Einstein’s Mistake on the Example of Venus Orbit
His wrong postulate about distances and meter changing…
We are familiar with this drawing, based on Einstein’s theory,
representing distances and even the meter unit changing near the Sun. Einstein understood that time slows down near massive objects like the Sun (time near the Sun runs 1.000002 times slower than far away from the Sun), but then his math failed when working with Newton’s mechanics. To fix that, he postulated metric change (changes in lengths/distances/meter), as-if caused by changes in time flow. Here we will focus on the part that failed, and we will show that no metric changes are needed at all, for example, in Venus orbit. I picked Venus because it is located between the Earth’s orbit and the Sun, thus, time on the Earth runs faster than time on Venus, and time on the Sun runs slower than time on Venus, so we get both perspectives, from slower and faster time.
Einstein called time dilation the ratio of time slowness in some area compared to the observer’s time. We denote time dilation by D. From the Earth perspective, D = 1.000002 on the Sun. This ratio just indicates how slower or faster clocks run in different areas:
- D > 1 means that clocks in an area run slower than the observer’s clock
- D < 1 means that clocks in an area run faster than observer’s clock
- D = 1 indicates no difference in speed of time.
Now, to consider both slower and faster time cases, let’s take Venus’ time speed as the baseline (D = 1 for the observer on Venus itself):
and let’s see if time difference causes any problem with radius R of Venus’ orbit, from whatever perspective.
Mechanics works with velocities and accelerations. Velocity is the change in an object’s location per second. What happens if your clock ticks D-times-faster than clocks ticking in the area where the object is located? In our/observer D-times-shorter second, object makes D-times-smaller move than in the local-to-the-object longer second, thus:
★ Apparent or observed velocity = Local velocity / D.
(We use terms apparent and observed interchangeably). Let’s take it further: acceleration is the change in an object’s velocity per second. In D-times-shorter second any change is proportionally smaller. Change in velocities in the observer’s-shorter-second, aka shorter time period, is D-times-smaller than locally observed change in velocities. And that comes on top of already D-times-smaller apparent/observed velocity than the local velocity, thus:
★★ Apparent or observed acceleration = Local acceleration / D².
All relativity in mechanics can be reduced just to these 2 formulas above, which I originally introduced in chapter 1 of Classical Physics Beyond Einstein’s (plus variation of the first formula for the speed of light, which won’t be needed here).
Now, to Newton’s mechanics: it gives us the acceleration formula g = GM/R² on the Venus’ orbit (where G is gravitational constant, M is the mass of the Sun, and R is the radius of the Venus’ orbit), and balance between centripetal and centrifugal forces on the Venus orbit gives us orbital velocity formula: v = √(GM/R). (See explanation below). We use symbol √ for square root. Change in time flow changes velocities and accelerations, thus, change in time speed (time dilation) might throw off the balance between forces acting on the Venus, and shift Venus inside or outside the classical orbit, changing radius R and length of the Venus orbit. That, allegedly, will change the position of Venus, both from the Earth’s and the Sun’s perspective. For this weirdness not to happen, Einstein postulated metric changes (meter unit change). According to Einstein, radiuses increase near the Sun because of curvature in metrics:
Let’s show that time dilation actually does not cause any metric changes in Venus’ orbit, and to see that through, two formulas for apparent speed and apparent acceleration are enough.
- Local-to-Venus acceleration g = GM/R² becomes g = GM/R² / D² (according to ★★), when observed either from the Earth or the Sun (for either D>1 or D<1). Let’s group constants together as G/D²:
★★★ g = (G/D²)×M/R². - Newton’s circular velocity v = √(GM/R) comes from the balance between centrifugal acceleration (which is v²/R, check Wikipedia) and gravity (g = GM/R²): v²/R = GM/R² =>
★★★★ v = √(GM/R).
With G replaced with G/D² in gravitational formula ★★★, we will need to replace G with G/D² in balanced speed formula ★★★★, which gives us formula v = √( (G/D²)×M/R) = √(GM/R ) / D. Therefore, balanced velocity ★★★★ decreases by factor D, because of ★★★. The same decrease by factor D happens to apparent/observed velocity (according to ★) in comparison to local velocity.
With still matching balanced and observed velocities (even when D≠1 — for whatever nonlocal observer), Venus trajectory remains balanced and is viewed the same (as for D=1, by a classical time-dilation-unaware observer), both from the Earth and the Sun, and no need to change the Venus orbit radius and length.
P.S. If you are interested in optical effects caused by time dilation (for example, why light refracts near the Sun, or why remote Universe appears red/infrared), or if you are interested in Mercury orbit precession (elliptical orbits shift, unlike stable circular orbits we have discussed here),
explained without metric changes (just by ★★ and variable D on an elliptical orbit vs. constant D on a circular orbit), check my short 40-page free eBook:
