On energy and its positivity in spacetimes with an expanding flat de Sitter background

This paper establishes a definition of quasi-local energy for initial data sets with an umbilic second fundamental form in an expanding de Sitter background and proves its positivity for certain bounded values of the cosmological constant by adapting the Liu-Yau energy framework.

The Gist

Here is an explanation of the paper "On Energy and Its Positivity in Spacetimes with an Expanding Flat de Sitter Background," translated into simple, everyday language with creative analogies.

The Big Picture: Measuring the Weight of the Universe

Imagine you are trying to weigh a single apple. In a normal kitchen, you just put it on a scale. But what if the kitchen itself is a giant, expanding balloon? And what if the "scale" you are using relies on the floor being perfectly flat and still?

This is the problem physicists face when trying to calculate the energy (or mass) of a gravitational system, like a black hole or a star, in our actual universe.

For a long time, physicists used a "standard kitchen" model called Minkowski space. This is a universe that is empty, flat, and doesn't expand. In this flat, still universe, we have a perfect, well-defined way to measure energy (called the ADM energy). It works great for isolated systems in a static universe.

But here's the catch: Our real universe isn't a flat, still kitchen. It's an expanding balloon. We live in a universe with a positive "cosmological constant" (a kind of dark energy pushing everything apart). In this expanding universe, the old rules for weighing energy break down because the "floor" is stretching, and there is no single, global "scale" that works for the whole universe.

The New Idea: A Quasi-Local Scale

The authors of this paper, Rodrigo Avalos, Eric Ling, and Annachiara Piubello, are asking: "How do we weigh a star if the universe is expanding?"

They propose a new solution: instead of trying to weigh the whole universe at once (which is impossible because the expansion gets in the way), we should weigh small, bounded chunks of the universe. They call this "Quasi-Local Energy."

Think of it like this:

• Old Method (Global): Trying to weigh the entire ocean to see how heavy a single fish is. Impossible because the ocean is moving and expanding.
• New Method (Quasi-Local): Putting the fish in a small, clear bucket of water and weighing just that bucket.

The "De Sitter" Background

To make this work, they chose a specific type of expanding universe as their background model, called de Sitter space.

• The Analogy: Imagine a loaf of raisin bread rising in the oven. The raisins are galaxies, and the dough is space. As the bread rises, the raisins move apart.
• In this paper, they look at a specific way of slicing this rising bread (called "flat-expanding coordinates"). In this slice, the space looks flat (like a sheet of paper), but it is expanding over time.

The Main Challenge: The "Cosmic Horizon"

In an expanding universe, there is a limit to how far you can see. This is called the Cosmological Horizon. It's like a foggy wall that moves away from you as you travel. You can never see or interact with things beyond this wall.

This creates a problem for their "bucket" analogy:

1. To weigh the energy, they have to draw a boundary (a sphere) around the object.
2. This boundary must stay inside the "foggy wall" (the horizon). If the boundary crosses the horizon, the math breaks, and the energy becomes undefined.
3. The authors had to figure out: How big can our "bucket" be before it hits the edge of the universe?

The Solution: A New Formula

They created a new formula for energy (let's call it $E_\lambda$) that works specifically for this expanding, flat universe.

• The Ingredients: The formula compares two things:
  1. How curved the surface of our "bucket" is in the real, expanding universe.
  2. How curved that same surface would be if it were sitting in a perfect, flat, non-expanding Euclidean space (like a standard sheet of paper).
• The Result: They proved that as long as the "cosmological constant" (the rate of expansion) isn't too huge, this new energy value is always positive.

Why does "Positive Energy" matter?
In physics, negative energy is weird and often implies instability (like a ball rolling up a hill instead of down). Proving that energy is always positive is a fundamental "sanity check" for the laws of physics. It confirms that our new way of measuring energy in an expanding universe makes sense and is stable.

The "Goldilocks" Zone

The paper finds a "Goldilocks" zone for this new energy definition:

• If the universe expands too fast (the cosmological constant is too high), the "bucket" hits the cosmic horizon too quickly, and the math fails.
• If the universe expands slowly (which is true for our actual universe, as the expansion rate is tiny on human scales), the math works perfectly, and the energy is positive.

The "Rigidity" Mystery

The authors also touched on a "rigidity" question. In the old flat universe, if the energy is zero, it means the space is perfectly empty and flat (like a calm, empty ocean).
In their new expanding model, they found that if the energy is zero, the boundary of the "bucket" must be a perfect sphere. They suspect (but haven't fully proven yet) that if the energy is zero, the whole region inside must be a perfect piece of the expanding de Sitter universe.

Summary in a Nutshell

1. The Problem: We can't use old methods to weigh gravity in an expanding universe because the "floor" is moving.
2. The Fix: Weigh small, local regions (Quasi-Local Energy) instead of the whole universe.
3. The Constraint: These regions must stay inside the "Cosmic Horizon" (the edge of the observable universe).
4. The Discovery: The authors created a new formula for this energy and proved it is always positive (stable) as long as the universe isn't expanding too violently.
5. The Takeaway: This gives us a solid mathematical foundation for understanding energy in the real, expanding universe we live in, rather than just in the theoretical, static universes of the past.

It's like finally inventing a scale that works on a moving train, proving that even while the world is rushing by, the laws of physics still hold steady.

Technical Summary

Here is a detailed technical summary of the paper "On Energy and Its Positivity in Spacetimes with an Expanding Flat de Sitter Background" by Rodrigo Avalos, Eric Ling, and Annachiara Piubello.

1. Problem Statement

The paper addresses the fundamental challenge of defining and proving the positivity of energy for isolated gravitational systems within an expanding cosmological universe characterized by a positive cosmological constant ($\Lambda > 0$).

• Limitations of Traditional Approaches: Classical Positive Energy Theorems (Schoen–Yau, Witten) rely on asymptotically flat spacetimes (Minkowski background). These definitions assume the metric approaches the Euclidean metric at infinity and utilize global timelike Killing fields (time-translation symmetry) to define conserved quantities like ADM mass.
• The Cosmological Obstacle: In a universe with $\Lambda > 0$, the background is de Sitter space, not Minkowski space.
  • De Sitter space possesses a cosmological horizon ($r = 1/\lambda$, where $\lambda = \sqrt{\Lambda/3}$).
  • There are no global timelike Killing fields; the natural time-like vector field $\partial_t$ becomes spacelike beyond the horizon.
  • Consequently, a global definition of energy at "infinity" is obstructed, and standard conservation laws derived from time-translation symmetry do not apply globally.
• Goal: To establish a physically meaningful notion of energy for initial data sets in an expanding de Sitter background and prove that this energy is non-negative under appropriate conditions.

2. Methodology and Framework

A. Geometric Setting

The authors adopt the flat-expanding coordinate system for de Sitter space. In these coordinates:

• The spacetime is foliated by spatial hypersurfaces that are isometric to Euclidean 3-space ($\mathbb{R}^3$).
• The second fundamental form of these slices is umbilic ($k_0 = \lambda g$), representing the uniform expansion of the universe.
• The background metric is $g_0 = -(1-\lambda^2 r^2)dt^2 - 2\lambda r dr dt + dr^2 + r^2 d\Omega^2$.

B. Quasi-Local Energy Definition

Since global energy is ill-defined, the authors propose a quasi-local energy $E_\lambda$ defined on a bounded region $\Omega$ with boundary $\Sigma = \partial \Omega$.

• Reference: They adapt the Liu–Yau energy (originally defined for asymptotically flat spacetimes) to the de Sitter setting.
• Construction:
  1. Let $(\Omega, g, k)$ be a compact initial data set with boundary $\Sigma$ having positive Gauss curvature.
  2. By the Weyl Embedding Theorem, $\Sigma$ is isometrically embedded into Euclidean space $\mathbb{R}^3$ as the boundary of a convex domain. Let $H_0$ be the mean curvature of this embedding in $\mathbb{R}^3$.
  3. Let $H$ be the mean curvature of $\Sigma$ within the physical spacetime $(\Omega, g, k)$.
  4. The energy is defined as:
     $$E_\lambda := \frac{1}{8\pi} \int_\Sigma \left( \sqrt{H_0^2 - 4\lambda^2} - \sqrt{H^2 - (\text{tr}_\Sigma k)^2} \right) d\mu$$
  • Note: The term $\sqrt{H_0^2 - 4\lambda^2}$ represents the norm of the mean curvature vector in the reference de Sitter space, adjusted for the cosmological constant.

C. Constraints and Conditions

• Dominant Energy Condition (DEC): The initial data must satisfy $\rho \ge |J|_g$ with respect to $\Lambda$. In the umbilic case ($k=\lambda g$), this reduces to the non-negativity of the scalar curvature ($R_g \ge 0$).
• Horizon Constraint: The isometric embedding of $\Sigma$ into the reference space must lie entirely within the cosmological horizon ($r < 1/\lambda$) to ensure the reference Killing field remains timelike.

3. Key Contributions and Results

Main Theorem (Theorem 3.3)

The authors prove a Positive Energy Theorem for the quasi-local energy $E_\lambda$ under specific constraints on the cosmological parameter $\lambda$.

• Statement: For a compact initial data set $(\Omega, g, k)$ satisfying the DEC, if $\lambda$ is sufficiently small (specifically $\lambda \le \alpha$, where $\alpha$ is a constant derived from the geometry of $\Sigma$), then:
  1. The surface $\Sigma$ can be isometrically embedded inside the cosmological horizon.
  2. The energy is non-negative: $E_\lambda \ge 0$.
  3. If $0 < \lambda < \alpha$, then$E_\lambda > 0$.
  4. Rigidity: If $E_\lambda = 0$ at the critical value $\lambda = \alpha$, the boundary $\Sigma$ is isometric to a disjoint union of round spheres.

Corollary 3.4 (Riemannian Case)

For the specific case of umbilic initial data ($k = \lambda g$), which corresponds to the vacuum constraint equations with $\Lambda = 3\lambda^2$:

• There exists a constant $\lambda_{\max}$ (dependent only on the Riemannian geometry $(\Omega, g)$) such that for all $\lambda \in [0, \lambda_{\max}]$, the energy $E_\lambda$ is well-defined and non-negative.
• This result relies on the Shi–Tam rigidity for the Brown–York energy in the limit $\lambda \to 0$.

Technical Lemmas

• Lemma 5.1 (Embedding Condition): Establishes a sufficient condition on the Gauss curvature ($K \ge \frac{3}{8\pi^2}\lambda^2$) to ensure the Weyl embedding fits within the cosmological horizon. This uses Jung's Theorem and the Bonnet–Myers theorem to bound the diameter of the embedded surface.
• Lemma 5.2 (Positivity): Proves that $E_\lambda \ge 0$ by comparing it to the standard Liu–Yau energy ($E_{LY}$) and showing that the "cost" of the cosmological constant does not exceed the geometric energy density for small $\lambda$.

4. Examples and Applications

The paper validates the theory through two main examples:

1. Schwarzschild–de Sitter Space: The authors compute $E_\lambda$ for coordinate spheres in this exact solution. They show that for sufficiently large spheres (where the horizon condition holds), the energy is strictly positive.
2. Conformal Blow-ups: Using Schoen's blow-up argument, they construct a family of $\Lambda$-vacuum initial data sets (scalar flat metrics) where the theorem applies, demonstrating the existence of many such systems.

5. Significance and Discussion

• Physical Relevance: The work bridges the gap between mathematical relativity (positive energy theorems) and observational cosmology (accelerating expansion, $\Lambda > 0$). It provides the first candidate for a positive energy theorem in an expanding de Sitter background.
• Quasi-Local Necessity: It rigorously justifies the shift from global to quasi-local energy definitions in cosmological settings, acknowledging the obstruction posed by the cosmological horizon.
• Optimality and Future Directions:
  • The authors note that the upper bound on $\lambda$ ($\lambda \le \alpha$) is likely not optimal due to the use of Jung's theorem estimates, but it is physically meaningful given the extremely small observed value of the cosmological constant.
  • Rigidity: A full rigidity theorem (proving $E_\lambda = 0 \implies$ data is exactly de Sitter) remains an open conjecture; the current result only proves rigidity for the boundary geometry (round spheres).
  • Embedding Dependence: The paper highlights that $E_\lambda$ depends on the choice of isometric embedding (foliation). It suggests future work should explore variational approaches (minimizing over embeddings, similar to Wang–Yau energy) to define an "optimal" energy in de Sitter space.

Conclusion

This paper successfully extends the Positive Energy Theorem to expanding cosmological spacetimes by defining a quasi-local energy $E_\lambda$. It proves that for initial data sets with a positive cosmological constant, provided the expansion rate is not too large relative to the local geometry, the energy remains non-negative. This establishes a foundational step toward understanding the energetics of isolated systems in our actual universe.