In this paper, we show that the time dilation field of mass-energy is sufficient to predict gravitational effects on light. Essay written for the Gravity Research Foundation 2021 Awards for Essays on Gravitation. Free-falling into Black Hole We start with a black hole, B, possessing a Schwarzschild radius (rs) of 3000 m, giving it a mass of roughly one solar-mass (~2*10^30 kg). We take a body, A, of negligible mass, initially resting at a great distance (PEi = KEi = 0), and allow it to free-fall towards B. To calculate A’s coordinate velocity at a given distance, r, from the center of B, we start with Please Register or Log in to view the hidden image! (1) Please Register or Log in to view the hidden image! (2) Please Register or Log in to view the hidden image! (3) Please Register or Log in to view the hidden image! (4) Please Register or Log in to view the hidden image! (5) so Please Register or Log in to view the hidden image! (6) Integrating gives us: Please Register or Log in to view the hidden image! (7) We evaluate the equation for the final 1000 meters of A’s path before reaching the event horizon (i.e. r = 4000..3000, see Fig 2) Please Register or Log in to view the hidden image! (8) Please Register or Log in to view the hidden image! (9) So Please Register or Log in to view the hidden image! (10) Here we take note of the proper velocity of A at r = 4000 (see Fig 2) Please Register or Log in to view the hidden image! (11) We also note that Please Register or Log in to view the hidden image! (12) where t0 is the proper time of events for A, tf is the coordinate time of those same events (for a distant observer), and the radical value is the time dilation factor which approaches 0 as r approaches the event horizon at rs. Please Register or Log in to view the hidden image! <continued...>

<part 2 of 3> Light passing through a graded refractive index A refractive index of a medium is defined as the dimensionless number Please Register or Log in to view the hidden image! (13) where v is the measured velocity of light through that medium. In other words, n can be treated as the reciprocal of an apparent time dilation factor (from the remote observer’s point of view). If we consider a gravitational field as the medium being traversed, then we can use (12) to represent that medium’s refractive index as Please Register or Log in to view the hidden image! (14) where ys is analogous to the Schwarzschild radius of B above. We see that as a light ray R approaches a height of ys the “time dilation factor” approaches zero, and n diverges to infinity. Light in this area is effectively frozen, and, for all intents and purposes, the horizontal boundary of y = ys is an event horizon. Now we take Snell’s Law Please Register or Log in to view the hidden image! (15) where k is a constant determined by the initial angle and location in the medium of an incident ray. Combining (14) with (15) we now have Please Register or Log in to view the hidden image! (16) so Please Register or Log in to view the hidden image! (17) In Figure 1 we are considering theta to be the angle between R and the normal to the x-axis, therefore Please Register or Log in to view the hidden image! (18) We can combine (16-18) to get Please Register or Log in to view the hidden image! (19) We now want to determine k. Since we know from (11) that the body A is approaching B at a velocity of .866025c at r = 4000 (see Fig 2), we choose theta such that the light ray R is approaching B at the same rate at y = 4000. A light ray with a vertical component moving downward at .866025c is doing so at pi/6 radians off the y-axis Please Register or Log in to view the hidden image! (20) Please Register or Log in to view the hidden image! (21) Please Register or Log in to view the hidden image! (22)

<part 3 of 3> All such k-values will be unity when the vertical component of R is equal to a radial free-fall velocity, such as A’s, at a given height. Plugging k = 1 into (19) we have Please Register or Log in to view the hidden image! (23) We now calculate the length of R’s spatial path from y = 4000 to the so-called event horizon ys = 3000. Please Register or Log in to view the hidden image! (24) Please Register or Log in to view the hidden image! (25) Please Register or Log in to view the hidden image! (26) Please Register or Log in to view the hidden image! (27) Please Register or Log in to view the hidden image! (28) So Please Register or Log in to view the hidden image! (29) which is the same value we calculated in (10). Gravity and refraction equivalence We can verify the relationship between (8) and (27) by adjusting (7) Please Register or Log in to view the hidden image! (30) Please Register or Log in to view the hidden image! (31) Please Register or Log in to view the hidden image! (32) where of course the constants of integration are irrelevant to the integral. Conclusion In conclusion, we have shown that the time dilation field of mass-energy predicts gravitational effects on light. Parsimony and equivalence would suggest that this mechanism is sufficient to explain gravitational forces on massive objects as well, possibly in the form of EM mass.

Einstein's 1911 paper on GR Mk I underestimated the gravitational deflection of light precisely because he only accounted for the temporal metric contribution. Something remedied in the final 2015-16 version where the spatial metric now made an equal contribution. It being understood the net result was on a coordinate i.e. 'as seen from afar' basis.

Well, I've actually updated my paper to give a more visual explanation of what's going on: https://docs.google.com/document/d/1RCmoSXd5YbkMHuYT8OwV_gW8uY5nl8BrBTELQevVfNE/edit# Viewed in this manner, light isn't deflected twice as much as matter, but rather matter is deflected "somewhere between 1/2 and 1 times the amount that light is" depending upon velocity.

That statement cannot be generally correct. Over a given infinitesimal time interval a light particle and matter particle each at the same initial location and moving transverse to g direction in a gravitational field, both experience the same infinitesimal fractional change in radially directed momentum. It's only as the matter particle becomes ultrarelativistic that it's trajectory approaches that of light. At low velocities the radius of curvature is many orders of magnitude less. An orbiting satellite is an obvious counterexample to the "somewhere between 1/2 and 1 times the amount that light is" claim. https://home.fnal.gov/~syphers/Education/Notes/lightbend.pdf And for overkill, good old Wikipedia: https://en.wikipedia.org/wiki/Schwarzschild_geodesics

That was terribly worded. I was trying to describe the reason GR predicts twice the curvature of light as Newtonian gravity around the Sun but I need to elucidate this.

Only the temporal component of light curvature can at least in principle be locally measured. Corresponding very closely to Newtonian gravity predicted deflection in weak gravity situations. To get the full deflection as determined from afar, you need to include the spatial metric component. Your opening sentence this thread claims otherwise. Good luck!

Please read the paper if you believe my claim isn’t possible. The math is very straightforward. Here’s the updated paper: https://docs.google.com/document/d/1RCmoSXd5YbkMHuYT8OwV_gW8uY5nl8BrBTELQevVfNE/edit

Sorry RJB but your purely Newtonian expressions in (2) and (3) are already in conflict with the opening premise of a Schwarzschild exterior spacetime, and relativistic energy-momentum, respectively.