Chapter 4: The Flux Law

The Heart of Lapse-First GR: The flux law connects energy flow to temporal evolution. It tells us that only actual energy flux can change the flow of time, making the temporal nature of gravity explicit.

The Fundamental Equation

At the core of lapse-first GR lies a beautiful and simple equation:

\[ \boxed{\partial_t \Phi = -\frac{4\pi G}{c^4} r T^{tr}} \]

This is the flux law. Let's understand what each piece means and why this equation is so profound.

Breaking Down the Equation

Left Side: ∂_t Φ

∂_t Φ = Rate of change of the temporal potential with time

How fast the "temporal landscape" is changing
Positive: Temporal potential increasing (gravity weakening)
Negative: Temporal potential decreasing (gravity strengthening)
Zero: No change (static spacetime)

Right Side Components

The Constants: -4πG/c⁴

G: Newton's gravitational constant
c: Speed of light
4π: Geometric factor from spherical geometry
Minus sign: Ensures correct physical behavior

The Radius: r

The areal radius - defined so spheres have area 4πr². This factor accounts for the spherical geometry.

The Energy Flux: T^{tr}

T^{tr} = Radial energy flux density

Physical Meaning of T^{tr}:

T^{tr} measures the flow of energy through spherical surfaces at radius r. It's the amount of energy crossing unit area per unit time in the radial direction.

T^{tr} > 0: Energy flowing outward (radiation, explosion)
T^{tr} < 0: Energy flowing inward (accretion, collapse)
T^{tr} = 0: No radial energy flow (static or purely tangential motion)

Physical Interpretation

The flux law tells us something profound about the nature of gravity:

Core Insight: Only actual energy flow can change the temporal potential. Static matter, no matter how dense, cannot cause ∂_t Φ ≠ 0.

What This Means

1. Energy Flow Sculpts Time

Think of the temporal potential Φ as a landscape. Energy flux acts like water flow that actively carves and reshapes this landscape:

Outward energy flow: "Erodes" the temporal well (makes Φ less negative)
Inward energy flow: "Deepens" the temporal well (makes Φ more negative)
No flow: Landscape remains unchanged

2. Birkhoff's Theorem Emerges

In vacuum (T^{tr} = 0), we get ∂_t Φ = 0:

\[ T^{tr} = 0 \Rightarrow \partial_t \Phi = 0 \Rightarrow \Phi = \Phi(r) \text{ only} \]

This is Birkhoff's theorem: spherically symmetric vacuum spacetimes must be static (in this gauge).

3. Causality and Information Flow

The flux law is local - the change in Φ at radius r depends only on the energy flux through that specific radius. Information about energy flow propagates at the speed of light.

Examples and Applications

Example 1: Spherical Star Collapse

Consider a collapsing star with matter falling inward:

Energy flow: T^{tr} < 0 (inward)
Flux law: ∂_t Φ > 0
Physical result: Temporal potential increases, gravity strengthens
Observable effect: Time dilation increases as collapse proceeds

Example 2: Supernova Explosion

During a supernova explosion:

Energy flow: T^{tr} > 0 (outward)
Flux law: ∂_t Φ < 0
Physical result: Temporal potential decreases, gravity weakens
Observable effect: Time dilation decreases as energy escapes

Example 3: Static Neutron Star

For a perfectly static neutron star:

Energy flow: T^{tr} = 0 (no radial energy transport)
Flux law: ∂_t Φ = 0
Physical result: Temporal potential constant in time
Observable effect: Gravitational redshift remains constant

Derivation Outline

The flux law emerges from Einstein's field equations through the ADM formalism. Here's the conceptual derivation:

Step 1: Start with Einstein's Equations

\[ G_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu} \]

Step 2: Focus on Mixed Component

In our diagonal gauge, the key equation is the mixed component:

\[ G^t_r = \frac{8\pi G}{c^4} T^t_r \]

Step 3: Compute Left Side Geometrically

Using our metric with Φ, the Einstein tensor component gives:

\[ G^t_r = \frac{2}{r} \partial_t \Phi \]

Step 4: Equate and Solve

\[ \frac{2}{r} \partial_t \Phi = \frac{8\pi G}{c^4} T^{tr} \]

Solving for ∂_t Φ:

\[ \partial_t \Phi = \frac{4\pi G}{c^4} r T^{tr} \]

The minus sign in our final form comes from index conventions (T^{tr} vs T^t_r).

Connection to Other Formulations

Relationship to Mass Function

The flux law can be rewritten in terms of a mass function m(t,r):

\[ \frac{\partial m}{\partial t} = -4\pi r^2 T^{tr} \]

This shows that energy flux changes the "mass within radius r."

Connection to Vaidya Solution

The famous Vaidya solution for null radiation corresponds to:

\[ T^{tr} = \frac{\dot{M}(v)}{4\pi r^2} \]

where v is advanced time and M(v) is the time-dependent mass.

Why the Flux Law is Remarkable

1. Simplicity

Compare the flux law to the full Einstein field equations (10 coupled nonlinear PDEs). The flux law is just one linear PDE relating energy flow to temporal evolution.

2. Physical Transparency

The equation directly connects cause (energy flux) to effect (change in time flow). No complicated tensor manipulations needed to see the physics.

3. Computational Efficiency

Numerical simulations can evolve Φ using this simple equation rather than dealing with the full metric tensor evolution.

4. Conceptual Clarity

It makes manifest that gravity is fundamentally about temporal geometry - the flow of time itself responds to energy flow.

Limitations and Scope

Spherical Symmetry Required

The flux law in this simple form applies only to spherically symmetric situations. For general spacetimes, we need the full tensor formalism.

Diagonal Gauge Assumption

We've assumed zero shift (N^i = 0). Rotation and frame-dragging require including the shift vector evolution.

The γ_rr = e^{-2Φ} Constraint

This links radial space to time. For some matter types, this constraint might not be maintainable, requiring a more general approach.

Energy Conservation and the Flux Law

The flux law is intimately connected to energy conservation. In the weak field limit:

\[ \frac{\partial}{\partial t}\left(\rho c^2\right) + \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2 T^{tr}\right) = 0 \]

This is the continuity equation for energy density. Combined with the flux law, it ensures that energy is neither created nor destroyed - only transported.

The Intuitive Picture

Here's the big picture that the flux law reveals:

Intuitive Understanding:

Gravity isn't a force pulling on objects. Instead, energy flow actively sculpts the rate at which time passes. Objects then follow the "natural" paths through this dynamically evolving temporal landscape, which we perceive as gravitational attraction.

The flux law makes this precise: ∂_t Φ tells us how fast the temporal landscape is changing, and T^{tr} tells us what's causing that change.

Summary

The flux law ∂_t Φ = -(4πG/c⁴) r T^{tr} is the heart of lapse-first GR because it:

Connects energy flow to temporal evolution in the simplest possible way
Makes Birkhoff's theorem automatic (no flux → no evolution)
Provides computational advantages over full tensor equations
Offers physical intuition about how gravity works
Emerges directly from Einstein's equations - it's not approximate

In the next chapter, we'll explore how the value of Φ is determined in different physical situations, combining both static contributions from mass distributions and dynamic contributions from the flux law.

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