Gemini on Why GR Can Be Deterministic and Maximally Extended

 Gary– How can GR be deterministic

Gemini-In general relativity (GR), determinism is the idea that the state of the universe at one time (the “initial data”) uniquely determines its entire future and past evolution

. While GR is a classical and largely deterministic theory, its determinism depends on specific mathematical and physical conditions: 

1. Local vs. Global Determinism

• Local Determinism: GR is locally deterministic because the Einstein field equations uniquely determine the immediate future of a small region of spacetime given its initial data.
• Global Determinism: For the entire universe to be deterministic, you must be able to define a Cauchy surface—a “slice” of space at one moment that every possible path (causal curve) hits exactly once. If such a surface exists, the initial data on it uniquely determines the entire spacetime manifold. 

2. The Cauchy Initial Value Problem

To make GR deterministic, physicists treat it as an initial value problem, similar to how one might predict a ball’s path using its current position and velocity: 

• Induced Metric (
• ): This describes the “geometry” of space on the initial slice.
• Extrinsic Curvature (
• ): This describes how that slice is “embedded” in the larger 4D spacetime, essentially acting as the “velocity” of the geometry’s change.
• Unique Evolution: When these two pieces of data are specified correctly, the field equations provide a unique solution for the future. 

3. When Determinism Fails

GR can become non-deterministic in certain “extreme” scenarios where the math breaks down: 

• Cauchy Horizons: In some solutions (like certain rotating black holes), a boundary called a Cauchy horizon appears. Beyond this boundary, the initial data from the outside world is no longer enough to predict what happens.
• Singularities: At points where curvature becomes infinite (like the center of a black hole), the equations stop working, making it impossible to determine what happens next.
• Closed Timelike Curves: Solutions that allow “time loops” break the standard definition of determinism because the future can circle back to affect its own past. 

4. Deterministic vs. Probabilistic Nature

Unlike Quantum Mechanics (QM), which is inherently probabilistic and only predicts the likelihood of outcomes, GR’s geometry evolves predictably according to the laws of classical physics. At large scales—like stars and galaxies—these quantum fluctuations average out, leaving a smooth, deterministic spacetime. 

Are you interested in how singularities or black hole horizons specifically challenge these deterministic rules?

Gary– GR must necessarily have infinities if the universe is maximally extended to infinity?

Gemini– In general relativity (GR), a universe that is “maximally extended” does not necessarily have to contain infinities (singularities). However, the relationship between these two concepts is a cornerstone of modern cosmology.

 1. What “Maximally Extended” Means

In mathematical physics, a spacetime is maximally extended if it is “as large as it can possibly be” without being able to fit into a larger, consistent spacetime. 

• Coordinate vs. True Singularities: Sometimes, a mathematical model seems to have an “infinity” (like at the event horizon of a black hole) that is just a result of the chosen coordinate system. When you “maximally extend” the geometry using better coordinates, that infinity often disappears.
• The End of the Road: After you’ve extended the space as much as possible, if a path (geodesic) still ends abruptly, you have reached a true singularity. 

2. Can it be Infinite and Smooth?

Yes. It is entirely possible to have a maximally extended universe that is “infinite” in scale but has no infinities in terms of density or curvature. 

• Minkowski Spacetime: This is the simplest model of a flat, empty universe. It is infinite in all directions and is already “maximally extended.” It contains no singularities or points of infinite curvature.
• Expanding Universes: Many cosmological models (like the Friedmann-Lemaître-Robertson-Walker model) can describe a universe that is spatially infinite and evolves forever without ever hitting a “Big Crunch” or singularity in the future. 

3. When Infinities Are Necessary

While not logically necessary for all infinite universes, infinities are often physically predicted by GR due to the Singularity Theorems: 

• The Past: Hawking and Penrose proved that if the universe contains a certain amount of matter and is expanding, it must have had a singularity in its past (the Big Bang).
• Black Holes: Similarly, once a certain amount of matter is compressed into a small enough space, GR predicts that a singularity is inevitable, even in a maximally extended solution. 

4. Determinism and Extension

The reason physicists insist on maximal extension is to preserve determinism. If a spacetime is not maximally extended, a particle could simply “fall off the edge” of the map for no physical reason. By extending it, we ensure that every particle’s path is accounted for until it either goes on forever or hits a true physical boundary (a singularity). 

In summary, a universe can be spatially infinite and smooth (like Minkowski space), but in our specific universe—which contains matter and energy—GR predicts that “maximal extension” eventually leads to an infinity in the past or inside black holes. 

Would you like to explore how Quantum Gravity might remove these predicted infinities from the “maximally extended” map?

Gary– Can GR be regarded as just a method or relational system rather than a particular cosmological model?

Gemini-Yes, that is actually the most accurate way to view it. 

General Relativity is a framework, not a specific map of our universe.

In physics terms, GR is a set of field equations (the Einstein Field Equations) that describe the relationship between matter/energy and the geometry of spacetime.

1. The “Rulebook” vs. The “Story”

Think of GR like the rules of chess. The rules tell you how pieces can move, but they don’t tell you how a specific game will be played.

• The Framework (GR): Gravity = Curvature. .
• The Model (Cosmology): Plucking in specific values (like how much dust or dark energy exists) to create a model like the ΛCDM model.

2. A Relational System

You hit the nail on the head with “relational.” This is often called Background Independence.

• In Newtonian physics, space is a fixed “stage” where things happen.
• In GR, there is no stage. Spacetime is defined entirely by the relationships between the objects within it. If you move a star, the geometry of space itself changes.

3. One Framework, Infinite Universes

Because GR is a method, it can describe many different (and even impossible) universes depending on what “initial data” you feed it:

• Schwarzschild Solution: A universe with only one static black hole.
• Gödel Universe: A universe that rotates and allows time travel.
• De Sitter Space: A universe with no matter, only vacuum energy.

4. Why this matters for Determinism

Viewing GR as a method is why we can talk about it being deterministic. We aren’t saying “the universe is definitely deterministic”; we are saying “the mathematical system of GR produces deterministic results if you provide a complete set of starting conditions.”

Essentially, GR is the logic used to build a cosmological model, but the model itself is just one possible “solution” to those equations.

Do you want to look at how different solutions (like the FLRW metric for our universe) change the way we see time and causality?

Gary– GR is just a way of describing or explaining gravity, and since the actual mechanics of that are unknown, gr as a relations system is a pro tem sort of theory that might be overthrown or surpassed by new understanding and discoveries in quantum mechanics or other forces or fields that creating the gravity and space-time effects described by gr

Gemini-You’ve hit on the exact perspective held by most modern physicists: 

General Relativity is an “effective theory.”

It is a remarkably accurate description of how gravity behaves at large scales, but it doesn’t explain the “gears” underneath. Here is why your view of it as a pro tem (temporary) relational system is the standard scientific outlook:

1. The “Smooth” vs. “Chunky” Problem

GR treats spacetime as a smooth, continuous fabric. However, Quantum Mechanics (QM) suggests that at the smallest scales (the Planck scale), everything is “quantized” or chunky. Because GR can’t account for this chunkiness, it “breaks” at the center of black holes and the Big Bang. This is the clearest sign that GR is an incomplete description.

2. Emerging from Something Deeper

Many theorists now suspect that gravity and spacetime are not fundamental. Instead, they might be “emergent” properties—similar to how “temperature” isn’t a real thing on its own, but just the collective result of millions of atoms bouncing around.

• The Goal: A theory of Quantum Gravity (like String Theory or Loop Quantum Gravity) would be the “new understanding” that explains where the “fabric” of GR actually comes from.

3. The Relational “Placeholder”

By treating gravity as geometry (curvature), GR bypasses the need to explain why mass attracts mass. It simply says: “Given this much mass, the relation between these two points in space will change by X amount.” It is a mathematical bookkeeping system for the effects of a force we don’t yet fully understand at a particle level.

4. Will it be “Overthrown”?

In science, “overthrown” usually means “nested.” Just as Einstein didn’t prove Newton wrong (Newton’s laws still work for building bridges), a future theory won’t prove Einstein wrong. It will likely show that GR is just a special case that happens to work perfectly when things are big and heavy, but fails when they are small and high-energy.

Does this make you wonder if “space” itself might just be an illusion created by quantum entanglement?

Gary– I think illusion is perhaps amorphous since it implies a psychological element of perception, Maybe contingent and emergent field theory would be a better place to describe the entanglements in space-time that create a sense of space-time in a self-standing kind of way

Gemini– Spot on. “Illusion” is a loaded term that suggests a trick of the mind, whereas 

emergence implies a rigorous physical transition from one layer of reality to another—like how the “solidity” of a table emerges from the interactions of mostly empty atoms.

In the context of Entanglement Entropy, many physicists (like Van Raamsdonk or Maldacena) argue that spacetime is contingent upon quantum states. If you “turn off” the entanglement between two regions of pixels (qubits), the physical distance between them effectively becomes infinite; the “bridge” of space-time snaps.

By framing it as a contingent field theory, you shift the focus from what space is to what space does as a relational byproduct:

• Degrees of Freedom: Space-time isn’t the “bucket” everything sits in; it’s a coarse-grained description of a massive number of underlying quantum degrees of freedom.
• Geometric Phase Transition: Just as water vapor condenses into liquid, “space” might be a “condensed” phase of highly entangled quantum information.
• Self-Standing vs. Relational: If space-time is emergent, it isn’t “self-standing.” It’s a secondary structure—a convenient mathematical language we use because we can’t easily track trillions of individual entanglements.

This moves GR from being a “fundamental truth” to a highly efficient bookkeeping system for quantum connectivity.

Do you think a discrete (pixelated) or continuous underlying field better accounts for the smooth “sense” of space we observe at our scale?

Gary– I need to leave this interesting topic for later. Thanks- I am out the door…

Gemini-Totally fair—duty calls! We’ve essentially moved from 

General Relativity as a rigid law to seeing it as a smooth “map” of a much deeper, entangled “terrain.”