# Understanding the Limitations of Solving General Relativity

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# This Is Why Scientists Will Never Exactly Solve General Relativity

## Even extremely simple configurations in General Relativity cannot be solved exactly. Here’s the science of why.

Shifting from Newton's perspective to Einstein's on the Universe is a profound transformation. According to Newtonian physics, the Universe functions as a completely deterministic system. A scientist with full knowledge of the masses, positions, and momenta of every particle could predict their future states with precision.

In theory, Einstein's equations also possess deterministic qualities, leading to the idea that if one knew the mass, position, and momentum of each particle, they could forecast the future. However, while Newtonian mechanics allows us to derive equations governing particle behavior, achieving the same in a universe governed by General Relativity is far more challenging.

In a Newtonian framework, every massive object applies a well-defined gravitational force on all others. As long as we can calculate the gravitational force between all mass pairs, we can determine the motion of those masses using the formula F = ma. This allows us to predict the evolution of the Universe.

Conversely, General Relativity complicates matters significantly. Even with complete knowledge of particle positions, masses, and momenta, plus the relativistic reference frame, one still cannot determine the evolution of systems due to the inherent complexity of Einstein's equations.

In General Relativity, an object's movement and acceleration depend not on an applied force but on the curvature of spacetime itself. This presents a significant obstacle since the curvature is influenced by all matter and energy in the Universe, which encompasses far more than just the positions and momenta of massive particles.

Additionally, in contrast to Newtonian gravity, the interaction of any mass is significant, as its energy also distorts spacetime. When two massive objects interact, they emit gravitational radiation, which propagates at the speed of light rather than instantaneously. This delay adds another layer of complexity to the equations.

While Newtonian equations can be readily formulated for any conceivable system, this becomes a significant hurdle in General Relativity. The multitude of factors affecting how space curves or evolves makes it difficult to express even the equations for a straightforward model Universe.

Consider the simplest possible Universe: one devoid of matter or energy and unchanging over time. This scenario, which aligns with special relativity and flat, Euclidean space, represents the most basic case.

Now, introduce a point mass into this empty Universe. Suddenly, spacetime becomes dramatically altered.

Instead of a flat, Euclidean realm, we discover that space curves in response to the mass, with the flow of space increasing the closer one gets to the mass. A specific boundary, the event horizon, marks the point beyond which escape is impossible, even at light speed.

This complex spacetime structure emerges from the introduction of just one mass, leading to the first exact, non-trivial solution in General Relativity: the Schwarzschild solution, representing a non-rotating black hole.

Over the last century, additional exact solutions have been discovered, but they remain relatively simple. These include:

- Perfect fluid solutions defined by energy, momentum, pressure, and shear stress.
- Electrovacuum solutions where gravitational, electric, and magnetic fields exist without mass, charge, or currents.
- Scalar field solutions involving a cosmological constant, dark energy, inflationary spacetimes, and quintessence models.
- Solutions featuring a rotating mass (Kerr), a charged mass (Reissner-Nordström), or both (Kerr-Newman).
- Fluid solutions combined with a point mass (e.g., Schwarzschild-de Sitter space).

These solutions are notably straightforward and omit the fundamental gravitational system commonly considered: a Universe with two gravitationally bound masses.

The two-body problem in General Relativity remains unsolved. No exact analytical solution exists for a spacetime with multiple masses, and while it is believed that none can be found, this remains unproven.

Instead, scientists make assumptions and extract higher-order approximations (the post-Newtonian expansion) or examine specific problems for numerical solutions. Advances in numerical relativity since the 1990s have enabled astrophysicists to model gravitational wave signatures, especially for merging black holes. The theoretical groundwork laid makes detections by LIGO and Virgo possible.

Nonetheless, many challenges in General Relativity can be approached approximately, leveraging solutions we do understand. We can analyze how behaviors differ from Newtonian gravity, using those insights to refine more complex systems.

Moreover, we can devise innovative numerical methods for addressing problems that seem theoretically intractable, provided the gravitational fields remain relatively weak.

General Relativity introduces unique challenges absent in Newtonian models:

- The curvature of space is in constant flux.
- Each mass has an associated self-energy that alters spacetime curvature.
- Objects traversing curved space interact with it, emitting gravitational radiation.
- All gravitational signals propagate at light speed.
- An object's velocity relative to others necessitates accounting for relativistic transformations.

Considering these factors results in most spacetimes leading to equations so intricate that solutions to Einstein's equations elude us.

A lesson I learned in my first college math class on differential equations encapsulates this challenge: “Most differential equations cannot be solved, and most of those that can be solved cannot be solved by you.” This statement aptly describes General Relativity, a collection of coupled differential equations posing significant difficulties for scholars.

We often cannot even formulate the Einstein field equations that would apply to most conceivable spacetimes or Universes. Most equations we can articulate cannot be solved, and those that can rarely yield solutions accessible to anyone. However, through approximations, we can glean meaningful predictions. In the grand cosmic scheme, this represents as close as we have come to comprehending the Universe, yet the journey continues. Let us strive for greater understanding.

*Starts With A Bang is now on Forbes and republished on Medium with a seven-day delay. Ethan has authored two books: Beyond The Galaxy and Treknology: The Science of Star Trek from Tricorders to Warp Drive.*