# Why is this true? General relativity

• Replusz
In summary, the conversation discusses the validity of the scalar product for a general manifold. The speaker expresses confusion about its general applicability and mentions that it makes sense intuitively in Euclidean space. The conversation then shifts to discussing the use of basis vectors in defining the scalar product and the importance of considering manifolds with metric.
Replusz
Homework Statement
I dont understand why this is true generally. I can see it in Euclidean space, but this is for a general manifold.
Relevant Equations
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"their scalar product is..." why?

Why would you think it is not true for a general manifold?

Chestermiller said:
Why would you think it is not true for a general manifold?
At the moment I don't think anything, and its Maths, not what people think. This is a lecture notes so I assume its correct. But I don't see where it comes from, I.e. Why its true
Even for euclidean space i see it intuitively not mathematically

Replusz said:
Homework Statement: I don't understand why this is true generally. I can see it in Euclidean space, but this is for a general manifold.

Write ##\mathbf{u}## and ##\mathbf{v}## in terms of the basis vectors ##\left\{ \partial / \partial x^\mu \right\} ##, and substitute these expressions into ##g \left( \mathbf{u} , \mathbf{v} \right)##.

A little nitpicking about terminology. There is no "general manifold" statement really accurate.
The sensible way to make the scalar product of ##\mathbb{R}^n## "more general" is to consider manifolds with metric, aka (pseudo-)Riemann-ian manifolds. A general manifold needn't have a metric.

## 1. Why is general relativity considered a fundamental theory?

General relativity is considered a fundamental theory because it provides a comprehensive explanation for the behavior of gravity and its effects on space and time. It has been extensively tested and has consistently been found to accurately describe the universe on both small and large scales.

## 2. What evidence supports the validity of general relativity?

There is a wealth of evidence that supports the validity of general relativity, including the observation of gravitational lensing, the accurate prediction of the precession of Mercury's orbit, and the detection of gravitational waves. Additionally, general relativity has been incorporated into other areas of physics, such as cosmology, and has been found to be consistent with other well-established theories.

## 3. How does general relativity differ from Newton's theory of gravity?

General relativity differs from Newton's theory of gravity in several ways. Firstly, general relativity explains gravity as a curvature of space and time, rather than a force between masses. It also accounts for the effects of gravity on the flow of time. Additionally, general relativity predicts that gravity can bend light, while Newton's theory does not account for this phenomenon.

## 4. Why is general relativity important for our understanding of the universe?

General relativity is important for our understanding of the universe because it provides a framework for understanding the behavior of gravity, which is a fundamental force in the universe. It also allows us to make accurate predictions about the behavior of objects in the universe, such as the movement of planets and the existence of black holes.

## 5. Can general relativity be tested and potentially proven wrong?

Yes, general relativity can be tested and potentially proven wrong. Scientists are constantly conducting experiments and observations to test the predictions of general relativity and see if they hold up in various scenarios. If an experiment were to produce results that contradict the predictions of general relativity, it would indicate a flaw in the theory and could potentially lead to its revision or replacement.

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