# Metric Tensor of the Reissner–Nordström Metric

• Philosophaie
In summary: You can find a derivation of the Schwarzschild metric here:http://math.stackexchange.com/questions/156508/derivation-schwarzschild-metricIn summary, the Reissner-Nordstrom metric can be derived from the Schwarzschild metric using the Harrison transformation. The Schwarzschild metric is a difficult calculation to make on its own, so finding a derivation that is easy to follow is helpful.

#### Philosophaie

I am looking for the Metric Tensor of the Reissner–Nordström Metric.$$g_{μv}$$
I have searched the web: Wiki and Bing but I can not find the metric tensor derivations.

As far as deriving it goes, you can do this easily yourself if you've seen the standard derivation of the Schwarzschild solution before. The only extra thing you would have to do is solve the source-free Maxwell equations ##\nabla_{[\gamma}F_{\mu\nu]} = 0## and ##\nabla^{\mu} F_{\mu\nu} = 0## simultaneously with the electrovacuum field equations ##G_{\mu\nu} = 8\pi T^{EM}_{\mu\nu}## but because we are dealing with a spherically symmetric static source, you can easily deduce ##F_{\mu\nu}## by working in the coordinates adapted to all the symmetries of the space-time and then solve ##G_{\mu\nu} = 8\pi T^{EM}_{\mu\nu}## in said coordinates.

Alternatively, you can derive the solution without using coordinates at all; you would be doing all of your calculations (right before you write down the actual solution) in a coordinate-free manner. This is harder but in my opinion much more insightful than just going through the mindless coordinate computations.

Last edited:
WannabeNewton said:
As far as deriving it goes, you can do this easily yourself if you've seen the standard derivation of the Schwarzschild solution before.

Alternatively, you can derive the solution without using coordinates at all; you would be doing all of your calculations (right before you write down the actual solution) in a coordinate-free manner. This is harder but in my opinion much more insightful than just going through the mindless coordinate computations.

I have never seen the Reissner–Nordström Metric Tensor derived before from its metric. The Schwarzschild Metric Tensor is difficult enough.

Is there any websites that go thru the and simplifies the derivation of either of these Non-rotating Charged or Uncharged Metric Tensor Components from its Metric?

Oops, sorry. I didn't notice that you wanted a derivation, not just the metric itself.

Philosophaie said:
I have searched the web: Wiki and Bing but I can not find the metric tensor derivations.
Google "Reissner-Nordstrom derivation" immediately turns up several, including:

http://arxiv.org/pdf/physics/0702014.pdf

Actually the easiest way to obtain the Reissner-Nordstrom metric is not to start from scratch, but to derive it from Schwarzschild using the Harrison transformation.

## 1. What is the Metric Tensor of the Reissner-Nordström Metric?

The Metric Tensor of the Reissner-Nordström Metric is a mathematical tool used in general relativity to describe the geometry of spacetime in the presence of a charged, non-rotating black hole. It is a symmetric tensor field that contains information about the curvature and distances in the vicinity of the black hole.

## 2. How is the Metric Tensor of the Reissner-Nordström Metric derived?

The Metric Tensor of the Reissner-Nordström Metric is derived from the Einstein field equations, which relate the curvature of spacetime to the distribution of matter and energy. It is a solution to these equations in the case of a black hole with both mass and electric charge.

## 3. What does the Metric Tensor of the Reissner-Nordström Metric tell us about the black hole?

The Metric Tensor of the Reissner-Nordström Metric tells us about the curvature of spacetime around the black hole, as well as the behavior of particles and light near the event horizon. It also contains information about the electric field surrounding the black hole.

## 4. What is the significance of the charge parameter in the Reissner-Nordström Metric?

The charge parameter in the Reissner-Nordström Metric represents the electric charge of the black hole. It is an important factor in determining the properties of the black hole, such as its event horizon and the strength of its electric field. It also affects the behavior of objects and particles near the black hole.

## 5. How is the Metric Tensor of the Reissner-Nordström Metric used in practical applications?

The Metric Tensor of the Reissner-Nordström Metric is used in practical applications to model and study the behavior of charged black holes in astrophysics and cosmology. It is also used in the study of gravitational waves and the effects of strong gravitational fields on matter and light.