Vector fields, metrics and two forms on a spacetime.

Your Name]In summary, the conversation discusses two parts regarding spacetime (M,g). In part (a), it is shown that if two vector fields A and A' have the same inner product with any future-pointing timelike vector field Y, then they must be equal. This can be proven using the tensorial transformation law for the metric tensor g. In part (b), it is stated that if two two-forms w and w' have the same contraction with any future-pointing timelike vector field A, then they must be equal. This can also be proven using a similar approach as in part (a).
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Homework Statement


Let (M,g) be a spacetime.
(a) Let A and A' be vector fields on M such that g(A,B)=g(A',B) for any future-pointing timelike vector field Y. Show that X=X'.
(b) Let w and w' be two two-forms on M. Suppose that i¬A w = i¬A w' for any future -pointing timelike vector field A on M, where i¬x denotes the contraction with A. Show that w=w'


Homework Equations





The Attempt at a Solution


I think that part (a) is based on the transformation law between two metrics. So if we are going from g(A,B) -> g(A',B') we use the tensorial transformation law:
g(A,B) = d(A',A)d(B',B) g(A',B')
where d(A',A) mean the partial derivative of A' with respect to A, etc.

But g(A',B')=g(A,B) (given)
so g(A,B) = d(A',A)d(B',B) g(A,B)

Thus d(A',A)d(B',B) = 1

But B'=B , so d(B',B) = Kroneckerdelta (B',B) = 1

So d(A',A)=1, thus A'=A.

I'm not sure if this is even the right way of tackling this question for part (a). As for Part (b), I'm completely lost :S
 
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  • #2


Thank you for your post. Your solution for part (a) is on the right track, but there are a few minor errors.

For part (a), the key concept to use is that of a timelike vector field. Remember that a vector field is timelike if and only if g(A,A) < 0 for all points in M. This means that we can choose A and A' to be future-pointing timelike vector fields, and then use the given condition g(A,B) = g(A',B) for all such vector fields B.

You correctly use the tensorial transformation law, but it should be noted that this only applies to tensor fields, not to vector fields. So instead of using A' and B' as the new vector fields, you should use A' and B' as the new basis vectors for the tangent space at each point, and then use the transformation law for the metric tensor g. This will give you a similar result, but without the error of using the tensorial transformation law for vector fields.

For part (b), you are on the right track in thinking about the contraction of two-forms with vector fields. Remember that the contraction of a two-form w with a vector field A is defined as (i_A w)(B) = w(A,B) for all vector fields B. So the given condition i_A w = i_A w' means that w(A,B) = w'(A,B) for all future-pointing timelike vector fields A. This is similar to part (a), so you can use a similar approach to show that w=w'.

I hope this helps. Please let me know if you have any further questions.
 

1. What is a vector field?

A vector field is a mathematical concept used in physics and engineering to describe a quantity that has both magnitude and direction at every point in space. This quantity can represent physical quantities such as velocity, force, or electric fields.

2. How is a metric used in a vector field?

A metric is a mathematical tool used to define distances and angles in a vector field. It assigns a numerical value to each point in space, allowing for the calculation of distances between points and the determination of the angle between two vectors.

3. What is a two-form in a vector field?

A two-form is a mathematical object that describes the orientation and magnitude of a vector field at each point in space. It consists of two vectors, or one vector and one scalar, which represent the direction and magnitude of the vector field at that point.

4. How are vector fields, metrics, and two-forms used in spacetime?

In spacetime, these mathematical concepts are used to describe the behavior of physical quantities such as energy, momentum, and mass. They allow for the calculation of distances and angles in curved spacetime, which is essential for understanding the effects of gravity on the movement of objects.

5. What are some real-world applications of vector fields, metrics, and two-forms?

These mathematical concepts have various real-world applications, such as in physics, engineering, and computer graphics. They are used to model fluid dynamics, weather patterns, and electric and magnetic fields. Additionally, they are crucial in the study of general relativity and the behavior of black holes.

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