Yes, Einstein's equations can be solved without invoking the concept of the "aether." In fact, Einstein's theory of general relativity, which is based on these equations, was specifically developed to provide a more comprehensive and consistent description of gravity without the need for an aether.
In classical physics, the concept of the aether was proposed as a hypothetical medium through which light and other electromagnetic waves were thought to propagate. However, experimental evidence, including the famous Michelson-Morley experiment in the late 19th century, failed to detect any evidence of such a medium. This led to the rejection of the aether concept.
Einstein's theory of general relativity, formulated in 1915, introduced a new understanding of gravity based on the concept of spacetime curvature. According to this theory, massive objects like stars and planets curve the fabric of spacetime around them, and this curvature influences the motion of other objects in their vicinity. The equations of general relativity, known as Einstein's field equations, describe the relationship between the curvature of spacetime and the distribution of matter and energy.
Solving Einstein's field equations involves finding the mathematical descriptions of spacetime curvature in the presence of matter and energy. This can be a challenging task and often requires numerical methods or approximations for complex systems. However, the solutions to these equations do not require the assumption of an aether. Instead, they describe the behavior of spacetime and the motion of objects within it purely based on the distribution of matter and energy.