Understanding Metric Tensor: Time & Spatial Coordinates and Indices

In summary, the conversation discusses the use of metric tensor g=diagonal(1,-1,-1,-1) in a fields course and the orientation of the time coordinate being opposite to the spatial coordinates. It also delves into the concept of upper and lower indices and their role in classical field theory. The metric tensor, as defined, allows for "null distances" of length zero and is an important concept in the General Theory of Relativity. The use of upper and lower indices is a way to incorporate the relationship between tensors and their components in different bases.
  • #1
nolanp2
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in my fields course we are using the metric tensor g=diagonal(1,-1,-1,-1), off diagonal(0)
i'm looking for an explanation of why the time coordinate has to be orientated oppositely to the spatial coordinates. can anyone give me an explanation of this?

i'm also lost with upper and lower indices. i don't understand why multiplying a lower index by the metric tensor gives an upper index.

any help appreciated, thanks
 
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  • #2
The simple answer is that by defining the metric tensor that way, there are "null distances" of length zero which correspond to path that light rays travel. It does not *have* to be defined that way, but in the General Theory of Relativity, it can be defined that way.

The upper and lower indices are more subtle- they are not, in general, movable. Upper indices correspond to vectors, lower indices correspond to forms. If a metric tensor can be defined on a generic geometry, then it is possible to move them up and down.

Misner, Thorne, and Wheeler's book "Gravitation" is an excellent way to get started understanding this stuff.
 
  • #3
so the upper and lower indices are like dual spaces? what advantage is there to raising indices, why is it necessary that they be used in Classical field theory?
 
  • #4
The basic reason for the "indefinite metric", with time having the opposite sign from space, is that the combination t^2-x^2-y^2-z^2 is an invariant under Lorentz transformation.
The two types of indices, upper and lower, is a relatively simple way to incorporate this.
 
  • #5
nolanp2 said:
so the upper and lower indices are like dual spaces? what advantage is there to raising indices, why is it necessary that they be used in Classical field theory?

Yes the upper indexed components are the components of some vector in some basis. The lower indexed components are the components of it's dual vector in some dual basis.

The metric tensor maps a vector to it's dual.

Index gymnastics is just a way in that the relationship between tensors refelcts the relationship between their components in different bases.
 

1. What is a metric tensor?

A metric tensor is a mathematical object used in the study of differential geometry and general relativity. It is a symmetric matrix that describes the local geometry of a spacetime, including the relationships between time and spatial coordinates.

2. How does the metric tensor relate to time and spatial coordinates?

The metric tensor contains information about the curvature of a spacetime, which affects the measurement of time and distances between points in space. It allows us to define a consistent way to measure time and distances in a curved spacetime.

3. What is the significance of the indices in the metric tensor?

The indices in the metric tensor represent the different components of the tensor. In a four-dimensional spacetime, there are four indices, one for each dimension. These indices are used to calculate the metric tensor and determine the geometry of the spacetime.

4. How is the metric tensor used in general relativity?

In general relativity, the metric tensor is used to describe the curvature of spacetime caused by the presence of mass and energy. It is a key component in Einstein's field equations, which relate the curvature of spacetime to the distribution of matter and energy.

5. How does the metric tensor affect our understanding of the universe?

The metric tensor is crucial in our understanding of the universe as it allows us to accurately describe and measure the curvature of spacetime. This is essential in understanding the behavior of matter and energy, as well as the dynamics of the universe on a large scale.

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