Chapter 5: What Determines Φ?

The Complete Picture

The value of the temporal potential Φ at any point is determined by Einstein's equations, which connect geometry to matter and energy. Unlike electromagnetism where you might "choose" the potential, in gravity Einstein's equations determine Φ based on:

1. Where the matter/energy is located (static contribution)
2. How the matter/energy is moving (dynamic contribution)
3. Boundary conditions (what happens at infinity or other boundaries)

Let's explore each of these systematically.

1. Static Contribution: The Poisson Equation

When matter is present but not flowing radially, Φ satisfies a Poisson-like equation:

This is remarkably similar to:

• Electrostatics: ∇²V = -ρₑ/ε₀ (electric potential from charge density)
• Newtonian gravity: ∇²φ = 4πGρ (gravitational potential from mass density)

Physical Interpretation

Solution for Spherical Mass

For a spherically symmetric mass M, the solution is:

This is the exact Schwarzschild solution! In the weak field limit (far from the mass):

Key Properties

• Φ → 0 as r → ∞: Potential vanishes at infinity
• Φ < 0 everywhere: Masses create "temporal wells"
• 1/r falloff: Same as Newtonian gravity
• Logarithmic divergence: Φ → -∞ at r = 2GM/c² (Schwarzschild radius)

Superposition in Weak Fields

For multiple masses in the weak field limit, potentials approximately add:

This makes calculations much simpler than working with the full nonlinear Einstein equations.

2. Dynamic Contribution: The Flux Law

When energy flows, Φ changes according to the flux law we learned in Chapter 4:

Integration Over Time

If we know the energy flux history, we can integrate to find how Φ evolves:

Physical Examples

Accretion onto Black Hole

• Energy flux: T^{tr} < 0 (inward flow)
• Evolution: ∂_t Φ > 0 (Φ increases)
• Result: Black hole mass grows, event horizon expands

Supernova Explosion

• Energy flux: T^{tr} > 0 (outward flow)
• Evolution: ∂_t Φ < 0 (Φ decreases)
• Result: Effective mass decreases as energy escapes

3. Boundary Conditions

To solve for Φ uniquely, we need boundary conditions. The choice depends on the physical situation:

Asymptotic Flatness (Isolated Objects)

For isolated systems like stars or black holes:

This ensures spacetime becomes flat far from the source.

Cosmological Setting (Homogeneous Universe)

In cosmology, we typically normalize at the present time:

where t₀ is the present cosmic time.

Surface Boundary Conditions

At the surface of a star or other object, Φ must match smoothly to the interior solution.

Practical Algorithm to Find Φ

Here's a step-by-step approach to determine Φ in any situation:

Step 1: Identify Your Setup

• Static mass distribution? → Use Poisson equation
• Energy flowing? → Use flux law
• Cosmological problem? → Use scale factor relation
• Time-dependent masses? → Combine both approaches

Step 2: Write the Relevant Equation

For Static Problems:

For Dynamic Problems:

For Combined Problems:

Solve the Poisson equation at each time slice, then evolve using the flux law.

Step 3: Apply Boundary Conditions

• Usually Φ → 0 at infinity for isolated systems
• Or normalize at some reference point/time
• Ensure continuity across boundaries

Step 4: Solve

• Analytically: For simple cases (spherical masses, etc.)
• Numerically: For complex matter distributions or time-dependent problems

Specific Solutions

1. Vacuum Around Spherical Mass (Schwarzschild)

2. Weak Field Approximation

3. Uniform Density Sphere (Interior)

Inside a uniform density sphere of radius R:

4. Cosmological (Friedmann Universe)

In an expanding universe with scale factor a(t):

5. Vaidya Solution (Null Radiation)

For a mass function M(t) with null radiation:

The Role of Matter Types

Different types of matter contribute differently to Φ:

Dust (Pressureless Matter)

• Static contribution: ρc² in Poisson equation
• Dynamic contribution: Only if flowing radially
• Example: Dark matter, slow-moving stellar material

Radiation (Photons, Neutrinos)

• Static contribution: Energy density ρc²
• Dynamic contribution: Large T^{tr} for radial propagation
• Example: Light from stars, cosmic microwave background

Perfect Fluid

• Static contribution: (ρ + 3P/c²)c² in general
• Dynamic contribution: Depends on flow pattern
• Example: Stellar interiors, relativistic gases

Computational Considerations

Numerical Evolution

For complex problems, we can evolve Φ numerically:

1. Start with initial Φ distribution (from Poisson equation)
2. Compute energy fluxes T^{tr} from matter dynamics
3. Evolve Φ using flux law: Φⁿ⁺¹ = Φⁿ - Δt × (4πG/c⁴) r T^{tr}
4. Update matter dynamics in new gravitational field
5. Repeat

Convergence and Stability

The flux law evolution is generally more stable than full metric evolution because:

• It's first-order in time (not second-order like metric evolution)
• It's linear in Φ (not nonlinear like Einstein equations)
• Gauge issues are minimized in the diagonal form

Connection to Energy Conservation

The determination of Φ respects energy conservation automatically. The continuity equation:

ensures that energy density and energy flux are consistently related. When combined with the flux law, this guarantees that energy is conserved as Φ evolves.

Quick Reference: Common Scenarios

Scenario	Equation to Use	Key Physics
Static star	∇²Φ = 4πGρ/c²	Mass creates temporal well
Stellar collapse	∂_t Φ = -(4πG/c⁴) r T^{tr}	Infall deepens well
Supernova	∂_t Φ = -(4πG/c⁴) r T^{tr}	Outflow shallows well
Black hole accretion	Both equations	Growing mass + infall
Expanding universe	Φ = -ln(a)	Scale factor evolution

The Key Insight

The beautiful thing about the lapse-first approach is that this determination becomes much more transparent:

• Static matter: Solve one Poisson equation
• Energy flow: Integrate one flux law equation
• Combined: Use both equations together

Compare this to solving 10 coupled nonlinear Einstein field equations!

Summary

The value of Φ is determined by a combination of:

• Static contribution: Mass distributions via ∇²Φ = 4πGρ/c²
• Dynamic contribution: Energy flows via ∂_t Φ = -(4πG/c⁴) r T^{tr}
• Boundary conditions: Usually Φ → 0 at infinity

The resulting temporal potential then determines all gravitational effects: time dilation, redshift, orbital dynamics, and event horizons. This makes Φ the central quantity that encodes gravity as temporal geometry.

In the next chapter, we'll see how these principles work out in real-world applications, from GPS satellites to neutron stars to cosmology.