Finite mass two throat wormholes: global light rings, branch resolved strong lensing, and scalar transmission

Imagine the universe as a giant, stretchy trampoline. Usually, when we think of a "wormhole," we picture a tunnel connecting two distant points, like a shortcut through a mountain. But in this new study, physicists Anirudh Pradhan and his team have designed a very specific, mathematically perfect version of a wormhole that looks less like a simple tunnel and more like a figure-eight made of space itself.

Here is the big picture: They built a model of a wormhole with two throats (the narrowest parts of the tunnel) and a middle equator (a bulge in the middle), all sitting inside a universe that has a finite mass. This means the wormhole isn't just a mathematical trick with infinite weight; it has a specific, calculable weight that stays the same no matter which end of the wormhole you look at.

The "Figure-Eight" Universe

Think of the shape of this wormhole like a dumbbell or a figure-eight.

• The Throats: These are the two narrow "waists" of the figure-eight. In this model, there are two of them, symmetrically placed.
• The Equator: Between the two waists, the space bulges out, creating a middle section that is wider than the throats.
• The Mass: The authors calculated that if you were to weigh this object from either end of the universe, you would get the exact same number. It's a "finite mass" object, meaning it's heavy but not infinitely heavy, and it behaves like a normal star or planet when you are far away from it.

The "Light Rings" and the Traffic Jam

One of the coolest things about this model is how light behaves around it. In physics, light doesn't always travel in straight lines; gravity can bend it. Around this wormhole, light can get stuck in circular orbits called light rings.

The authors discovered that depending on the specific settings of their model (which they call parameters $m$ and $\chi$), the light rings behave in three different "phases":

1. Phase I: The light gets stuck right at the two narrow throats.
2. Phase II: This is the most surprising part. The light gets stuck in four different rings: two near the throats and two further out. Even though all four rings are at the same distance from the center and look the same to a distant observer, they are not the same. Two of them are "unstable" in a way that makes light spiral away quickly, while the others are "unstable" in a different, slower way. It's like having four identical-looking traffic circles where cars in two of them spin out of control instantly, while cars in the other two take a little longer to crash.
3. Phase III: The light rings move to the outside, and the middle bulge becomes a place where light can get stuck too.

The paper explicitly rules out the idea that all these rings are the same just because they look the same. They proved mathematically that even if the rings share the same "impact scale" (how close the light gets), they have different "Lyapunov exponents." In plain English, this means the instability rates are different. Some light rings are more chaotic than others.

The "Cross-Throat" Lensing Effect

When light passes near a black hole or a wormhole, it bends, creating a phenomenon called gravitational lensing. Usually, we think of light bending back to the same side it came from. But because this wormhole has two ends, light can do something special: it can enter one side, zip through the middle, and pop out the other side.

The authors calculated exactly how much the light bends in these two scenarios:

• Same-side bending: Light turns around and comes back.
• Cross-throat bending: Light goes all the way through.

They found a "magic number" (a logarithmic coefficient) that tells us how strong this bending is. For the "cross-throat" scenario, the bending is much stronger because the light has to pass through four different critical rings (in Phase II) instead of just one. It's like a runner having to sprint through four different wind tunnels instead of just one; the total resistance adds up. The paper confirms this with direct numerical integration, meaning they ran the numbers on a computer and the results matched their math perfectly.

The "Sound" of the Wormhole

To test if their model was real, the authors didn't just look at light; they also simulated how scalar waves (a type of theoretical wave, similar to sound or ripples in a pond) would travel through it.

They found that the wormhole acts like a series of barriers.

• In some phases, the wave hits a double barrier and gets trapped, bouncing back and forth.
• In other phases, the wave can "tunnel" through, but only at very specific frequencies. This is called resonant transmission.

Think of it like pushing a child on a swing. If you push at just the right rhythm, the swing goes high. If you push at the wrong rhythm, nothing happens. The wormhole only lets waves through if they hit the "right rhythm" (frequency). The paper shows that these resonances happen because of the multi-barrier shape of the wormhole, and the computer simulations confirmed that the waves behave exactly as the math predicted.

What This Model is NOT

It is important to know what this paper says this wormhole is not:

• It is not a black hole. There is no event horizon (the point of no return). You can theoretically travel through it.
• It is not a model with infinite mass. The mass is finite and calculated to be the same at both ends.
• It is not a model that ignores energy conditions. The authors proved that while the local energy conditions are violated (which is required for wormholes to exist), the total energy violation across the whole wormhole is strictly negative and exact. They didn't just guess; they derived an exact formula for this.

How Sure Are They?

The authors are very confident in their results because they didn't just suggest ideas; they derived them mathematically and verified them with computer simulations.

• They calculated the exact shape of the wormhole and the mass.
• They classified every possible phase of light rings for all positive masses.
• They derived exact formulas for how light bends and how waves scatter.
• They ran direct numerical integrations (computer simulations) to check their math, and the numbers matched perfectly.

For example, they calculated that for a specific setup (where the mass parameter $m = 0.3$ and the redshift deformation $\chi = 0.45$), the "cross-throat" bending coefficient is roughly 10.8, which is more than five times stronger than the "same-side" bending. This isn't a guess; it's a number they calculated and then proved with a computer.

The Bottom Line

This paper presents a complete, self-contained "toy universe" with a two-throat wormhole. It shows that if you build a wormhole with a finite mass and a specific shape, you get a complex system where light and waves behave in surprising ways. The biggest discovery is that looking the same doesn't mean acting the same: four light rings can look identical to a distant observer, but they actually have different levels of chaos and instability. The authors have provided a single, consistent model where the geometry, the mass, the light, and the waves all fit together perfectly, offering a new way to understand how these exotic objects might work if they ever exist in our universe.