic data and theoretical solar models. Helioseismology observes oscillation frequencies on the Sun's surface, which are sensitive to internal properties like density. Then they using theoretical models, based on equations of stellar structure (hydrostatic equilibrium, energy transport, and the equation of state), to get an initial density profile. These models are iteratively refined by comparing predicted and observed oscillation frequencies.

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