Weyl double copy in Lifshitz spacetimes

This paper validates a regularization prescription for the Weyl double copy by demonstrating its consistency with the Kerr-Schild formulation across three distinct Lifshitz black hole examples that previously posed challenges for reconciling the two classical double copy approaches.

The Gist

Imagine the universe is built on a set of hidden blueprints. Physicists have long suspected that the complex, messy force of Gravity (which bends space and time) is actually just a "squared" version of a much simpler, cleaner force called Electromagnetism (which deals with light and electricity). This idea is called the Double Copy.

Think of it like this: If you take the recipe for a simple soup (Electromagnetism) and "square" the ingredients, you get the recipe for a complex, rich stew (Gravity). Usually, this works perfectly. But sometimes, when you try to apply this recipe to specific, exotic types of black holes called Lifshitz black holes, the math breaks down. The ingredients seem to disappear, leaving you with an empty bowl instead of a stew.

This paper by Alkac, Gumus, and Olpak is like a group of chefs testing a new "fix-it" tool to see if they can salvage the recipe for these tricky black holes.

The Problem: The "Vanishing Act"

The authors are testing two different ways to prove that Gravity and Electromagnetism are linked:

1. The Kerr-Schild Method: A geometric approach that treats gravity as a background space with a specific "ripple" on top of it.
2. The Weyl Double Copy: A more abstract, mathematical approach using "spinors" (a type of mathematical object) to map the curvature of space directly to electric and magnetic fields.

Usually, these two methods agree. But with Lifshitz black holes, the second method (Weyl) hits a snag. In these specific black holes, certain parts of the math that should represent the "electric field" or the "curvature" turn into zero.

It's like trying to bake a cake, but the recipe says you need 2 eggs. You look in the fridge, and the egg carton is empty. If you just stop there, you can't make the cake, and the two methods (geometric vs. mathematical) no longer match.

The Solution: The "Regularization" Trick

In previous work, the authors found a clever workaround called regularization. Imagine you are calculating a limit in math where a number gets closer and closer to zero. Sometimes, if you approach zero from a slightly different angle, you don't get zero; you get a tiny, non-zero number that saves the equation.

The "regularization" trick is essentially saying: "Don't just plug in the number that makes the result zero. Pretend the number is slightly different, do the math, and then gently nudge it back to the original value." This often reveals a hidden, non-zero value that was previously hiding behind the zero.

What They Tested

The authors decided to put this "fix-it" tool to the test on three new, difficult examples of Lifshitz black holes that no one had tried before:

1. The "Double Trouble" Case: They looked at a black hole where two different parts of the math were vanishing at the same time. It was like having two empty egg cartons instead of one.

   • Result: The fix worked. They applied the trick to both missing parts, and the recipe was restored.

2. The "Heavy Gravity" Case: They looked at a black hole that exists in a universe where gravity is slightly different (it has an extra "R-squared" correction term, which is like adding a heavy spice to the gravity recipe).

   • Result: Even with this extra complexity, the math vanished in a way that broke the link. The regularization trick fixed it, showing the link between gravity and electromagnetism still holds.

3. The "Spinning" Case: Most black holes they study are static (standing still). They tested a black hole that is spinning (stationary). This is harder because the math gets messy with rotation.

   • Result: Surprisingly, even though the black hole was spinning, the math behaved nicely. The "fix" worked here too, confirming that the link between the two forces holds even when the black hole is rotating.

The Big Takeaway

The paper concludes that the "regularization" trick is a robust tool. It successfully fixed the broken math in all three new, complex scenarios.

In simple terms: The authors proved that even when the math for these exotic black holes looks like it's falling apart (with numbers turning to zero), there is a consistent way to patch it up. This confirms that the deep connection between Gravity and Electromagnetism (the Double Copy) is likely a fundamental truth of the universe, even in these weird, anisotropic (time and space scaling differently) black holes.

They didn't find a way to build a time machine or a new battery; they simply confirmed that the theoretical blueprint of the universe remains consistent, even in its most complicated corners.

Technical Summary

Technical Summary: Weyl Double Copy in Lifshitz Spacetimes

Problem Statement
The classical double copy (CDC) establishes a correspondence between solutions in gravity and gauge theories. Two primary formulations exist: the Kerr-Schild double copy (KSDC), which relies on metrics admitting Kerr-Schild coordinates, and the Weyl double copy (WDC), which utilizes spinorial techniques to relate the Weyl spinor to the field strength spinor of a Maxwell solution. While these formulations agree for vacuum solutions admitting Kerr-Schild coordinates, inconsistencies arise when applied to non-vacuum solutions or spacetimes with specific symmetries, such as Lifshitz black holes.

Specifically, Lifshitz black holes present challenges because their metric functions often contain terms that, when analyzed separately, lead to vanishing Weyl scalars (due to conformal flatness) or vanishing electric fields in the single copy. These vanishing terms prevent the standard WDC consistency condition, $\Psi_2 \propto Z^2/\phi$, from being satisfied term-by-term, creating a discrepancy between the KSDC and the WDC. Previous work suggested that a regularization prescription could restore consistency, but this required further testing on more complex and novel examples.

Methodology
The authors test the validity of a specific regularization prescription on three distinct examples of Lifshitz black hole solutions found in the literature. The methodology involves:

1. Kerr-Schild Formulation: Expressing the Lifshitz black hole metrics in Kerr-Schild coordinates ($g_{\mu\nu} = \bar{g}_{\mu\nu} + \phi k_\mu k_\nu$) relative to a Lifshitz background. This allows the derivation of the single copy gauge field ($A_\mu = \phi k_\mu$) and the zeroth copy scalar ($\phi$) via the trace-reversed Einstein equations.
2. Weyl Double Copy Analysis: Calculating the Weyl scalar ($\Psi_2$) and the single copy field strength scalar ($Z$) for the same solutions. The authors apply the sourced Weyl double copy (SWDC) framework, which treats each term in the metric function separately.
3. Regularization Procedure: For terms where the critical exponent $n$ causes $\Psi_2$ or $Z$ to vanish (specifically $n^* = z, 2z-2$ for $\Psi_2$ and $n^* = 2z$ for $Z$), the authors apply the regularization method proposed in [31]. This involves:
   • Treating the exponent $n$ as a variable rather than the critical value $n^*$.
   • Scaling the coefficient $a_{n^*}$ by $(n-n^*)^{-1}$.
   • Calculating the resulting term in the consistency condition.
   • Setting $n = n^*$ to recover a non-zero, finite contribution that restores the consistency condition.

Key Contributions and Results
The paper validates the regularization prescription across three novel scenarios:

• Example I: Two Regularized Terms: The authors analyze a charged Lifshitz topological black hole solution [32] with a metric function containing multiple terms. They identify two distinct terms requiring regularization: one arising from the uncharged solution and another from the electric charge coupling. Both terms correspond to critical exponents where the Weyl scalar or electric field would naively vanish. Applying the regularization procedure to both terms simultaneously successfully restores the term-by-term consistency between the KSDC and SWDC.
• Example II: $R^2$-Correction: The study examines a Lifshitz black hole solution [33] arising from an Einstein-Hilbert action with an $R^2$ correction. The authors treat the higher-curvature correction as an effective matter source. For the specific Lifshitz exponent $z=3/2$, a term in the metric function leads to a vanishing single copy electric field ($Z=0$). The regularization procedure is applied to this term, demonstrating that the SWDC remains consistent with the KSDC even when the gravitational solution is derived from higher-curvature gravity.
• Example III: Stationary Lifshitz Black Hole: The authors investigate a stationary Lifshitz black hole solution [34] obtained via a local coordinate transformation from a static planar horizon solution. This is the first application of the SWDC to a stationary Lifshitz solution. The analysis confirms that despite the introduction of rotation and a magnetic field component in the single copy, the consistency condition holds. The authors note that in this specific case, the field strength spinor remains real, and the regularization of critical exponents ($n^*=2z-2, 2z$) ensures the match between the two formulations.

Significance and Claims
The paper claims that the regularization prescription introduced in [31] is robust and applicable to a broader class of Lifshitz black hole solutions than previously tested. The authors demonstrate that:

1. The prescription works even when multiple distinct terms in the metric function require regularization.
2. It is effective for solutions arising from higher-curvature gravity theories (treated as effective matter).
3. It extends to stationary Lifshitz black holes, bridging a gap in the literature regarding time-dependent or rotating solutions in the context of the WDC.

The authors conclude that the consistency between the Kerr-Schild and Weyl double copy formulations can be restored for these Lifshitz spacetimes by employing the regularization scheme. They note that while the specific stationary solution studied has special properties due to its construction via coordinate transformation, the results provide further evidence for the validity of the regularization approach. The paper suggests that future work should explore alternative background spacetimes for the KSDC in Lifshitz contexts and test the regularization scheme on generic stationary solutions not derived from simple coordinate transformations.