Geometric Times: Clocks of Spacetime Metric

“Geometric time can be understood as the metric’s projection onto the observer.”

🎯 Core Proposition

In general relativity, “time” has multiple geometric realizations. In the GLS theoretical framework, they are classified as the unified time scale equivalence class $[\tau ]$:

Theoretical Proposition: Under appropriate conditions, these times are related to each other through affine transformations:

$$t_1=\alpha t_2+\beta ,\alpha >0$$

Thus mathematically belonging to the same equivalence class $[T]$.

graph TB
    T["Unified Time Scale<br/>[τ]"] --> K["Killing Time<br/>t_K"]
    T --> A["ADM Time<br/>N·t"]
    T --> L["null parameter<br/>λ"]
    T --> E["Conformal Time<br/>η"]
    T --> M["Modular Time<br/>s_mod"]

    K -.->|"affine transformation"| A
    A -.->|"affine transformation"| L
    L -.->|"affine transformation"| E

    style T fill:#fff4e1,stroke:#ff6b6b,stroke-width:4px

💡 Intuitive Image: Rhythms of Different Clocks

Analogy: Multiple Clocks

Imagine a scenario:

• Ground Clock: Second hand rotates uniformly (Killing time)
• Mountain Clock: Faster at higher altitudes (ADM lapse)
• Photon Clock: Massless, moves infinitely fast (null parameter)
• Cosmic Clock: Slows down with cosmic expansion (conformal time)

They all serve as measures of time, but with different rhythms.

GLS Theory proposes: These clocks are related by simple rescaling, potentially pointing to the same underlying time concept.

📐 Four Geometric Times Explained

1. Killing Time (Static Spacetimes)

Definition:

In static spacetimes, there exists a Killing vector ${\xi }^{\mu }$:

$${\mathcal{L}}_{\xi }{g}_{\mu \nu }=0$$

That is, the metric is invariant along $\xi$.

Time Coordinate: Choose a coordinate system such that $\xi =\partial /\partial t$, then:

$$d{s}^2=-V(\mathbf{x})d{t}^2+h_{ij}d{x}^{i}d{x}^j$$

where $V=-g_{tt}>0$ (timelike).

Proper Time Relation:

For a stationary observer ($d{x}^i=0$), the proper time:

$$d\tau =\sqrt{-g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }}=\sqrt{V}dt$$

Physical Meaning:

• $V$ describes the “time dilation factor”
• Where the gravitational field is strong ($V$ small), time runs slow
• Redshift Formula: ${\nu }_{\infty }/{\nu }_0=\sqrt{V(0)}$

Example: Schwarzschild Metric

$$d{s}^2=-\left(1-\frac{2M}{r}\right)d{t}^2+{\left(1-\frac{2M}{r}\right)}^{-1}d{r}^2+r^{2}d{\Omega }^2$$

$V(r)=1-2M/r$, for a stationary observer:

$$d\tau =\sqrt{1-\frac{2M}{r}}dt$$

At the horizon $r=2M$: $d\tau =0$ (time freezes)

graph LR
    K["Killing Vector<br/>ξ^μ"] --> G["Metric Invariant<br/>£_ξ g = 0"]
    G --> V["Time Dilation<br/>V = -g_tt"]
    V --> TAU["Proper Time<br/>dτ = √V dt"]
    TAU --> RED["Gravitational Redshift<br/>ν ∝ √V"]

    style K fill:#e1f5ff
    style V fill:#fff4e1,stroke:#ff6b6b,stroke-width:3px
    style RED fill:#e1ffe1

2. ADM Lapse ($(3+1)$ Decomposition)

ADM Decomposition:

Decompose 4-dimensional spacetime into 3-dimensional space + 1-dimensional time:

$$d{s}^2=-N^{2}d{t}^2+h_{ij}(d{x}^i+N^{i}dt)(d{x}^j+N^{j}dt)$$

where:

• $N$: lapse function (ratio of coordinate time to proper time)
• $N^i$: shift vector (coordinate system drift)
• $h_{ij}$: spatial 3-metric

Orthogonal Observer:

For an observer along the slice normal ($d{x}^i+N^{i}dt=0$):

$$d\tau =Ndt$$

Physical Meaning:

• $N$ controls the “time flow rate”
• $N>1$: coordinate time faster than proper time
• $N<1$: coordinate time slower than proper time

ADM Equations:

Einstein’s equations can be written as constraint equations + evolution equations:

Constraints (Hamiltonian + Momentum):

$$\mathcal{H}=0,{\mathcal{H}}_i=0$$

Evolution:

$$\frac{\partial h_{ij}}{\partial t}=\cdots ,\frac{\partial K_{ij}}{\partial t}=\cdots$$

where $K_{ij}$ is the extrinsic curvature.

Relation to Killing Time:

In static spacetimes, $N=\sqrt{V}$, they are equivalent.

graph TB
    ADM["ADM Decomposition<br/>(3+1)"] --> L["lapse function<br/>N"]
    ADM --> S["shift vector<br/>N^i"]
    ADM --> H["spatial metric<br/>h_ij"]

    L --> TAU["Proper Time<br/>dτ = N dt"]
    TAU --> CON["Constraint Equations<br/>H = 0"]
    H --> EV["Evolution Equations<br/>∂_t h ~ K"]

    style ADM fill:#e1f5ff
    style L fill:#fff4e1,stroke:#ff6b6b,stroke-width:3px
    style TAU fill:#e1ffe1

3. Null Affine Parameter (Null Geodesics)

Null Geodesics:

Worldlines of light rays or massless particles, satisfying $d{s}^2=0$.

Geodesic Equation:

$$k^a{\nabla }_a{k}^b=0$$

where $k^a=d{x}^a/d\lambda$ is the tangent vector, $\lambda$ is the affine parameter.

Why is an Affine Parameter Needed?

For null geodesics, $ds=0$, so we cannot parameterize with $s$, must introduce $\lambda$.

Bondi Coordinates (Schwarzschild exterior):

Tortoise Coordinate:

$$r^{\ast }=r+2M\ln \left|\frac{r}{2M}-1\right|$$

Retarded Time: $u=t-r^{\ast }$

Advanced Time: $v=t+r^{\ast }$

Outgoing Null Surface: $u=\text{constant}$

Incoming Null Surface: $v=\text{constant}$

Physical Meaning:

• $u,v$ are natural “boundary times”
• In gravitational scattering, $u$ corresponds to asymptotic outgoing state time
• Bondi mass $M(u)$ is monotonically non-increasing along $u$ (energy radiation)

Conformal Time in FRW:

$$d\eta =\frac{dt}{a(t)}$$

Null Geodesics: $d{s}^2=0\Rightarrow -d{t}^2+a^{2}d{\chi }^2=0$

In $\eta$ coordinates: $d{\eta }^2=d{\chi }^2$

Straightening: Null geodesics appear as straight lines in conformal time.

graph LR
    NULL["Null Geodesics<br/>ds² = 0"] --> AFF["Affine Parameter<br/>λ"]
    AFF --> BONDI["Bondi Coordinates<br/>u = t - r*"]
    AFF --> CONF["Conformal Time<br/>η = ∫dt/a"]

    BONDI --> MASS["Bondi Mass<br/>M(u)↓"]
    CONF --> STRAIGHT["Straightening<br/>dη² = dχ²"]

    style NULL fill:#e1f5ff
    style AFF fill:#fff4e1,stroke:#ff6b6b,stroke-width:3px
    style BONDI fill:#ffe1e1
    style CONF fill:#e1ffe1

4. Conformal Time (FRW Universe)

FRW Metric:

$$d{s}^2=-d{t}^2+a(t{)}^2{\gamma }_{ij}d{x}^{i}d{x}^j$$

where $a(t)$ is the scale factor, ${\gamma }_{ij}$ is the unit 3-sphere/plane/hyperboloid metric.

Conformal Time Definition:

$$d\eta =\frac{dt}{a(t)}$$

Integrating:

$$\eta (t)={\int }_0^t\frac{d{t}^{\prime }}{a(t^{\prime })}$$

Metric Rewritten:

$$d{s}^2=a(\eta {)}^2\left[-d{\eta }^2+{\gamma }_{ij}d{x}^{i}d{x}^j\right]$$

Physical Meaning:

• Null geodesics are straight lines in $(\eta ,x^i)$ coordinates
• Comoving observer: $\tau =t$ (proper time = cosmic time)
• Particle horizon: $\eta =\infty$ corresponds to the visible universe boundary

Redshift-Time Relation:

For photons, $\nu \propto 1/a$:

$$1+z=\frac{a(t_0)}{a(t_e)}$$

Phase Rhythm Ratio (Chapter 1):

$$1+z=\frac{(d\varphi /dt{)}_e}{(d\varphi /dt{)}_0}$$

Cosmological redshift can be viewed as a global rescaling of the time scale.

graph TB
    FRW["FRW Metric<br/>ds² = -dt² + a²dΣ²"] --> A["Scale Factor<br/>a(t)"]
    FRW --> ETA["Conformal Time<br/>dη = dt/a"]

    ETA --> STRAIGHT["Straighten Null Geodesics<br/>dη² = dχ²"]
    A --> Z["Redshift<br/>1+z = a₀/a_e"]
    Z --> PHASE["Phase Rhythm Ratio<br/>(dφ/dt)_e / (dφ/dt)₀"]

    style FRW fill:#e1f5ff
    style ETA fill:#fff4e1,stroke:#ff6b6b,stroke-width:3px
    style Z fill:#ffe1e1
    style PHASE fill:#e1ffe1

🔑 Unification of Time Equivalence Classes

Theoretical Proposition: Under appropriate conditions, the following time parameters belong to the same equivalence class:

$$[T]\sim \{\tau ,t_K,N,\lambda ,u,v,\eta ,{\omega }^{-1},z,s_{\text{mod}}\}$$

Related to each other through affine transformations $t_1=\alpha t_2+\beta$.

Proof Outline:

1. Killing ↔ ADM: In static spacetimes, $N=\sqrt{V}$, $d\tau =Ndt=\sqrt{V}dt$

2. ADM ↔ null: The normal to ADM slices defines $\lambda$, $d\lambda \propto Ndt$

3. null ↔ conformal: In FRW, $\eta$ straightens null geodesics, $d\eta =d\lambda /a$

4. conformal ↔ redshift: $1+z=a_0/a_e={\eta }_e/{\eta }_0$ (with appropriate normalization)

All these transformations are affine.

graph TB
    KILLING["Killing Time<br/>dτ = √V dt"] --> ADM["ADM Time<br/>dτ = N dt"]
    ADM --> NULL["null parameter<br/>k^a∇_a k^b = 0"]
    NULL --> CONF["Conformal Time<br/>dη = dt/a"]
    CONF --> Z["Redshift<br/>1+z = a₀/a_e"]

    KILLING -.->|"N = √V"| ADM
    ADM -.->|"dλ ∝ N dt"| NULL
    NULL -.->|"straightening"| CONF
    CONF -.->|"scale factor"| Z

    ALL["Time Equivalence Class<br/>[T]"] --> KILLING
    ALL --> ADM
    ALL --> NULL
    ALL --> CONF
    ALL --> Z

    style ALL fill:#fff4e1,stroke:#ff6b6b,stroke-width:4px

📊 Connection to Unified Time Scale

Core Connection: Geometric times connect to the unified scale through the proper time-phase relation (Chapter 1):

$$\varphi =\frac{m{c}^2}{\hslash }\int d\tau$$

Various Geometric Times:

1. Killing Time: $d\tau =\sqrt{V}dt$ $$\varphi =\frac{m{c}^2}{\hslash }\int \sqrt{V}dt$$

2. ADM Time: $d\tau =Ndt$ $$\varphi =\frac{m{c}^2}{\hslash }\int Ndt$$

3. Conformal Time: $d\tau =ad\eta$ (comoving observer) $$\varphi =\frac{m{c}^2}{\hslash }\int ad\eta$$

Theoretically, they should give the same phase (along the same worldline).

Connection to Time Scale Identity:

$$\kappa (\omega )=\frac{{\phi }^{\prime }(\omega )}{\pi }=\frac{1}{2\pi }\text{tr}Q(\omega )$$

Geometric Interpretation: $\kappa$ is the “local time density,” integrating it gives any geometric time.

🤔 Exercises

1. Conceptual Understanding:

   • What is the difference between Killing time and ADM time?
   • Why is a null affine parameter necessary?
   • How does conformal time straighten null geodesics?

2. Calculation Exercises:

   • Schwarzschild metric: Calculate $d\tau /dt$ at $r=3M$
   • ADM decomposition: Prove $N=\sqrt{V}$ (static case)
   • FRW universe: Calculate $\eta (t)$ for matter-dominated era

3. Physical Applications:

   • Which geometric times are involved in GPS satellite time corrections?
   • How does Bondi mass evolve with $u$?
   • What is the relationship between cosmological horizon and $\eta$?

4. Advanced Thinking:

   • Can we define a global Killing time in non-static spacetimes?
   • What is the relationship between ADM energy conservation and time translation invariance?
   • What physical process corresponds to conformal time singularities?

Navigation:

• Previous: 04-time-scale-identity_en.md - Time Scale Identity
• Next: 06-modular-time_en.md - Modular Time
• Overview: 00-time-overview_en.md - Unified Time Overview
• GLS theory: unified-time-scale-geometry.md