The Landscape of Time: Clock Rates on the Slope Planets Fall Down

 J. Rogers, SE Ohio

Working draft. Companion to the simulation “Solar System as Moving Deformations.”

Abstract

Planets fall down a slope. This paper asks what the slope is a slope of, and takes the answer to be the rate at which interactions run. The height of the landscape at a coordinate is the rate of interactions there. The slope is how that rate changes from one coordinate to the next. Falling is moving toward slower rates. In this picture an orbiting clock's rate is assembled from two descents: the descent from the network's rest, which the height gives, and the descent along the clock's own loop, which the slope gives. The assembly reproduces the standard weak-field clock numbers: the 3/2 on any circular orbit, the GPS offset of 38.5 microseconds per day, and the periodic eccentricity correction that receivers apply, to the digits shown. The numbers are standard. What is new is the bookkeeping. The 3/2 can be cut as 1 + ½ (height plus speed) or as 2 − ½ (a distance term plus a fixed loop term). Both are correct, and they differ in what they say speed is: a second cause in the first, one side of a trade of height for speed in the second. The paper states what the picture says, what it does not yet do (Mercury's perihelion, and the delay that would have to stand in for space), and how it relates to existing work.

1. Framing: adding the landscape of time

Falling down is free. A body in free fall does nothing, and an accelerometer attached to it reads zero. The one that reads something is the one held still on the ground. What moves a free body is the network of deformations it sits in.

The companion simulation builds that network as one landscape of moving deformations and gives every object a single question: which way is downhill at my coordinates? There is no force between pairs, no tidal term and no separate straight-line default. Closed orbits, equal areas, the period and distance relation and the periods of twenty-one moons follow from the landscape alone.

What that picture leaves open is the slope itself. A slope of what? This paper proposes: of time. The height of the landscape at a coordinate is the rate at which interactions run there, whether those interactions are clocks, chemistry or orbital change. The slope is the change of that rate between neighboring coordinates. Two neighbors that run at different rates cannot stay in step, and what we call a change of speed is that mismatch. Falling is moving toward slower rates. We call the result the landscape of time.

Converting a height, which has units of speed squared, into a rate requires one scale. We use the speed of light, c. That is the only scale this paper adds to the simulation.

2. The rule

For a compact deformation of depth GM, the height at distance r is H = −GM/r. Two readings are available at a coordinate:

• the descent from the network's rest, v_d = √(−2H) = √(2GM/r), which the height gives;
• the slope, GM/r², which holds a body on a loop.

For a clock moving at speed v relative to the local static frame, the rate relative to the network's rest is

ρ = 1 − (v_d² + v²)/(2c²) = 1 + H/c² − v²/(2c²).

A clock held static has v = 0. A clock on a free circular loop has v² = r × slope = GM/r. In the weak field the two Lorentz factors multiply, so the squares of the speeds add.

This is the standard weak-field clock rate. It adds no ingredient. What changes is where each term comes from: v_d² comes from the height at the coordinate, and v² comes from the loop the slope holds the clock on.

3. Circular orbits

On a circular loop the slope holds the speed at v² = r × slope = GM/r, which is half the descent, v_d²/2. Equivalently, a circular orbit at r has the speed of a body falling from rest at 2r down to r, since 2GM(1/r − 1/2r) = GM/r. A circular loop is half a descent.

The clock rate on a circular orbit is therefore ρ = 1 − 3GM/(2rc²), three halves of the height term. Between two circular orbits the offset is

Δρ = (3/2)(GM/c²)(1/r₁ − 1/r₂).

We checked this two ways for pairs of circular orbits: assembling the rate from the height and the slope at each coordinate, and using the closed form. The two agreed to 1 part in 10¹⁶, which is machine precision. The orbit radii were 6,771 km (low Earth orbit), 26,571 km (GPS), 42,164 km (geostationary) and 384,400 km (the Moon).

From	To	Offset (μs per day)
Low Earth orbit	GPS	63.26
Low Earth orbit	Geostationary	71.26
Low Earth orbit	Moon	83.39
GPS	Geostationary	8.00
GPS	Moon	20.14
Geostationary	Moon	12.14

Because the loop speed is fixed by position, a clock on a circular orbit is a function of position alone.

4. One number, two cuts

Take a satellite on a circular orbit at radius r. Its rate shift is 3/2 of GM/(rc²). That number can be cut two ways.

Cut A, 1 + ½. This is the convention in GPS practice. The height gives 1: the clock is deep. The speed gives ½, since v²/2c² = GM/(2rc²). There are two causes. For the GPS clock against the ground they are +45.7 and −7.2 microseconds per day.

Cut B, 2 − ½. The orbit relation v² = GM(2/r − 1/a), where a is the size of the loop, gives v²/(2c²) = (GM/c²)(1/r − 1/(2a)). Adding the height term gives a total of (GM/c²)(2/r − 1/(2a)). That is a distance term with coefficient 2 and a fixed term of −½ that depends only on the size of the loop. On a circle a = r, and the total is 3/2.

Both cuts are correct. Which one you read depends on how you cut the problem. They agree numerically for every free orbit tested here. They disagree about what speed is. In cut A, speed is a second cause that the orbit happens to tie to height. In cut B, speed on a free loop is not a cause at all. It is the other side of a trade of height for speed, and the only entries are the distance and the loop.

The two cuts come apart for a clock whose speed was not bought by falling. A clock held off its loop by lift, such as a flown clock (Hafele and Keating, 1972), or pushed by thrust, has a speed term that is not a trade. In cut B it is a separate entry added to the books. Speed on a loop is free. Speed from a push is its own term.

5. Ellipses trade speed for height

On a loop of size a, the quantity E = ½v² + H = −GM/(2a) is constant. The clock shift is

(−H + v²/2)/c² = (E − 2H)/c² = (2GM/r − GM/(2a))/c².

This is a fixed term plus a term that follows the distance between centers. Three things follow.

• The mean. The time average of 1/r on a Kepler orbit is 1/a, so the mean shift is 3GM/(2ac²) for any eccentricity. Between any two orbits the offset is (3/2)(GM/c²)(1/a₁ − 1/a₂), with a the size of each orbit.
• The periodic part. The clock's departure from its mean is 2(GM/c²)(1/r − 1/a), twice how far the current height sits from the orbit's mean height. Its amplitude is 2√(GMa)·e/c², where e is the eccentricity.
• What position alone gives. Reading the shift as 3GM/(2rc²) with the instantaneous r gives the same mean but misses the periodic part. At e = 0.1 the gap reaches 9.3 × 10⁻¹² in rate, about 200 ns over a half orbit. At e = 0.4 it reaches 5.6 × 10⁻¹¹, about 1.2 μs.

We checked the periodic part by integrating an orbit with a = 26,571 km in the landscape with a leapfrog step of one second, and accumulating the clock. The direct sum agreed with (E − 2H)/c² to 10⁻¹⁸. The periodic amplitude matched the standard receiver correction for eccentric orbits (Ashby, 2003) to the digits shown:

Eccentricity	Periodic amplitude
0.01	22.9 ns
0.1	229.0 ns
0.4	916.1 ns

On a circle, position alone is enough. On an ellipse the clock also needs E, the loop's total. E is not independent of the gradient: it is what speed and height trade within.

6. Where the ground sits

The ground clock is not on a loop. It is held off the loop, so its speed relative to the local static frame is zero and the loop term is absent: ρ = 1 − GM/(Rc²). Against a GPS clock the offset is

(GM/c²)(1/R − 3/(2r)) = 4.458 × 10⁻¹⁰, or 38.5 μs per day,

with R = 6,371 km and r = 26,571 km. Earth's rotation, ignored here, is of order 0.1 μs per day. Treating the ground as an orbit at R and applying the two-orbit form would give 68.6 μs per day. The measured GPS offset is about 38.5, on the side of the held-off reading. Weight is the feeling of being held off the loop, and here it appears as a missing term.

7. What a push changes

A push, such as a rocket splitting off exhaust, changes E and nothing else. The depth-weighted center of the rocket and its exhaust is unchanged. We checked that this holds to second order in the separation: for two parts started side by side, the center stayed within 5.94 × 10⁻⁹, 5.94 × 10⁻⁷ and 5.97 × 10⁻⁵ AU of the single path for separations of 10⁻³, 10⁻² and 10⁻¹ AU after 0.02 years, a factor of one hundred for each factor of ten. For the clock, a push changes the fixed term: a burn that moves a craft to a larger loop a′ changes its mean shift from 3GM/(2ac²) to 3GM/(2a′c²). The standard accounting gives the same number. What the landscape adds is that the push touches only the fixed term.

8. Limits and open problems

1. Weak field. The rate is expanded to first order in GM/(rc²).
2. Inputs. The depth and the profile of each deformation are set by hand, and c is the scale that converts height into rate. None of these is derived.
3. Same numbers as the standard accounting. The clock numbers in this paper are the standard weak-field numbers. The paper does not claim a different clock prediction. The difference lies in how the 3/2 is assembled, and in the distinction between speed that is traded and speed that is pushed. Within the weak field the two cuts cannot be told apart by a clock reading alone.
4. Mercury. The plain sum of deformations gives an advance of 519 arcseconds per century at the simulation's own step, against 574 observed. In the standard parametrized accounting, a theory in which only the clock rate varies and space is flat gives one third of the 43 arcseconds that relativity adds. So the landscape of time alone cannot carry the remainder.
5. Length as delay. The proposed route is that length is a delay in now. The extra delay a signal picks up crossing a deeper region would then have to equal twice what the clock rate alone gives, which is the measured Shapiro delay, confirmed to about 2 parts in 10⁵ by the Cassini radio link. This has not been derived here.
6. The now of a moving deformation. A moving deformation's motion is its own descent in the landscape one level up, so Earth reads the Sun where it is and first-order aberration cancels. Second order is untested.

9. Relation to existing work

The weak-field clock rate, including the 38.5 μs per day for GPS and the eccentricity correction, is standard (Ashby, 2003). Time dilation as speed relative to space falling from rest at infinity is the river model of black holes (Hamilton and Lisle, 2008), which uses the Painlevé–Gullstrand coordinates (Painlevé, 1921; Gullstrand, 1922). The descent from the network's rest in this paper is that speed, and what we add is the loop term and the two cuts of the 3/2. The accounting of how much of Mercury's advance comes from the rate and how much from space is the parametrized post-Newtonian one (Will, 2014). The extra delay across a deep region was first measured by Shapiro (1964).

References

• Ashby, N. (2003). Relativity in the Global Positioning System. Living Reviews in Relativity, 6, 1.
• Bertotti, B., Iess, L. and Tortora, P. (2003). A test of general relativity using radio links with the Cassini spacecraft. Nature, 425, 374–376.
• Gullstrand, A. (1922). Allgemeine Lösung des statischen Einkörperproblems in der Einsteinschen Gravitationstheorie. Arkiv för Matematik, Astronomi och Fysik, 16(8), 1–15.
• Hafele, J. C. and Keating, R. E. (1972). Around-the-world atomic clocks: predicted relativistic time gains. Science, 177, 166–168.
• Hamilton, A. J. S. and Lisle, J. P. (2008). The river model of black holes. American Journal of Physics, 76, 519–532.
• Painlevé, P. (1921). La mécanique classique et la théorie de la relativité. Comptes Rendus de l'Académie des Sciences, 173, 677–680.
• Shapiro, I. I. (1964). Fourth test of general relativity. Physical Review Letters, 13, 789–791.
• Will, C. M. (2014). The confrontation between general relativity and experiment. Living Reviews in Relativity, 17, 4.