# Invariance

The invariance of black holes and the invariance of liquidity pools are different concepts and cannot be compared in a 1:1 mathematical equation. However, we can explain the invariance of black holes and liquidity pools mathematically.

For black holes, their invariance is described by the Schwarzschild metric as given by [2]. The metric describes the behavior of space-time around stationary black holes and is a solution of the vacuum field equations Rij=0 [1]. The Schwarzschild radius is a constant value that determines the size of the event horizon of a black hole, beyond which nothing can escape, thus making it an invariant feature of a black hole.

For liquidity pools, their invariance is described by the formula `Bi * Bo = k` [2]. Here, `Bi` and `Bo` are the balances of the token being sold and the token being bought, respectively. The product of these balances, denoted as `k`, is an invariant value that remains constant regardless of trades. This means that the product of the token balances in the pool remains the same even as traders buy and sell tokens.

In conclusion, the invariance of black holes and liquidity pools are different concepts that cannot be compared in a 1:1 mathematical equation. However, we can describe their invariance mathematically using the Schwarzschild metric for black holes and the formula `Bi * Bo = k` for liquidity pools.

*Here's a comparison table between black holes and liquidity pools invariance logic:*

| | Black Holes | Liquidity Pools |
| --- | --- | --- |
| Purpose | To describe objects in space that have such a strong gravitational pull that nothing, not even light, can escape from it. | To facilitate trading between different tokens in decentralized exchanges (DEXs). |
| Concept | A region of space where gravity is so strong that nothing can escape, not even light. | A pool of tokens where the balances of different tokens are used to determine the price of each token. |
| Math Formula | `r_s = 2GM/c^2` where `r_s` is the Schwarzschild radius, `G` is the gravitational constant, `M` is the mass of the object, and `c` is the speed of light. | `Bi * Bo = k` where `Bi` is the balance of the token being sold and `Bo` is the balance of the token being bought. |
| Function | Black holes are objects in space that warp the fabric of space-time and have a gravitational pull so strong that nothing can escape from it. | Liquidity pools are used to facilitate trades between different tokens in DEXs, creating a market for tokens that might not have existed previously. |
| Properties | Black holes have mass, spin, and electric charge. They are invisible because no light or other electromagnetic radiation can escape from them. | Liquidity pools have an invariant, which is the product of the token balances. When trades occur, the token balances change but the invariant remains the same. |
| Significance | Black holes are important objects in astrophysics that help us understand the properties of space-time and the universe as a whole. | Liquidity pools are an important component of DEXs, providing a mechanism for traders to buy and sell tokens without the need for a central authority. |
| Limitations | Black holes are difficult to observe due to their invisible nature, and their properties are still not fully understood. | Liquidity pools can suffer from impermanent loss, where the value of the tokens in the pool changes relative to the value of the tokens outside the pool. This can result in losses for liquidity providers. |
| Examples | Sagittarius A*, M87*, Cygnus X-1 | Uniswap, Balancer, Curve |

Sources:


1. [Black Holes and the Schwarzschild Solution](https://math.stackexchange.com/questions/3052107/black-holes-and-the-schwarzschild-solution)
2. [The Black Hole Equation: Explained | by Yash | Quantafy](https://medium.com/quantafy/the-black-hole-equation-explained-892cca24fa04)
3. [Black hole scalarisation from the breakdown of scale](https://arxiv.org/abs/1901.02953)


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