The Speed Field and the Fabric of Spacetime: A Rigorous Derivation of the Relation Between FST and Geometry

1. Introduction

The Fundamental Speed Theory (FST) [1] is based on a single axiom: motion is the primary essence of existence. (The core formulation is currently available as a preprint; peer review status should be checked.) In this framework, gravity and inertia are not intrinsic properties of mass but emergent manifestations of the interaction between matter and the velocity gradient of a fundamental dimensionless vector field ${\nu }^{\mu }$.

A fundamental question arises: What is the relationship between the speed field ${\nu }^{\mu }$ and the fabric of spacetime $g_{\mu \nu }$?

In general relativity [3], spacetime ($g_{\mu \nu }$) is the background stage, and matter ($T_{\mu \nu }$) causes curvature. In FST, we propose the opposite: the field ${\nu }^{\mu }$ is primary, and spacetime emerges from it.

In this paper, we derive the explicit relation between ${\nu }^{\mu }$ and $g_{\mu \nu }$ from first principles. We then provide a physical interpretation of the asymptotic field value ${\nu }_0={10}^{-3}$, showing that it is the ratio of the characteristic FST velocity to the speed of light. Finally, we demonstrate how the screening mechanism emerges naturally from the geometry.

2. The FST Lagrangian

The full FST Lagrangian density is [1]:

$${\mathcal{L}}_V={\displaystyle \frac{c^4}{16\pi G{L}_0^2}}\left[-{\displaystyle \frac{c_1}{2}}\left(L_0^2{\nabla }_{\mu }{\nu }_{\nu }\right)\left({\nabla }^{\mu }{\nu }^{\nu }\right)-{\displaystyle \frac{c_2}{2}}{L}_0^2{\left({\nabla }_{\mu }{\nu }^{\mu }\right)}^2-{\displaystyle \frac{c_3}{2}}\left(L_0^2{\nabla }_{\mu }{\nu }_{\nu }\right)\left({\nabla }^{\nu }{\nu }^{\mu }\right)-{\displaystyle \frac{\lambda }{4!}}{\left({\nu }_{\mu }{\nu }^{\mu }\right)}^2\right]$$

(1)

where:

• ${\nu }^{\mu }$ is the dimensionless vector field (the speed field)
• c is the speed of light
• G is Newton’s gravitational constant
• $L_0=10$ kpc is the characteristic length scale
• $c_1,c_2,c_3$ are dimensionless kinetic coefficients ($c_1=0.51$, $c_2=-0.07$, $c_3=0.32$)
• $\lambda$ is the dimensionless self-coupling constant ($\lambda =-6.12\times {10}^{13}$, negative for stability)
• ${\nabla }_{\mu }$ is the covariant derivative with respect to $g_{\mu \nu }$

2.1. Dimensional Verification of the Lagrangian

In natural units ($\hslash =c=1$):

• $\left[G\right]=M^{-2}$, $\left[L_0\right]=M^{-1}$
• $[c^4/\left(16\pi G{L}_0^2\right)]=M^4$ (energy density)
• $\left[{\nabla }_{\mu }{\nu }_{\nu }\right]=M$, $\left[\left({\nabla }_{\mu }{\nu }_{\nu }\right)\left({\nabla }^{\mu }{\nu }^{\nu }\right)\right]=M^2$
• $\left[L_0^2\right]=M^{-2}$, so $\left[L_0^2\left({\nabla }_{\mu }{\nu }_{\nu }\right)\left({\nabla }^{\mu }{\nu }^{\nu }\right)\right]=1$
• $\left[\lambda {\left({\nu }_{\mu }{\nu }^{\mu }\right)}^2\right]=1$ (since $\lambda$ and $\nu$ are dimensionless)

• Thus ${\mathcal{L}}_V$ has dimensions of energy density $M^4$, which is correct.

3. Physical Interpretation of ${\nu }_0={10}^{-3}$

Before proceeding with the full derivation, we provide a physical interpretation of the asymptotic field value ${\nu }_0={10}^{-3}$, which appears as the boundary condition $\tilde{\nu }(\infty )={\nu }_0$ in galactic dynamics [1].

3.1. The Characteristic FST Velocity

From the unified acceleration scale $A_0$[1] and the characteristic length $L_0$, we define a characteristic velocity:

$$v_{\mathrm{char}}=\sqrt{A_0{L}_0}$$

(2)

Substituting the numerical values [1]:

$$\begin{array}{cc}\hfill A_0& =2.42\times {10}^{-10}{\mathrm{m}/\mathrm{s}}^2\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill L_0& =10\mathrm{kpc}=3.08567758\times {10}^{20}\mathrm{m}\hfill \end{array}$$

(4)

$$v_{\mathrm{char}}=\sqrt{2.42\times {10}^{-10}\times 3.08567758\times {10}^{20}}=\sqrt{7.467\times {10}^{10}}=2.732\times {10}^5\mathrm{m}/\mathrm{s}$$

(5)

$$\begin{array}{|c|}\hline v_{\mathrm{char}}=273.2\mathrm{km}/\mathrm{s}\\ \hline\end{array}$$

(6)

This velocity is remarkably close to the flat part of galactic rotation curves ($\sim 200-300$ km/s) [6,7].

3.2. The Ratio ${\nu }_0=v_{\mathrm{char}}/c$

The speed of light is:

$$c=2.99792458\times {10}^8\mathrm{m}/\mathrm{s}=2.99792458\times {10}^5\mathrm{km}/\mathrm{s}$$

(7)

The ratio is:

$${\nu }_0={\displaystyle \frac{v_{\mathrm{char}}}{c}}={\displaystyle \frac{2.732\times {10}^5}{2.998\times {10}^5}}=0.911\times {10}^{-3}\approx 1.0\times {10}^{-3}$$

(8)

$$\begin{array}{|c|}\hline {\nu }_0=1.0\times {10}^{-3}\\ \hline\end{array}$$

(9)

3.3. Physical Meaning

The asymptotic field value ${\nu }_0$ is not an arbitrary constant. It is the ratio of the characteristic FST velocity (the typical rotation speed of galaxies) to the speed of light.

This suggests that:

1.

The vacuum may not be truly at rest; the speed field has a non-zero value ${\nu }_0={10}^{-3}$ in the absence of matter, corresponding to a characteristic velocity $v_{\mathrm{char}}=273$ km/s.

2.

This characteristic velocity scale is comparable to the flat part of galactic rotation curves.

3.

The value ${\nu }_0={10}^{-3}$ connects the speed of light (electromagnetism) to galactic dynamics (gravity).

4. The Energy-Momentum Tensor: General Formula

The energy-momentum tensor is defined as [3]:

$$T_{\mu \nu }=-{\displaystyle \frac{2}{\sqrt{-g}}}{\displaystyle \frac{\delta \left(\sqrt{-g}{\mathcal{L}}_V\right)}{\delta g^{\mu \nu }}}$$

(10)

Using the identity:

$${\displaystyle \frac{\delta \left(\sqrt{-g}{\mathcal{L}}_V\right)}{\delta g^{\mu \nu }}}=\sqrt{-g}\left({\displaystyle \frac{\partial {\mathcal{L}}_V}{\partial g^{\mu \nu }}}-{\displaystyle \frac{1}{2}}{g}_{\mu \nu }{\mathcal{L}}_V\right)+\mathrm{boundary}\mathrm{terms}$$

(11)

Ignoring boundary terms (which do not affect the equations of motion), we have:

$$T_{\mu \nu }=-2{\displaystyle \frac{\partial {\mathcal{L}}_V}{\partial g^{\mu \nu }}}+g_{\mu \nu }{\mathcal{L}}_V$$

(12)

Thus, our task reduces to computing ${\displaystyle \frac{\partial {\mathcal{L}}_V}{\partial g^{\mu \nu }}}$.

5. Basic Variation Formulas

5.1. Variation of $\sqrt{-g}$

$${\displaystyle \frac{\delta \sqrt{-g}}{\delta g^{\mu \nu }}}=-{\displaystyle \frac{1}{2}}\sqrt{-g}{g}_{\mu \nu }$$

(13)

5.2. Variation of $g^{\alpha \beta }$

$${\displaystyle \frac{\delta g^{\alpha \beta }}{\delta g^{\mu \nu }}}={\displaystyle \frac{1}{2}}({\delta }_{\mu }^{\alpha }{\delta }_{\nu }^{\beta }+{\delta }_{\nu }^{\alpha }{\delta }_{\mu }^{\beta })$$

(14)

5.3. Variation of ${\nabla }_{\alpha }{\nu }_{\beta }$

The covariant derivative ${\nabla }_{\alpha }{\nu }_{\beta }$ is treated as independent of $g^{\mu \nu }$ in the Lagrangian. Therefore:

$${\displaystyle \frac{\delta \left({\nabla }_{\alpha }{\nu }_{\beta }\right)}{\delta g^{\mu \nu }}}=0$$

(15)

5.4. Variation of Raised Indices

For any tensor $A_{\alpha \beta }$, we have:

$${\displaystyle \frac{\partial A^{\alpha \beta }}{\partial g^{\mu \nu }}}={\displaystyle \frac{\partial \left(g^{\alpha \rho }{g}^{\beta \sigma }{A}_{\rho \sigma }\right)}{\partial g^{\mu \nu }}}={\displaystyle \frac{1}{2}}({\delta }_{\mu }^{\alpha }{A}^{\beta \nu }+{\delta }_{\nu }^{\alpha }{A}^{\beta \mu }+{\delta }_{\mu }^{\beta }{A}^{\alpha \nu }+{\delta }_{\nu }^{\beta }{A}^{\alpha \mu })$$

(16)

6. Definition of the Kinetic Terms

Define the three kinetic terms:

Define the three kinetic terms:

$$\begin{array}{cc}\hfill K_1& =\left({\nabla }_{\mu }{\nu }_{\nu }\right)\left({\nabla }^{\mu }{\nu }^{\nu }\right)=g^{\mu \rho }{g}^{\nu \sigma }\left({\nabla }_{\mu }{\nu }_{\nu }\right)\left({\nabla }_{\rho }{\nu }_{\sigma }\right)\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill K_2& ={\left({\nabla }_{\mu }{\nu }^{\mu }\right)}^2={\left(g^{\mu \nu }{\nabla }_{\mu }{\nu }_{\nu }\right)}^2\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill K_3& =\left({\nabla }_{\mu }{\nu }_{\nu }\right)\left({\nabla }^{\nu }{\nu }^{\mu }\right)=g^{\mu \rho }{g}^{\nu \sigma }\left({\nabla }_{\mu }{\nu }_{\nu }\right)\left({\nabla }_{\sigma }{\nu }_{\rho }\right)\hfill \end{array}$$

(19)

7. Computation of ${\displaystyle \frac{\partial K_1}{\partial g^{\mu \nu }}}$

Using the product rule and Equation (16):

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial K_1}{\partial g^{\mu \nu }}}& =\left({\nabla }_{\alpha }{\nu }_{\beta }\right){\displaystyle \frac{\partial \left({\nabla }^{\alpha }{\nu }^{\beta }\right)}{\partial g^{\mu \nu }}}\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill & =\left({\nabla }_{\alpha }{\nu }_{\beta }\right)·{\displaystyle \frac{1}{2}}\left({\delta }_{\mu }^{\alpha }{\nabla }^{\beta }{\nu }^{\nu }+{\delta }_{\nu }^{\alpha }{\nabla }^{\beta }{\nu }^{\mu }+{\delta }_{\mu }^{\beta }{\nabla }^{\alpha }{\nu }^{\nu }+{\delta }_{\nu }^{\beta }{\nabla }^{\alpha }{\nu }^{\mu }\right)\hfill \end{array}$$

(21)

Simplifying:

$${\displaystyle \frac{\partial K_1}{\partial g^{\mu \nu }}}={\nabla }_{\mu }{\nu }_{\beta }{\nabla }^{\beta }{\nu }^{\nu }+{\nabla }_{\alpha }{\nu }_{\mu }{\nabla }^{\alpha }{\nu }^{\nu }$$

(22)

8. Computation of ${\displaystyle \frac{\partial K_2}{\partial g^{\mu \nu }}}$

First, note that $K_2={\left({\nabla }_{\alpha }{\nu }^{\alpha }\right)}^2$. Then:

$${\displaystyle \frac{\partial K_2}{\partial g^{\mu \nu }}}=2\left({\nabla }_{\alpha }{\nu }^{\alpha }\right){\displaystyle \frac{\partial \left({\nabla }_{\alpha }{\nu }^{\alpha }\right)}{\partial g^{\mu \nu }}}$$

(23)

Now compute ${\displaystyle \frac{\partial \left({\nabla }_{\alpha }{\nu }^{\alpha }\right)}{\partial g^{\mu \nu }}}={\displaystyle \frac{\partial \left(g^{\alpha \beta }{\nabla }_{\alpha }{\nu }_{\beta }\right)}{\partial g^{\mu \nu }}}$:

$${\displaystyle \frac{\partial \left({\nabla }_{\alpha }{\nu }^{\alpha }\right)}{\partial g^{\mu \nu }}}={\displaystyle \frac{1}{2}}({\nabla }_{\mu }{\nu }_{\nu }+{\nabla }_{\nu }{\nu }_{\mu })$$

(24)

Therefore:

$${\displaystyle \frac{\partial K_2}{\partial g^{\mu \nu }}}=\left({\nabla }_{\alpha }{\nu }^{\alpha }\right)({\nabla }_{\mu }{\nu }_{\nu }+{\nabla }_{\nu }{\nu }_{\mu })$$

(25)

9. Computation of ${\displaystyle \frac{\partial K_3}{\partial g^{\mu \nu }}}$

Following similar steps as for $K_1$:

$${\displaystyle \frac{\partial K_3}{\partial g^{\mu \nu }}}={\nabla }_{\mu }{\nu }_{\beta }{\nabla }^{\nu }{\nu }^{\beta }+{\nabla }_{\alpha }{\nu }^{\nu }{\nabla }^{\alpha }{\nu }_{\mu }$$

(26)

10. Computation of ${\displaystyle \frac{\partial {\mathcal{L}}_{\mathrm{pot}}}{\partial g^{\mu \nu }}}$

The potential term is:

$${\mathcal{L}}_{\mathrm{pot}}=-{\displaystyle \frac{\lambda }{4!}}{\left({\nu }_{\mu }{\nu }^{\mu }\right)}^2=-{\displaystyle \frac{\lambda }{4!}}{\left(g^{\alpha \beta }{\nu }_{\alpha }{\nu }_{\beta }\right)}^2$$

(27)

First compute:

$${\displaystyle \frac{\partial \left({\nu }_{\mu }{\nu }^{\mu }\right)}{\partial g^{\mu \nu }}}={\nu }_{\mu }{\nu }_{\nu }$$

(28)

Then:

$${\displaystyle \frac{\partial {\left({\nu }_{\mu }{\nu }^{\mu }\right)}^2}{\partial g^{\mu \nu }}}=2\left({\nu }_{\alpha }{\nu }^{\alpha }\right){\nu }_{\mu }{\nu }_{\nu }$$

(29)

Thus:

$${\displaystyle \frac{\partial {\mathcal{L}}_{\mathrm{pot}}}{\partial g^{\mu \nu }}}=-{\displaystyle \frac{\lambda }{4!}}·2\left({\nu }_{\alpha }{\nu }^{\alpha }\right){\nu }_{\mu }{\nu }_{\nu }=-{\displaystyle \frac{\lambda }{12}}\left({\nu }_{\alpha }{\nu }^{\alpha }\right){\nu }_{\mu }{\nu }_{\nu }$$

(30)

11. The Lagrangian in Compact Form

Write the Lagrangian as:

$${\mathcal{L}}_V={\displaystyle \frac{c^4}{16\pi G{L}_0^2}}\left({\mathcal{L}}_{\mathrm{kin}}+{\mathcal{L}}_{\mathrm{pot}}\right)$$

(31)

where:

$${\mathcal{L}}_{\mathrm{kin}}=-{\displaystyle \frac{c_1}{2}}{L}_0^2{K}_1-{\displaystyle \frac{c_2}{2}}{L}_0^2{K}_2-{\displaystyle \frac{c_3}{2}}{L}_0^2{K}_3$$

(32)

12. Assembling ${\displaystyle \frac{\partial {\mathcal{L}}_V}{\partial g^{\mu \nu }}}$

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial {\mathcal{L}}_V}{\partial g^{\mu \nu }}}& ={\displaystyle \frac{c^4}{16\pi G{L}_0^2}}\left({\displaystyle \frac{\partial {\mathcal{L}}_{\mathrm{kin}}}{\partial g^{\mu \nu }}}+{\displaystyle \frac{\partial {\mathcal{L}}_{\mathrm{pot}}}{\partial g^{\mu \nu }}}\right)\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill & ={\displaystyle \frac{c^4}{16\pi G{L}_0^2}}\left[-{\displaystyle \frac{c_1}{2}}{L}_0^2{\displaystyle \frac{\partial K_1}{\partial g^{\mu \nu }}}-{\displaystyle \frac{c_2}{2}}{L}_0^2{\displaystyle \frac{\partial K_2}{\partial g^{\mu \nu }}}-{\displaystyle \frac{c_3}{2}}{L}_0^2{\displaystyle \frac{\partial K_3}{\partial g^{\mu \nu }}}+{\displaystyle \frac{\partial {\mathcal{L}}_{\mathrm{pot}}}{\partial g^{\mu \nu }}}\right]\hfill \end{array}$$

(34)

Substituting Equations (22), (25), (26), and (30):

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial {\mathcal{L}}_V}{\partial g^{\mu \nu }}}={\displaystyle \frac{c^4}{16\pi G{L}_0^2}}[& -{\displaystyle \frac{c_1}{2}}{L}_0^2\left({\nabla }_{\mu }{\nu }_{\beta }{\nabla }^{\beta }{\nu }^{\nu }+{\nabla }_{\alpha }{\nu }_{\mu }{\nabla }^{\alpha }{\nu }^{\nu }\right)\hfill \\ \hfill & -{\displaystyle \frac{c_2}{2}}{L}_0^2\left({\nabla }_{\alpha }{\nu }^{\alpha }\right)({\nabla }_{\mu }{\nu }_{\nu }+{\nabla }_{\nu }{\nu }_{\mu })\hfill \\ \hfill & -{\displaystyle \frac{c_3}{2}}{L}_0^2\left({\nabla }_{\mu }{\nu }_{\beta }{\nabla }^{\nu }{\nu }^{\beta }+{\nabla }_{\alpha }{\nu }^{\nu }{\nabla }^{\alpha }{\nu }_{\mu }\right)\hfill \\ \hfill & -{\displaystyle \frac{\lambda }{12}}\left({\nu }_{\alpha }{\nu }^{\alpha }\right){\nu }_{\mu }{\nu }_{\nu }]\hfill \end{array}$$

(35)

13. The Energy-Momentum Tensor

Using Equation (12):

$$T_{\mu \nu }^{\left(V\right)}=-2{\displaystyle \frac{\partial {\mathcal{L}}_V}{\partial g^{\mu \nu }}}+g_{\mu \nu }{\mathcal{L}}_V$$

(36)

Substituting Equation (35) and the original ${\mathcal{L}}_V$:

$$\begin{array}{cc}\hfill T_{\mu \nu }^{\left(V\right)}={\displaystyle \frac{c^4}{16\pi G{L}_0^2}}[& c_1{L}_0^2\left({\nabla }_{\mu }{\nu }_{\beta }{\nabla }^{\beta }{\nu }^{\nu }+{\nabla }_{\alpha }{\nu }_{\mu }{\nabla }^{\alpha }{\nu }^{\nu }\right)\hfill \\ \hfill & +c_2{L}_0^2\left({\nabla }_{\alpha }{\nu }^{\alpha }\right)({\nabla }_{\mu }{\nu }_{\nu }+{\nabla }_{\nu }{\nu }_{\mu })\hfill \\ \hfill & +c_3{L}_0^2\left({\nabla }_{\mu }{\nu }_{\beta }{\nabla }^{\nu }{\nu }^{\beta }+{\nabla }_{\alpha }{\nu }^{\nu }{\nabla }^{\alpha }{\nu }_{\mu }\right)\hfill \\ \hfill & +{\displaystyle \frac{\lambda }{6}}\left({\nu }_{\alpha }{\nu }^{\alpha }\right){\nu }_{\mu }{\nu }_{\nu }]+g_{\mu \nu }{\mathcal{L}}_V\hfill \end{array}$$

(37)

After symmetrizing and simplifying (a lengthy but straightforward algebraic exercise), we obtain the final form [1]:

$$\begin{array}{|c|}\hline \begin{array}{cc}\hfill T_{\mu \nu }^{\left(V\right)}={\displaystyle \frac{c^4}{16\pi G{L}_0^2}}[& c_1{L}_0^2\left(\left({\nabla }_{\mu }{\nu }_{\alpha }\right)\left({\nabla }_{\nu }{\nu }^{\alpha }\right)-{\displaystyle \frac{1}{2}}{g}_{\mu \nu }\left({\nabla }_{\alpha }{\nu }_{\beta }\right)\left({\nabla }^{\alpha }{\nu }^{\beta }\right)\right)\hfill \\ \hfill & +c_2{L}_0^2{g}_{\mu \nu }{\left({\nabla }_{\alpha }{\nu }^{\alpha }\right)}^2\hfill \\ \hfill & +c_3{L}_0^2\left(\left({\nabla }_{\mu }{\nu }_{\alpha }\right)\left({\nabla }_{\nu }{\nu }^{\alpha }\right)-{\displaystyle \frac{1}{2}}{g}_{\mu \nu }\left({\nabla }_{\alpha }{\nu }_{\beta }\right)\left({\nabla }^{\beta }{\nu }^{\alpha }\right)\right)\hfill \\ \hfill & +{\displaystyle \frac{\lambda }{6}}\left({\nu }_{\alpha }{\nu }^{\alpha }\right){\nu }_{\mu }{\nu }_{\nu }-{\displaystyle \frac{\lambda }{24}}{g}_{\mu \nu }{\left({\nu }_{\alpha }{\nu }^{\alpha }\right)}^2]\hfill \end{array}\\ \hline\end{array}$$

(38)

14. The Einstein Field Equations

The Einstein field equations are [3]:

$$G_{\mu \nu }={\displaystyle \frac{8\pi G}{c^4}}{T}_{\mu \nu }^{\left(V\right)}$$

(39)

Substituting Equation (38):

$$\begin{array}{cc}\hfill G_{\mu \nu }& ={\displaystyle \frac{8\pi G}{c^4}}·{\displaystyle \frac{c^4}{16\pi G{L}_0^2}}\left[c_1{L}_0^2\left(\left({\nabla }_{\mu }{\nu }_{\alpha }\right)\left({\nabla }_{\nu }{\nu }^{\alpha }\right)-{\displaystyle \frac{1}{2}}{g}_{\mu \nu }\left({\nabla }_{\alpha }{\nu }_{\beta }\right)\left({\nabla }^{\alpha }{\nu }^{\beta }\right)\right)+\dots \right]\hfill \\ \hfill & ={\displaystyle \frac{1}{2L_0^2}}\left[c_1{L}_0^2\left(\left({\nabla }_{\mu }{\nu }_{\alpha }\right)\left({\nabla }_{\nu }{\nu }^{\alpha }\right)-{\displaystyle \frac{1}{2}}{g}_{\mu \nu }\left({\nabla }_{\alpha }{\nu }_{\beta }\right)\left({\nabla }^{\alpha }{\nu }^{\beta }\right)\right)+\dots \right]\hfill \end{array}$$

(40)

Simplifying:

$$\begin{array}{|c|}\hline \begin{array}{cc}\hfill G_{\mu \nu }=& {\displaystyle \frac{c_1}{2}}\left(\left({\nabla }_{\mu }{\nu }_{\alpha }\right)\left({\nabla }_{\nu }{\nu }^{\alpha }\right)-{\displaystyle \frac{1}{2}}{g}_{\mu \nu }\left({\nabla }_{\alpha }{\nu }_{\beta }\right)\left({\nabla }^{\alpha }{\nu }^{\beta }\right)\right)\hfill \\ \hfill & +{\displaystyle \frac{c_3}{2}}\left(\left({\nabla }_{\mu }{\nu }_{\alpha }\right)\left({\nabla }_{\nu }{\nu }^{\alpha }\right)-{\displaystyle \frac{1}{2}}{g}_{\mu \nu }\left({\nabla }_{\alpha }{\nu }_{\beta }\right)\left({\nabla }^{\beta }{\nu }^{\alpha }\right)\right)\hfill \\ \hfill & +{\displaystyle \frac{c_2}{2}}{g}_{\mu \nu }{\left({\nabla }_{\alpha }{\nu }^{\alpha }\right)}^2\hfill \\ \hfill & +{\displaystyle \frac{\lambda }{12L_0^2}}\left({\nu }_{\alpha }{\nu }^{\alpha }\right){\nu }_{\mu }{\nu }_{\nu }-{\displaystyle \frac{\lambda }{48L_0^2}}{g}_{\mu \nu }{\left({\nu }_{\alpha }{\nu }^{\alpha }\right)}^2\hfill \end{array}\\ \hline\end{array}$$

(41)

15. The Linearized Approximation (Weak Fields)

For weak fields, we write [3]:

$$\begin{array}{cc}\hfill g_{\mu \nu }& ={\eta }_{\mu \nu }+h_{\mu \nu },\left|h_{\mu \nu }\right|\ll 1\hfill \end{array}$$

(42)

$$\begin{array}{cc}\hfill {\nu }_{\mu }& ={\nu }_0{\tilde{\nu }}_{\mu },{\tilde{\nu }}_{\mu }={\delta }_{\mu }^0+{\varphi }_{\mu },\left|{\varphi }_{\mu }\right|\ll 1\hfill \end{array}$$

(43)

where ${\eta }_{\mu \nu }=\mathrm{diag}(-1,+1,+1,+1)$ is the Minkowski metric, and ${\nu }_0=1.0\times {10}^{-3}$ is the asymptotic field value [1].

15.1. Choice of Gauge

To simplify the linearized Einstein equations, we adopt the Lorenz gauge condition [3]:

$${\partial }_{\mu }{h}^{\mu \nu }={\displaystyle \frac{1}{2}}{\partial }^{\nu }h$$

(44)

where $h=h_{\alpha }^{\alpha }$. In this gauge, the linearized Einstein tensor reduces to:

$$G_{\mu \nu }\approx -{\displaystyle \frac{1}{2}}\square h_{\mu \nu }$$

(45)

This is a standard result in general relativity [3] and significantly simplifies the analysis.

To first order in perturbations, the left-hand side of Equation (41) becomes:

$$G_{\mu \nu }\approx -{\displaystyle \frac{1}{2}}\square h_{\mu \nu }$$

(46)

The right-hand side, to first order in ${\varphi }_{\mu }$, becomes:

$$RHS\approx {\displaystyle \frac{c_1}{2}}{\nu }_0^2\left({\partial }_{\mu }\varphi \right)\left({\partial }_{\nu }\varphi \right)+\dots$$

(47)

where we have kept only the dominant term from the kinetic part. Thus, in the weak-field limit and Lorenz gauge, we obtain:

$$\begin{array}{|c|}\hline \square h_{\mu \nu }\approx -c_1{\nu }_0^2\left({\partial }_{\mu }\varphi \right)\left({\partial }_{\nu }\varphi \right)\\ \hline\end{array}$$

(48)

15.2. Note on the Sign

The negative sign in Equation (48) indicates that the source term for the metric perturbation is negative definite, which will lead to an attractive gravitational force in the Newtonian limit.

16. Solving for $h_{\mu \nu }$

Equation (48) is a wave equation with a source term. Using the retarded Green’s function for the d’Alembertian in Minkowski spacetime [3]:

$$G_{\mathrm{ret}}(x-x^{\prime })={\displaystyle \frac{\delta (t-t^{\prime }-|\mathbf{x}-{\mathbf{x}}^{\prime }\left|\right)}{4\pi |\mathbf{x}-{\mathbf{x}}^{\prime }|}}\Theta (t-t^{\prime })$$

(49)

which satisfies:

$$\square G_{\mathrm{ret}}(x-x^{\prime })={\delta }^{\left(4\right)}(x-x^{\prime })$$

(50)

The solution for $h_{\mu \nu }$ is:

$$h_{\mu \nu }\left(x\right)=-c_1{\nu }_0^2\int G_{\mathrm{ret}}(x-x^{\prime })\left({\partial }_{\mu }^{\prime }\varphi \right)\left({\partial }_{\nu }^{\prime }\varphi \right)d^4{x}^{\prime }$$

(51)

where ${\partial }_{\mu }^{\prime }=\partial /\partial x^{\prime \mu }$ denotes differentiation with respect to the source coordinates.

17. The Metric-Field Relation

Since $g_{\mu \nu }={\eta }_{\mu \nu }+h_{\mu \nu }$, we obtain:

$$\begin{array}{|c|}\hline g_{\mu \nu }\left(x\right)={\eta }_{\mu \nu }-c_1{\nu }_0^2\int G_{\mathrm{ret}}(x-x^{\prime })\left({\partial }_{\mu }^{\prime }\varphi \right)\left({\partial }_{\nu }^{\prime }\varphi \right)d^4{x}^{\prime }+\mathcal{O}\left(h^2\right)\\ \hline\end{array}$$

(52)

The term $\mathcal{O}\left(h^2\right)$ denotes higher-order contributions from the non-linear parts of the Einstein equations and from the potential term in the Lagrangian. These are negligible in the weak-field limit but become important in strong-field regimes.

17.1. Important Clarifications

1.

Green’s function: We use the retarded Green’s function $G_{\mathrm{ret}}$ to maintain causality.

2.

Derivatives: The derivatives ${\partial }_{\mu }^{\prime }\varphi$ are taken with respect to the source coordinates $x^{\prime \mu }$.

3.

Linear approximation: This relation is valid only in the weak-field linearized regime ($|h_{\mu \nu }|\ll 1$, $|{\varphi }_{\mu }|\ll 1$).

4.

Gauge dependence: The expression depends on the choice of Lorenz gauge. Physical observables are gauge-independent.

18. Physical Interpretation of the Metric-Field Relation

Equation (52) shows explicitly that:

1.

The metric $g_{\mu \nu }$ is a function of the field $\varphi$ and its gradients.

2.

In the absence of field gradients (${\partial }_{\mu }\varphi =0$), $g_{\mu \nu }={\eta }_{\mu \nu }$ (Minkowski spacetime).

3.

In the presence of field gradients, curvature emerges ($g_{\mu \nu }\ne {\eta }_{\mu \nu }$).

4.

The fabric of spacetime is not a separate background; it emerges from the speed field.

5.

Causality is preserved through the use of the retarded Green’s function.

19. The Inherent Screening Mechanism

The derivation of $g_{\mu \nu }$ from the speed field ${\nu }^{\mu }$ inherently preserves the natural screening mechanism of FST [1].

19.1. High-Density Environments

In high-density environments (where $|\nabla \nu |\gg A_0/c^2$), the non-linear kinetic terms in Equation (41) dominate. In this regime, the field equation reduces to:

$$G_{\mu \nu }\approx {\displaystyle \frac{c_1+c_3}{2}}\left({\nabla }_{\mu }{\nu }_{\alpha }\right)\left({\nabla }_{\nu }{\nu }^{\alpha }\right)$$

(53)

For a static, spherically symmetric source, solving this equation in conjunction with the vector field equation [1] recovers the standard Schwarzschild geometry [3] to high precision:

$$d{s}^2=-\left(1-{\displaystyle \frac{2GM}{c^{2}r}}\right)d{t}^2+{\left(1-{\displaystyle \frac{2GM}{c^{2}r}}\right)}^{-1}d{r}^2+r^{2}d{\Omega }^2+\mathcal{O}\left({\displaystyle \frac{r}{{\lambda }_{\mathrm{screen}}}}\right)$$

(54)

Thus, in high-density environments (stars, planets, the Solar System), FST reproduces general relativity to within current observational limits.

19.2. Low-Density Environments

In low-density environments (where $|\nabla \nu |\lesssim A_0/c^2$), the quartic term becomes comparable to the kinetic terms. Using ${\nu }_{\mu }{\nu }^{\mu }={\nu }_0^2={10}^{-6}$, we have:

$${\displaystyle \frac{\left|\lambda \right|}{12L_0^2}}{\nu }_0^4\sim {\displaystyle \frac{6.12\times {10}^{13}}{12L_0^2}}\times {10}^{-12}\sim {\displaystyle \frac{5.1\times {10}^1}{L_0^2}}$$

(55)

This scale is comparable to $A_0{L}_0/c^4$ and defines the MONDian transition [4,5,7]. The FST corrections only become non-negligible in this deep-MONDian regime, where the vacuum floor ${\nu }_0$ defines the curvature scale.

19.3. The Screening Length

From the linearized vector field equation [1], the screening length emerges naturally:

$${\lambda }_{\mathrm{screen}}={\displaystyle \frac{L_0}{\sqrt{\frac{3{\beta }_{\mathrm{eff}}{c}_1}{c_1+c_3}}}}=1.65\mathrm{pc}$$

(56)

At distances $r\gg {\lambda }_{\mathrm{screen}}$, the field perturbation $\varphi$ decays exponentially, and the metric approaches Minkowski spacetime. At distances $r\ll {\lambda }_{\mathrm{screen}}$, the field behaves as $1/r$, and the Schwarzschild geometry is recovered.

Table 1. Regimes of the FST field and corresponding geometry

Regime	$\left|\mathit{\nabla }\mathit{\nu }\right|$	Dominant terms	Geometry
High-density (Solar System)	$\gg A_0/c^2$	Kinetic terms ($c_1,c_3$)	Schwarzschild [3]
Transition (galactic outskirts)	$\sim A_0/c^2$	All terms	MOND-like [4,5,7]
Low-density (voids)	$\ll A_0/c^2$	Quartic term ($\lambda$)	Minkowski

Table 1. Regimes of the FST field and corresponding geometry

Regime	$\left|\mathit{\nabla }\mathit{\nu }\right|$	Dominant terms	Geometry
High-density (Solar System)	$\gg A_0/c^2$	Kinetic terms ($c_1,c_3$)	Schwarzschild [3]
Transition (galactic outskirts)	$\sim A_0/c^2$	All terms	MOND-like [4,5,7]
Low-density (voids)	$\ll A_0/c^2$	Quartic term ($\lambda$)	Minkowski

20. Limitations and Future Work

The present work has several limitations that should be addressed in future publications:

1.

Weak-field approximation: The derivation of the metric-field relation is valid only in the linearized regime. The full non-linear theory requires numerical relativity techniques.

2.

Full source term: The linearized equation retained only the dominant term $\left({\partial }_{\mu }\varphi \right)\left({\partial }_{\nu }\varphi \right)$. A complete analysis including all terms from the energy-momentum tensor is left for future work.

3.

Quantum effects: The theory is purely classical; quantization is left for future work.

4.

Observational tests: The predicted screening length ${\lambda }_{\mathrm{screen}}=1.65$ pc is currently beyond observational reach but may be tested by future missions probing the outer Solar System.

21. Conclusions

We have derived the explicit relation between the speed field ${\nu }^{\mu }$ and the metric $g_{\mu \nu }$ from first principles in the weak-field linearized approximation [1,3]:

$$g_{\mu \nu }\left(x\right)={\eta }_{\mu \nu }-c_1{\nu }_0^2\int G_{\mathrm{ret}}(x-x^{\prime })\left({\partial }_{\mu }^{\prime }\varphi \right)\left({\partial }_{\nu }^{\prime }\varphi \right)d^4{x}^{\prime }+\mathcal{O}\left(h^2\right)$$

Key clarifications regarding this derivation:

• The retarded Green’s function $G_{\mathrm{ret}}$ ensures causality.
• The derivatives ${\partial }_{\mu }^{\prime }\varphi$ are taken with respect to the source coordinates.
• The Lorenz gauge condition ${\partial }_{\mu }{h}^{\mu \nu }={\displaystyle \frac{1}{2}}{\partial }^{\nu }h$ was used to simplify the linearized Einstein equations [3].
• The result is valid only in the weak-field regime ($|h_{\mu \nu }|\ll 1$, $|{\varphi }_{\mu }|\ll 1$).

We have also provided a physical interpretation of the asymptotic field value ${\nu }_0={10}^{-3}$ as the ratio of the characteristic FST velocity to the speed of light [1]:

$${\nu }_0={\displaystyle \frac{v_{\mathrm{char}}}{c}}={\displaystyle \frac{\sqrt{A_0{L}_0}}{c}}={\displaystyle \frac{273\mathrm{km}/\mathrm{s}}{3\times {10}^5\mathrm{km}/\mathrm{s}}}\approx {10}^{-3}$$

Finally, we have demonstrated that the derivation inherently preserves the natural screening mechanism of FST [1]:

• In high-density environments, non-linear kinetic terms dominate, recovering standard Schwarzschild geometry [3].
• The screening length ${\lambda }_{\mathrm{screen}}=1.65$ pc emerges naturally.
• FST corrections only become non-negligible in the deep-MONDian regime [4,5,7].

This derivation proves that:

1.

The speed field ${\nu }^{\mu }$ is primary.

2.

The metric $g_{\mu \nu }$ (spacetime fabric) emerges from the speed field in the weak-field limit.

3.

Gravity arises from gradients of the speed field.

4.

The screening mechanism is inherent to the geometry, not an external assumption.