The general theory of relativity constitutes one of the most essential foundations of the scientific world. The approach that this theory brings to events, and especially its success in explaining cosmological events, is of invaluable importance. It successfully explained complex events despite all the difficulties for over a century, making this theory even more indispensable. However, this does not mean that general relativity is not a problem-free theory.

Although there are many challenges physicists have to deal with in the general theory of relativity, in this article, we will try to find a solution for only one: the dependence of general relativity on the metric tensor and its elements. For many physicists, it may not sound like a problem, or it is not a problem. However, when you try to explain a phenomenon using general relativity and effort to find the metric tensor and its elements, you see this as a problem.

Conventionally, when general relativity is used for space-time, it is necessary to compute the metric tensor and its elements up to sixteen. Naturally, the event symmetries and properties studied can reduce this by a few, but more is needed. Determining these elements means solving differential equations as much as the number of unknowns according to the boundary conditions. The fact that there are so many unknowns is a reason that makes the situation even more complex, and the number of unknowns should be reduced somehow. If this can be accomplished, the procedures required to understand the phenomenon under investigation will be much easier.

In this study, we present a method that may solve the abovementioned difficulty: Expressing Einstein's equations in terms of basis vectors. Although some calculations have been made about basis vectors and tetrad fields (see [1, 2] and references therein), these calculations were not continued until expressing the Einstein tensor in terms of basis vectors and have yet to lead to the desired clear result. The use of metric tensors and their elements remained indispensable in this theory.

Our ultimate goal will be to express the Einstein tensor using basis vectors. Since each metric tensor is the inner product of some basis vectors, we calculate the Einstein tensor by writing the metric tensor as the scalar product of the basis vectors, not implicitly. As a result of simple but somewhat lengthy calculations, using the properties and symmetries these basis vectors must provide for four-dimensional space-time, we find the Einstein tensor only in terms of basis vectors. Thus, we successfully reduce the number of unknowns to four in general relativity or the most general case. In addition, we show that if these vectors or spaces are appropriately chosen, the Einstein tensor has the same form as the stress-energy tensor of electrodynamics in terms of the vectors.

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