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

[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?

 

[Matterwave]

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.

 

[roya]

thank you

 

[Chestermiller]

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.