# Varying determinant of a metric

In summary, to calculate δg, you can use Jacobi's formula and plug in the values for g and δgμν. However, there may be a sign error in the given expression and a quick check using a special case can help confirm the correct sign. Additionally, there may be some relabelling involved in the calculation.
Hi does anyone know how to calculate:

$\delta (det|g_{\mu\nu}|) or simply \delta g$

Or equivalently,

δg = g gμν δgμν

You should get δg=ggμνδ(gμν)
Actually I believe this answer has a sign error, and does not correspond to what I wrote, since δgμν = - gμα gνβ δgαβ.

A quick way to check the sign is to consider a special case, gμν = λ ημν under the variation δλ. For this case,

gμν = λ-1 ημν and g = -λ4, so δgμν = ημν δλ and δg = -4 λ3 δλ.

The minus sign agrees with the expression I gave, namely g gμν δgμν = (-λ4)(λ-1ημν)(ημν δλ).

However the other expression g gμν δ(gμν) = (-λ4)(λ ημν)(-λ-2) (ημν δλ) comes out positive, which is incorrect.

Bill_K said:
Actually I believe this answer has a sign error, and does not correspond to what I wrote, since δgμν = - gμα gνβ δgαβ.

A quick way to check the sign is to consider a special case, gμν = λ ημν under the variation δλ. For this case,

gμν = λ-1 ημν and g = -λ4, so δgμν = ημν δλ and δg = -4 λ3 δλ.

The minus sign agrees with the expression I gave, namely g gμν δgμν = (-λ4)(λ-1ημν)(ημν δλ).

However the other expression g gμν δ(gμν) = (-λ4)(λ ημν)(-λ-2) (ημν δλ) comes out positive, which is incorrect.

You're right. Thanks for the correction.

yeah thanks i knew the answer but had absolutely no idea how to get it, I end up with $g^{\nu\rho}g_{p\mu}\delta g^{\mu p}=-g^{\nu\rho}g^{\mu p} \delta g_{p\mu}$ and am not sure how to proceed from here, there is clearly relabelling but I am still a bit stuck.

## 1. What is a determinant of a metric?

A determinant of a metric is a mathematical concept that represents the volume distortion of a geometrical space. It is used to measure the change in distances between points when transforming from one coordinate system to another.

## 2. How does the determinant of a metric affect the curvature of a space?

The determinant of a metric is directly related to the curvature of a space. A higher determinant value indicates a more curved space, while a lower determinant value indicates a flatter space.

## 3. Can the determinant of a metric be negative?

Yes, the determinant of a metric can be negative. This indicates that the space has a negative curvature, such as a hyperbolic space.

## 4. What factors can cause the determinant of a metric to vary?

The determinant of a metric can vary due to changes in the coordinate system, the shape or size of the space, and the presence of matter or energy.

## 5. How is the determinant of a metric used in Einstein's theory of general relativity?

In general relativity, the determinant of a metric is used to calculate the curvature of spacetime, which is then used to determine the motion of objects under the influence of gravity. It is a crucial component in understanding the fabric of the universe.

• Special and General Relativity
Replies
1
Views
292
• Special and General Relativity
Replies
4
Views
668
• Special and General Relativity
Replies
8
Views
233
• Special and General Relativity
Replies
2
Views
632
• Special and General Relativity
Replies
17
Views
1K
• Special and General Relativity
Replies
62
Views
4K
• Special and General Relativity
Replies
5
Views
1K
• Special and General Relativity
Replies
4
Views
2K
• Special and General Relativity
Replies
5
Views
1K
• Special and General Relativity
Replies
9
Views
601