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Einstein's theory of relativity introduced complex mathematical equations due to its profound implications for spacetime and gravity. The equations involve concepts like tensors and non-Euclidean geometry, making them challenging to solve directly. Additionally, relativity deals with high velocities and extreme conditions, which require specialized mathematical techniques to fully understand and solve.
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All of this maybe true, but it is not an answer on the question: why do you need mathematics in the first place. Why cannot it be simpler.
To go in more detail why do you need tensors?
For example, we want to simulate a binary system of two Black Holes. Which are all the tensors involved and how are all the parameters measured. This is an important issue which can only solved when you start from one coordinate system.
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1. Tensors and Non-Euclidean Geometry: In Einstein's theory of general relativity, spacetime is described using a mathematical construct called tensors. Tensors are multidimensional arrays that represent the relationships between physical quantities in different directions and at different points in spacetime. These tensors, especially the metric tensor, define the curvature of spacetime due to the presence of mass and energy. Non-Euclidean geometry, which deviates from the familiar geometry of flat surfaces, is used to describe this curved spacetime. This introduces complex mathematical concepts that require a solid foundation in advanced mathematics to fully comprehend and manipulate.
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2. Challenging Equations: The field equations of general relativity, known as the Einstein field equations, describe how matter and energy curve spacetime, and how objects move in response to this curvature. These equations involve multiple variables, derivatives, and terms that are not linear or straightforward. Solving them requires using tools from differential geometry and advanced calculus. The equations are also coupled, meaning that changes in one variable can affect others, making them intricate to work with.
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3. High Velocities and Extreme Conditions: Relativity deals with scenarios involving high velocities, such as those near the speed of light, and extreme conditions like those near massive objects or in the early universe. These situations introduce nonlinear effects and can't be accurately described by classical physics. Solving equations under these circumstances requires specialized techniques like numerical simulations or perturbation theory.
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The most important part is a clear description what is physical assumed.
The problem is every mathematical model of the evolution of the universe is tricky. This is completely different of a model of the solar system or of the stars surrounding the Black Hole in the center of our Galaxy. This models can be tested based on current collected data. For a physical/mathematical model of the Universe, starting from the Big Bang, this is a whole different ball game, because it is already a nontestable assumption if anything can be observed from that initial moment. What makes this worse, is something called the Inflation Theory which tries to solve the homogeneity issue, at the same time introducing new physical issues.
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4. Space and Time Are Intertwined: Einstein's theories unified space and time into a single entity called spacetime.
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This is typical a mathematical solution, because t is multiplied by c, the speed of light. The speed of light is a physical concept and considered constant. The result the parameter ct now becomes a distance, which is neither t or c.
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This idea is central to relativity and requires a different way of thinking compared to classical physics, where space and time were treated as separate entities.
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The problem, as mentioned above, is that spacetime is mathematics not physics. Spacetime is a calculated quantity.
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The mathematics of spacetime involves four dimensions (three of space and one of time), and the interplay between these dimensions introduces complexities that can be challenging to grasp.
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The four dimensions are: three of space and the clock time multiplied by the speed of light.
Yes, this is difficult to grasp.
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5. Predicting Real-world Phenomena: General relativity has led to predictions such as gravitational time dilation (time passing at different rates depending on the strength of gravity) and the bending of light by gravity, known as gravitational lensing.
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Gravitational time dilation is nothing more than that the movement, the accuracy of any hardware clock is influenced by gravity.
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These predictions have been confirmed by experiments, but their understanding and accurate calculation require a deep understanding of the underlying mathematics.
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The concept that light is bended by gravity is "old" and in fact assumes that light, photons behave the same as objects. To calculate the trajectories requires a deep understanding of a physical model.
The physical problem is that what we observe from a star is a point source which reaches our eyes. In reality the star emits a much broader beam of light into out neighbourhood, which we don't observe. This extra light can influence the trajectory of the light we observe.
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