Describing 4-vectors in space-time

In summary, the conversation discusses the concept of a 4-vector being described locally in space-time by multiplying it with a metric. The idea of "local vector fields" on curved manifolds is also mentioned.
  • #1
jfy4
649
3
I have a question...

I would like to generically describe a 4-vector locally in space-time. Would i go about that by simply taking a 4-vector and multiplying it by a metric? like

[tex] u^{\alpha}=u_{\beta}g^{\alpha\beta} [/tex]

with [tex]u^{\alpha}[/tex] the new 4-vector in the space specified by the metric?
 
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  • #2
I'm not sure what you mean; what you've just written down is the notion of vector spaces and dual vector spaces being isomorphic, with the isomorphism given by the metric.

"Local vector fields" are vector fields which can change from point to point on your manifold. On curved manifolds these vectors live in the tangent space at that point.
 

FAQ: Describing 4-vectors in space-time

1. What is a 4-vector in space-time?

A 4-vector in space-time is a mathematical representation of a physical quantity that has four components - three spatial components and one temporal component. It is commonly used in special relativity to describe the position, velocity, and momentum of an object in a four-dimensional space-time continuum.

2. How is a 4-vector different from a regular vector?

A 4-vector is different from a regular vector in that it has four components instead of three. The fourth component represents the time coordinate, making it a representation of both space and time. Additionally, 4-vectors follow different transformation rules under Lorentz transformations, which are used to describe the effects of relativity.

3. How are 4-vectors used in physics?

4-vectors are commonly used in physics, particularly in the field of special relativity. They are used to describe the position, velocity, and momentum of objects in space-time, as well as other physical quantities such as energy and force. They are also used in other areas of physics, such as quantum field theory and general relativity.

4. What is the significance of the Minkowski diagram in describing 4-vectors?

The Minkowski diagram is a graphical representation of the space-time coordinates of a 4-vector. It is used to visualize the effects of special relativity and Lorentz transformations on an object's position and velocity. It also helps to understand the concept of space-time intervals and how they differ from Euclidean distances.

5. Can 4-vectors be used in other dimensions besides space-time?

While 4-vectors are most commonly used in the context of space-time, they can also be used in other dimensions. For example, in quantum mechanics, 4-vectors are used to represent the four components of the wave function in four-dimensional space. However, the concept of space-time and Lorentz transformations may not apply in these other dimensions.

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