# A basic question about the use of a metric tensor in general relativity

• roya
In summary, in order to understand general relativity, you need to first understand the concept of curved spacetime. This is done by extrapolating from lower dimensional examples. Once you have a basic understanding of curved spacetime, you can then start to understand GR by looking at specific examples.
roya
I have very little knowledge in general relativity, though I do have a decent understanding of
the theory of special relativity.

In special relativity, points in space-time can be represented in Minkowski space (or a hyperbolic space) so that the metric tensor (that is derived in order to preserve the invariant interval) has a unique form corresponding to that space.

From what I understand (and I say this with caution as I could easily be mistaken), since in the theory of general relativity light bends under the influence of gravity (does it?), this requires a more complicated set of coordinates hence the use of more complicated metric tensors to preserve the invariant interval in such curvilinear space.
Is that correct? how would such metric tensor look like?

Yes, basically, but one caveat is that one has to realize that the metric tensor's components (which many people simply call the "metric") is highly non-unique and is completely dependent on the choice of basis vectors. Notice that even in SR, the metric does not have such a nice structure as diag(-1,1,1,1) if we used polar coordinates (and corresponding coordinate basis vectors).

Also, even in GR, one can use what are so-called orthonormal bases (sometimes called tetrads or more esoterically vierbeins) such that the metric tensor always has the form it has in SR (i.e. diag(-1,1,1,1)). In coordinate bases, the metric will tend to have a different form.

For examples, you can see e.g. the Schwarzschild metric or the FLRW metric.

thank you

roya said:
I have very little knowledge in general relativity, though I do have a decent understanding of
the theory of special relativity.

In special relativity, points in space-time can be represented in Minkowski space (or a hyperbolic space) so that the metric tensor (that is derived in order to preserve the invariant interval) has a unique form corresponding to that space.

From what I understand (and I say this with caution as I could easily be mistaken), since in the theory of general relativity light bends under the influence of gravity (does it?), this requires a more complicated set of coordinates hence the use of more complicated metric tensors to preserve the invariant interval in such curvilinear space.
Is that correct? how would such metric tensor look like?

In order to begin to understand GR, you need to first get an understanding of the concept of curved spacetime. Most people are able to do this by extrapolating from lower dimensional examples. Consider the 2 dimensional surface of a sphere, and imagine that a race of beings is living within this 2D surface. They are unaware that there is a 3rd radial dimension present. They lay out perpendicular lines of constant longitude and constant latitude on the sphere surface, and, at least locally, think that they are using a flat rectangular Cartesian coordinate system. However, when they actually measure distances in their 2D world, and try to apply a Euclidean metric to their system, they run into trouble. This is because the sphere is curved, and not flat. The Euclidean metric only works to first order. To describe the metrical characteristics correctly on the surface of a sphere, they need to use curvilinear coordinates, such as Spherical.
Analogous to this is 4D Lorentzian spacetime which, for the most part is "flat", and amenable to the Minkowski metric (with constant metrical coefficients). However, in the vicinity of massive objects, 4D spacetime is curved, and cannot be described using a rectilinear orthogonal (Lorentzian) set of coordinates. It is as if the 4D spacetime had been deformed "out of plane" into the 5th dimension.
All accelerated frames of reference must be regarded as non-flat. Thus, the hyperbolic space you described above is curved "out of the plane" of Lorentzian space, and cannot be described using the Minkowski metric. Similarly, the Schwartzchild metric applies to an accelerated frame of reference.

## 1. What is a metric tensor in general relativity?

A metric tensor is a mathematical tool used in general relativity to describe the curvature of spacetime. It is a mathematical object that assigns a distance or interval between any two points in spacetime, allowing us to measure the geometry of the universe.

## 2. Why is a metric tensor important in general relativity?

In general relativity, gravity is described as the curvature of spacetime. To understand and calculate this curvature, we need a metric tensor. It is the fundamental mathematical concept that allows us to describe how objects move in the presence of gravity.

## 3. How is a metric tensor different from a regular tensor?

A metric tensor is a special type of tensor that is used to describe the geometry of spacetime. It is different from a regular tensor in that it has a specific set of properties and it is used for a specific purpose in general relativity. Regular tensors, on the other hand, can be used for a variety of mathematical and physical applications.

## 4. Can a metric tensor change in different regions of spacetime?

Yes, a metric tensor can change in different regions of spacetime. In general relativity, the curvature of spacetime is not constant and can vary depending on the presence and distribution of matter and energy. This means that the metric tensor can also change in different regions to accurately describe the curvature of spacetime.

## 5. How is a metric tensor related to the concept of spacetime curvature?

The metric tensor is the mathematical representation of spacetime curvature in general relativity. It describes how the geometry of spacetime changes in the presence of mass and energy. By using the metric tensor, we can calculate the curvature of spacetime and understand how gravity affects the motion of objects in the universe.

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