Mimetic gravity in the extended objects framework

This paper introduces Lovelock-type brane gravity (LBG) as a second-order extension of geodetic brane gravity within an elasticity framework, demonstrating how it can be reformulated as mimetic embedding gravity where a fictional current generates an effective energy-momentum tensor with perfect fluid characteristics.

The Gist

Here is an explanation of the paper "Mimetic gravity in the extended objects framework" using simple language, creative analogies, and metaphors.

The Big Picture: The Universe as a Floating Sheet

Imagine our entire universe isn't just a 3D room, but a giant, flexible sheet of rubber (a "brane") floating inside a much larger, empty, flat room (the "ambient space").

In standard physics (General Relativity), we usually think of gravity as the curvature of space itself. But this paper asks a different question: What if gravity is just the shape of this rubber sheet as it moves through the bigger room?

The author, Efraín Rojas, is trying to figure out the rules for how this sheet moves. Specifically, he wants to find a set of rules that are simple enough to be solvable (mathematically speaking) but complex enough to explain the weird stuff we see in the universe, like Dark Matter.

The Problem: Too Many Wiggles (The "Ghost" Problem)

When you try to write down the math for how a sheet moves, you usually run into a problem. If you include how the sheet bends and twists (its "curvature"), the math gets messy. It starts predicting things that shouldn't exist—like extra, invisible particles or "ghosts" that break the laws of physics.

Think of it like a guitar string. If you pluck it, it vibrates in a nice, predictable way. But if you try to write a rule that says the string's vibration depends on how fast it's changing its shape and how fast that change is changing, the string might start vibrating in impossible, chaotic ways.

The author's goal was to find a specific type of math (a "Lagrangian") that describes this sheet without creating those chaotic ghosts. He found a special family of equations called Lovelock-type Brane Gravity (LBG). These are like a "Goldilocks" set of rules: they are complex enough to be interesting, but simple enough to keep the physics stable.

The Surprise: The "Invisible Fluid" (Dark Matter)

Here is the coolest part of the paper.

When the author solved these equations, he found that the math naturally included a "bonus" term. It's like if you were calculating the path of a boat on a lake, and your math suddenly said, "Oh, and there's a hidden current pushing the boat, even though you can't see the water moving."

In physics, we call this invisible push Dark Matter.

Usually, scientists have to invent Dark Matter to explain why galaxies spin too fast. But in this theory, Dark Matter isn't a mysterious particle we haven't found yet. It's a side effect of the geometry. It's a "fictional" or "mimetic" matter that appears just because of how the universe-sheet is embedded in the bigger space.

The Analogy: The Elastic Band and the Internal Stress

To explain why this invisible matter exists, the author uses a concept from elasticity theory (the study of how rubber bands and springs stretch).

Imagine a giant, elastic band floating in space.

1. External Forces: If you push the band from the outside, it moves.
2. Internal Forces: Inside the band, the molecules are pulling on each other.

The author argues that the "Dark Matter" current ($T^a_\mu$) is like the internal tension of the band.

• If you look at the band from the outside, the internal forces cancel each other out. The center of the band doesn't move just because of its own internal tension.
• However, if you look at the shape of the band, that tension changes how it curves.

The paper suggests that what we perceive as Dark Matter is actually the universe's "internal stress." It's a force that exists inside the fabric of space-time, keeping the sheet taut, but it doesn't push the universe around like a rocket engine. It just changes the geometry.

The "Mimetic" Trick: Copying the Rules

The paper also connects this to something called Mimetic Gravity.

Think of "Mimetic" as "mimicking" or "copying."

• In standard gravity, we need to add "Dark Matter" as a separate ingredient to make the recipe work.
• In this paper, the author shows that if you change the ingredients slightly (by treating the position of the universe-sheet as the main variable instead of the metric), the "Dark Matter" ingredient mimics itself into existence.

It's like baking a cake where you don't need to add chocolate chips; the batter naturally turns into chocolate chips as it bakes because of the way the heat interacts with the flour. The "Dark Matter" is a natural byproduct of the geometry.

The "Lagrange Multiplier" (The Magic Glue)

Finally, the author shows how to write this mathematically using "Lagrange multipliers."

Imagine you are trying to tie a knot, but you have a rule: "The rope must stay exactly 1 meter long." You can't just let the rope stretch. You need a "magic glue" (the Lagrange multiplier) that forces the rope to obey that rule.

In this paper, the "magic glue" is the Dark Current. It's a mathematical tool that forces the universe-sheet to obey the rules of geometry. When you solve the equations, this "glue" turns out to look exactly like the energy and pressure of a fluid (Dark Matter).

Summary: What Did We Learn?

1. The Universe is a Sheet: We can model the universe as a sheet floating in a higher-dimensional space.
2. Stable Rules: The author found a specific set of rules (LBG) that describe this sheet without breaking physics.
3. Dark Matter is Geometry: The mysterious "Dark Matter" that holds galaxies together isn't a hidden particle. It's a natural consequence of the sheet's shape and internal tension.
4. No New Particles Needed: We might not need to find a new particle in a collider. The "Dark Matter" is already there, hiding in the geometry of space-time itself.

In a nutshell: The universe is like a trampoline. The "Dark Matter" isn't a heavy weight sitting on the trampoline; it's the tension in the fabric itself, which we can now describe mathematically without breaking the laws of physics.

Technical Summary

Here is a detailed technical summary of the paper "Mimetic gravity in the extended objects framework" by Efraín Rojas.

1. Problem Statement

The paper addresses two interconnected problems in theoretical physics:

1. The Nature of Dark Matter: Standard General Relativity (GR) requires the introduction of non-baryonic "dark matter" to explain cosmological observations. The author investigates whether dark matter can be an emergent geometric effect rather than a new physical field.
2. Higher-Order Derivative Pathologies: In theories of extended objects (branes), Lagrangians depending on extrinsic curvature ($K_{ab}$) typically lead to fourth-order equations of motion (EOM). These higher-order equations introduce non-physical degrees of freedom (ghosts) via the Ostrogradsky instability. The goal is to identify a class of geometric models that yield second-order EOMs, ensuring physical consistency while extending the known Geodetic Brane Gravity (GBG).

2. Methodology

The author employs a rigorous geometric and variational approach within the framework of extended objects (branes) embedded in a flat, codimension-one Minkowski ambient spacetime.

• Geometric Setup: The brane $\Sigma$ is described by embedding functions $X^\mu(x^a)$ mapping a $(p+1)$-dimensional world volume $m$ into an $N$-dimensional flat space. The geometry is defined by the induced metric $g_{ab}$ (first fundamental form) and the extrinsic curvature $K_{ab}$ (second fundamental form).
• Variational Analysis: The action is constructed as $S = \int \sqrt{-g} L(g_{ab}, K_{ab})$. The author performs a variation of this action to derive the general equations of motion.
• Matrix Discriminant Analysis: To ensure the resulting EOMs are second-order, the author analyzes the tensor $L_{ab} = \partial L / \partial K_{ab}$. Using matrix theory (specifically properties of generalized Kronecker deltas and discriminants), the author derives the necessary conditions for $L_{ab}$ to be divergence-free ($\nabla_a L_{ab} = 0$), which eliminates higher-order derivatives in the EOM.
• Mimetic Reformulation: The theory is reformulated by introducing a "dark current" $T^a_\mu$ as a shift in the conserved stress tensor. The author utilizes Lagrange multipliers to construct extended actions that enforce the constraints defining this current.
• Elasticity Analogy: The physical origin of the current is interpreted through the lens of elasticity theory, treating the brane as an elastic medium where internal stresses sum to zero.

3. Key Contributions

A. Derivation of Lovelock-Type Brane Gravity (LBG)

The paper derives the most general class of Lagrangians $L(g_{ab}, K_{ab})$ that yield second-order equations of motion. These are identified as Lovelock-type brane invariants (LBI), denoted $L_s$, which are constructed from the $s$-th discriminant of the extrinsic curvature matrix.

• The resulting action is:
  $$S_{LBG} = \int d^{p+1}x \sqrt{-g} \sum_{s=0}^{p} \alpha_s L_s(g_{ab}, K_{ab})$$
• This framework generalizes the Regge-Teitelboim (RT) model (which corresponds to the $s=2$ case, i.e., the Einstein-Hilbert term).
• The conserved stress tensor is shown to be entirely tangential to the brane, expressed via Lovelock-type brane tensors (LBT), $J^{(s)}_{ab}$.

B. Emergence of Mimetic Dark Matter

The author demonstrates that LBG can be reformulated as a mimetic gravity theory. By shifting the conserved stress tensor $f^a_\mu$ by an arbitrary vector current $T^a_\mu$ (where $\nabla_a T^a_\mu = 0$), the equations of motion remain unchanged.

• This current decomposes into a tangential component $T^{ab}$ and a normal component.
• To avoid altering the order of the EOMs, the normal component must vanish, leaving $T^a_\mu = T^{ab} \partial_b X^\mu$.
• The tensor $T^{ab}$ satisfies $\nabla_a T^{ab} = 0$ and $T^{ab}K_{ab} = 0$. This $T^{ab}$ behaves exactly like the energy-momentum tensor of a pressureless fluid (dust), effectively acting as fictitious dark matter (or embedding matter) arising purely from geometry.

C. Physical Origin via Elasticity Theory

The paper provides a mechanical interpretation for the dark current $T^a_\mu$. Drawing from elasticity theory, the current is identified as an internal stress tensor.

• Just as the sum of internal forces in a non-relativistic system is zero (leaving the center of mass motion unaffected), the net force exerted by $T^a_\mu$ on the brane boundary is zero.
• This explains why the current does not alter the brane's dynamics despite contributing to the energy-momentum balance.

D. Variational Formulation with Lagrange Multipliers

The author constructs two extended actions ($S_1$ and $S_2$) that incorporate the dark current $T^{ab}$ and the embedding functions $X^\mu$ via Lagrange multipliers.

• Action $S_1$: Uses $T^{ab}$ as a multiplier to enforce the definition of the induced metric ($g_{ab} = \partial_a X \cdot \partial_b X$).
• Action $S_2$: A bimetric-like action involving a square root of a matrix structure, which enforces the constraint $T^a \cdot \partial_a X = \text{Tr}(\sqrt{g^{-1}T \cdot T})$.
• Variation of these actions reproduces the conservation laws for the dark current and the induced metric constraints, providing a complete variational foundation for the mimetic interpretation.

4. Results

• Second-Order Dynamics: The LBG framework successfully avoids the Ostrogradsky instability by restricting the Lagrangian to Lovelock-type invariants, ensuring the EOMs remain second-order in derivatives of $X^\mu$.
• Geometric Dark Matter: The theory naturally generates a conserved energy-momentum tensor $T^{ab}$ that mimics dark matter. In the specific case of the RT model ($s=2$), the EOMs reduce to the standard GBG equations with an added dark matter term:
  $$G_{ab} - \kappa T^{matter}_{ab} = \kappa T^{dark}_{ab}$$
• Consistency: The derived equations satisfy the Bianchi identities and the conservation of the total stress tensor. The "dark" component is shown to be a geometric artifact of the embedding, not a new matter field.
• Embedding Constraints: The paper confirms that for a 4-dimensional FRW brane, a 5-dimensional Minkowski bulk is sufficient to host the geometry, validating the local existence of the embedding frame.

5. Significance

• Unification of Concepts: The paper bridges Geodetic Brane Gravity, Lovelock Gravity, and Mimetic Gravity. It shows that mimetic dark matter is not an ad-hoc addition but a natural consequence of the geometric constraints on extended objects in flat spacetime.
• Alternative to Particle Dark Matter: It offers a compelling alternative to particle dark matter, suggesting that the observed gravitational effects attributed to dark matter could be manifestations of the extra dimensions and the geometry of the embedding space.
• Cosmological Implications: The framework provides a tool to study cosmic acceleration and late-time cosmology without invoking a cosmological constant or exotic scalar fields, relying instead on the geometric properties of the brane and its extrinsic curvature.
• Theoretical Robustness: By grounding the "dark current" in elasticity theory and providing a rigorous variational formulation with Lagrange multipliers, the paper strengthens the theoretical viability of embedding gravity as a candidate for a quantum theory of gravity or a modification of GR.

In conclusion, Rojas establishes Lovelock-type Brane Gravity as a consistent, higher-dimensional geometric framework where "dark matter" emerges naturally as a conserved current associated with the internal stresses of the brane, offering a purely geometric explanation for phenomena usually attributed to unseen matter.