Checking http://tinyurl.com/zluy8x7
Accuracy of GR prediction is 10^-3. for Mercury
Using my criterion for the accuracy of the Geodesic equation, i.e.
r/(r^3/rs)^1/2 ~ 10^-3
r ~ 10^12 cm mean distance Mercury to Sun
rS ~ 10^5 cm (Sun)
This is same order of magnitude of the latest observed accuracy.
Therefore, future increases in accuracy of the advance of the perihelion of Mercury can Popper falsify my picture of the physical meaning of Einstein’s mathematics.
On Jan 26, 2016, at 2:47 PM, Academia.edu <This email address is being protected from spambots. You need JavaScript enabled to view it. > wrote:
Hi Jack,
Congratulations! You uploaded your paper 2 days ago and it is already gaining traction.
Total views since upload:
You got 118 views from the United Kingdom, Italy, Canada, the United States, Mexico, the Philippines, Croatia, India, Algeria, Australia, Hungary, Germany, Romania, Serbia, Slovenia, and Ghana on "BASIC PHYSICAL IDEAS OF EINSTEIN'S GENERAL RELATIVITY V3 - Sarfatti Lecture".
Thanks,
The Academia.edu Team
The Levi-Civita connection is mathematically a non-tensor local field at the position of the test particle. We are dealing with local differential equations in Einstein’s GR.
This is an idealization of course. In fact, the real observer’s separation from the test particle must be small compared to the local radii of spacetime curvature for the test particle local differential equations of motion to be an accurate description. For idealized non-rotating spherical source M
The geodesic equation is essentially Newton’s first law of motion seen from an arbitrarily accelerating Local Non-Inertial Frame Observer-Detector LNIF
DV/ds = 0
DV/ds = dV/ds - {Levi-Civita}V^2
Test particle’s local proper acceleration = test particle’s relative kinematical acceleration - detector’s actual local proper acceleration
This local differential field equation is more and more accurate the smaller the ratio of the separations of test particle from detector to the local radii of curvature at the test particle caused by the source Tuv.
V = 4-velocity of test particle relative to detector
In the weak field slow speed Newtonian limit
dV/ds —> d^2r/dt^2
{Levi-Civita}V^2 —> -GMr/r^3
A = 4pir^2
r = Schwarzschild radial coordinate =/= proper radius though the error is small here.
The observer is idealized to be at r looking at the test particle coincident with him.
“coincident” here means “approximately coincident,” i.e., the actual separation traveled by light signals from test object to detector is small compared to radii of curvature at the location of the test object caused by the source Tuv.
For a test particle on a timelike geodesic its local proper acceleration is zero. It is only the LNIF observer that has a real proper off-geodesic acceleration. Therefore, the test particle’s relative acceleration is actually that of the detector’s proper acceleration.
i.e. radii of curvature ~ [(Schwarzschild Radius)/(Area of Spherical Surface at position of test particle relative to center of mass of source)^3/2]^-1/2
For Sun :
Schwarzschild radius = 10^5 cm
“Distance” Earth from Sun = 10^13 cm
10^5/10^39 = 10^-34
Sun’s radii of curvature at Earth’s orbit ~ 10^17 cm ~ ten thousand AU
Therefore, the error in treating the geodesic equation as a local differential equation is ~ 10^-4
i.e. with Earth as the test particle, the separation of Earth from Sun is small compared to the local dominating radii of curvature caused by the Sun.
Next consider the orbit of the Moon around the Earth completely neglecting influence of Sun for now - that would be a three body problem BTW
Earth’s Schwarzschild radius = 0.5 cm
Distance Moon to Earth ~ 3.8 x 10^10 cm
~ (3.8 x 10^10)^3/2 ~ 10^16 cm
so the relative error is ~ 10^-6
For an artillery shell near Earth’s surface with observer a static LNIF at Earth’s surface
r ~ 6 x 10^8 cm
radii of curvature ~ 6^3/2 x 10^12 cm
so the relative error is ~ 10^-3.