Buzz
Viktor T. Toth:
For the same reason that multiple individuals can solve most other kinds of equations.
Unlike basic equations such as the quadratic formula typically studied in high school, most equations do not have easy, universal solutions. Instead, solutions are tailored to specific values, or certain sets of values, defined by the equation's parameters.
Einstein's field equations are quite complex. When fully written out, they form a set of 10 interrelated second-order differential equations with 10 unknown functions. Solving these isn't something you can do casually.
The solutions that are available represent specific, exceptional cases. The Schwarzschild solution is perhaps the most well-known among them. It represents a highly symmetrical situation: a vacuum solution (without any matter), independent of time, and spherically symmetric, depending only on the radial coordinate. Ultimately, this simplifies to two unknown functions, leading to two very straightforward differential equations that are easy to solve.
Other solutions, however, are far more complex. In many cases, neat and simple solutions in closed form are unavailable, requiring numerical methods to solve the equations. Even then, the task is difficult, as finding initial values for the unknown functions that correspond to physically meaningful and stable configurations of matter is challenging. A whole field, numerical relativity, focuses on this area.
In short: most equations lack easy, straightforward, general solutions, and Einstein's field equations are no different.
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