Off Topic: The True Gravitational Geodesic

• Academia / Research

Introduction

This is a followup on my "Refraction in a Varying Media" article [1] taking it from a refractive scale to a gravitational scale. It is the answer to my quest in the "Speculations" section to find a vector differential equation for the trajectory of a photon in a gravity field. It is also a correction to my earlier assertion that the natural log of the effective index of gravity, known as refractive density, is exactly proportional to gravitational potential. There is a small adjustment factor which makes them approximately proportional.

Index of Refraction

The index of refraction $n$ is the factor by which a theoretical refractive particle slows down from a maximum possible speed. The magnitude of the local velocity of light $\vec v$ is the local speed $c$, a fraction of the full speed in a vacuum $c_0$. $$ \|\vec v(x,y,z)\| = c(x,y,z) = \frac{c_0}{n(x,y,z)} \tag {1} $$ This gives the definition of a static refractive field.

Refractive Density

Refractive density $\rho$ is defined as the natural log of the index of refraction. The gradient of the refractive density can then be expressed in terms of the gradient of index of refraction normalized by the index of refraction. $$ \rho = \ln(n) \implies n = e^{\rho}, \; \vec \nabla \rho = \frac{ \vec \nabla n }{n} \tag {2} $$ The local speed of light can then be expressed as a function of the refractive density. $$ \|\vec v\| = c = c_0 e^{-\rho} \tag {3} $$

Ideal Refractive Acceleration

The Law of Ideal Refractive Acceleration is derived in my first Physics article [1]. It can be stated in both the index of refraction scale and the refractive density scale. $$ \begin{aligned} \frac{d \vec v}{dt} &= \left( \vec v \cdot \vec v \right) \frac{ \vec \nabla n }{n} - 2 \left( \frac{ \vec \nabla n }{n} \cdot \vec v \right) \vec v \\ &= \left( \vec v \cdot \vec v \right) \vec \nabla \rho - 2 \left( \vec \nabla \rho \cdot \vec v \right) \vec v \\ \end{aligned} \tag {4} $$ It describes the acceleration felt by a particle in the refractive field as a linear combination of the gradients and the velocity. This means that the acceleration will lay in the plane, or maybe line, formed by the gradient and the velocity.

Empirical Gravity Assumption

It is assumed when a photon goes by the Sun at the closest approach that the acceleration felt is exactly twice the local gravity gradient. This is based on empirical observation.

Applying (4) to a refractive particle traveling at the local speed of light nearest the source gives a conversion factor between the gravitational gradient and refractive density gradient. $$ \begin{aligned} 2 \vec g &= \left( \vec v \cdot \vec v \right) \vec \nabla \rho - 2 \left( \vec \nabla \rho \cdot \vec v \right) \vec v \\ &= \left( \| \vec v \|^2 \right) \vec \nabla \rho - 2 \left( 0 \right) \vec v \\ &= c^2 \vec \nabla \rho \end{aligned} \tag {5} $$ At the point of closest approach, the velocity is perpendicular to the gradient so it zeroes out the second term.

Gravitational Geodesic

(5) can be solved for the refractive density gradient. $$ \vec \nabla \rho = \frac{2}{c^2} \vec g \tag {6} $$ This can then be plugged back into (4). $$ \begin{aligned} \frac{d \vec v}{dt} &= \left( \vec v \cdot \vec v \right) \frac{2}{c^2} \vec g - 2 \left( \frac{2}{c^2} \vec g \cdot \vec v \right) \vec v \\ &= 2 \left[ \left( \frac{\vec v}{c} \cdot \frac{\vec v}{c} \right) \vec g - 2 \left( \vec g \cdot \frac{\vec v}{c} \right) \frac{\vec v}{c} \right] \\ &= 2 \left[ \left( \vec u_v \cdot \vec u_v \right) \vec g - 2 \left( \vec g \cdot \vec u_v \right) \vec u_v \right] \\ &= 2 \left[ \vec g - 2 \left( \vec g \cdot \vec u_v \right) \vec u_v \right] \\ & \text{ where } \vec u_v = \frac{\vec v}{c} \\ \end{aligned} \tag {7} $$ This is the Law of Ideal Refractive Acceleration expressed in gravitational terms. Notice that the speed doesn't matter, only the direction of the velocity. This is the true gravitational geodesic vector differential equation for a photon traveling through a gravity field.

There is a nice graphical interpretation shown in the Appendix. If a mirror is placed perpendicular to the trajectory, then the acceleration felt by the photon is twice the mirror image of the gradient. Therefore the magnitude of the acceleration vector is always twice the magnitude of the gradient, or $\|2\vec g\|$. This can be confirmed by calculating the magnitude directly from the definition. $$ \begin{aligned} \left\| \frac{d \vec v}{dt} \right\| &= \left( \frac{d \vec v}{dt} \cdot \frac{d \vec v}{dt} \right)^{1/2} \\ &= \left( 4 \left[ \vec g - 2 \left( \vec g \cdot \vec u_v \right) \vec u_v \right] \cdot \left[ \vec g - 2 \left( \vec g \cdot \vec u_v \right) \vec u_v \right] \right)^{1/2} \\ &= \left( 4 \left[ \vec g \cdot \vec g - 4 \left( \vec g \cdot \vec u_v \right) \vec g \cdot \vec u_v + 4 \left( \vec g \cdot \vec u_v \right)^2 \vec u_v \cdot \vec u_v \right] \right)^{1/2} \\ &= \left( 4 \; \vec g \cdot \vec g \right)^{1/2} \\ &= \| 2 \vec g \| \\ \end{aligned} \tag {8} $$

Gravitational Acceleration

(5) can be solved for the gravity acceleration instead and combined with (3). $$ \vec g = \frac{c^2}{2} \vec \nabla \rho = \frac{c_0^2}{2} e^{-2\rho} \vec \nabla \rho \tag {9} $$ This shows that the gravitational acceleration is dependent on the refractive density as well as its gradient.

True Gravitational Potential

There is a potential function for the gravitational acceleration, thus it is also a gradient. The constant of integration is set so that the potential function is zero when the refractive density is zero. $$ \begin{aligned} \vec \nabla Y &= \vec g = \frac{c_0^2}{2} e^{-2\rho} \vec \nabla \rho \\ Y &= \frac{c_0^2}{4} \left( 1 - e^{-2\rho} \right) = \frac{1}{4} \left( c_0^2 - c^2 \right) \\ \end{aligned} \tag {10} $$ $Y$ is being used instead of $\Phi$ because it is slightly different and the negative convention for potential energy is not used.

An interesting property is that the potential will reach a finite maximum as the limit of the refractive density approaches infinity and the local speed approaches zero.

Central Mass Sphere

Suppose that the refractive density imposed by a spherical mass in space decreases by the inverse of the distance to the center beyond the physical surface. The value of $k$ represents the field strength. $$ \rho(r) = \frac{k}{r} \tag {11} $$ This can then be plugged into the definition of the potential in (10). $$ Y(r) = \frac{c_0^2}{4} \left( 1 - e^{-2\frac{k}{r}} \right) \tag {12} $$ Observe that the linear Taylor approximation of the exponential has the same limit as the distance goes to infinity and the refractive density goes to zero. $$ \lim_{r \to \infty } e^{-2\frac{k}{r}} = \lim_{r \to \infty } 1 - 2\frac{k}{r} \tag {13} $$ This gets plugged into (12) and set equal to the Newtonian definition. $$ \lim_{r \to \infty } Y(r) = \lim_{r \to \infty } \frac{c_0^2}{4} \left( 2 \frac{k}{r} \right) = \frac{G_0 M}{r} \tag {14} $$ The limit can then be rearranged to isolate the field strength constant. $$ \lim_{r \to \infty } k = \frac{2 G_0 M}{c_0^2} \implies k = \frac{2 G_0 M}{c_0^2} = r_s \tag {15} $$ Since the constant is independent of $r$, the limit disappears and the value remains. This value turns out to be the standard definition of the Schwarzschild radius $r_s$.

Central Gravitational Gradient

The gradient of (11) can be taken giving an inverse square law. $$ \vec \nabla \rho = -\frac{k}{r^2} \vec u_r = -\frac{r_s}{r^2} \vec u_r \tag {16} $$ This can then be plugged into (9). $$ \vec g = \frac{c_0^2}{2} e^{-2\frac{r_s}{r}} \left(-\frac{r_s}{r^2} \vec u_r\right) = \frac{G_0 M}{r^2} e^{-2\frac{r_s}{r}} \left(-\vec u_r\right) = \frac{G M}{r^2} \left(-\vec u_r\right) \tag {17} $$ This is where the variation in Newton's constant comes in. $$ G = G_0 e^{-2\frac{r_s}{r}} = G_0 e^{-2 \rho } \tag {18} $$ $G_0$ is Newton's constant in zero refractive density and $G$ is a local lower value. They are called $G$ and $G'$ respectively in my Fluidic Physics article [2]. $$ r_s = \frac{2 G_0 M}{c_0^2} = \frac{2 G_0 e^{-2 \rho } M}{\left(c_0 e^{- \rho }\right)^2} = \frac{2 G M}{c^2} \tag {19} $$ The Schwarzschild radius can be calculated using either coordinates. This confirms that the gravitational constant needs to vary by the square of the speed of light factor.

Motion Enhanced Gravity

The first big result in my Fluidic Physics article [2], occurring prior to the fluidic hypothesis, is a new formula for gravity which includes motion relative to a static gravity field. First the velocity is normalized on a speed of light scale so the magnitude will vary from zero to one. $$ \vec D = \frac{\vec v}{c} \tag {20} $$ It is then presumed that a mass particle consists of refractive particles in structured bounded trajectories around each other. The net sum of the refractive particle accelerations form a gravitational acceleration of the mass particle. This acceleration is enhanced by motion relative to the gravity field. $$ \vec a = \left( 1 + \vec D \cdot \vec D \right) \vec g - 4 \left( \vec g \cdot \vec D \right) \vec D \tag {21} $$ When the particle is standing still the speed is zero and the acceleration is the gravity gradient. If it were able to get up to the speed of light it would behave according to the gravitational geodesic equation just like light. This means the math is the same for light having mass or not. $$ \begin{aligned} \| \vec D \| &= 0 \implies \vec a = \vec g \\ \| \vec D \| &= 1 \implies \vec a = 2 \vec g - 4 \left( \vec g \cdot \vec D \right) \vec D = 2 \left[ \vec g - 2 \left( \vec g \cdot \vec u_v \right) \vec u_v \right] \\ \end{aligned} \tag {22} $$ For any trajectory that a point moves to a different spot, the normalized velocity $\vec D$ has to be non-zero for some part and thus the the acceleration is no longer strictly the gradient. So even though a potential function exists, it is only the limit of what a real particle can do in a field as it moves ever slower. Path independence is lost and so is the concept of exact potential energy.

Conclusion

This is a new version of the fundamentals of Physics which completely upends the current paradigm. More is in the earlier article [2]. The big result here is the true gravitational geodesic equation with its remarkable graphical interpretation. The big surprise is that if refractive density follows an inverse square law, which is the natural presumption, then gravity is only approximately inverse square.

If the assumptions hold, I've upset Einstein and Newton. Ain't that something?

If you found this interesting, please contact your favorite Content Creator, or two, and ask them to make a nice presentation.

This article appears in the "Off Topic" section of my overview article.

References

[1] Dawg, C., "Off Topic: Refraction in a Varying Medium", 2018
[2] Dawg, C., "Off-Topic: A Fluidic Model of the Universe", 2022

Appendix - Gravitational Geodesic Diagram

The long green line is the trajectory of a photon in a gravity field. The blue line represents a mirror placed perpendicular to the trajectory at the current location. The short green vector $\vec u_v$ represents the unit velocity. The yellow vector $\vec g$ represents the gravity gradient at the current location. The gradient is then projected onto the unit velocity vector shown in outline orange $ ( \vec g \cdot \vec u_v ) \vec u_v$. This is then subtracted twice to give the mirror image of the gradient $[ \vec g - 2 ( \vec g \cdot \vec u_v ) \vec u_v ]$ shown in red. Finally, it is doubled to give the gravitational geodesic equation $\frac{d \vec v}{dt} = 2[ \vec g - 2 ( \vec g \cdot \vec u_v ) \vec u_v ]$.

The acceleration felt by a photon in a gravity field is twice the mirror image of the local gradient.

The acceleration can then be decomposed along the trajectory and the perpendicular shown as the pink outline vectors. The lateral acceleration is $2[ \vec g - ( \vec g \cdot \vec u_v ) \vec u_v ]$ shown in blue vectors and the directional acceleration is $-2 ( \vec g \cdot \vec u_v ) \vec u_v $.

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