# Einstein equation point mass solution

• espen180
In summary, the metric in spherical coordinates for a point mass at r=0 is symmetrical wrt angle and has entries for time and spatial dimensions. The spatial entry for radius goes to infinity at the Schwarzchild radius and photons can orbit the point mass at r=3M. It is not possible to compute the curvature of space and space-time at a specific radius without calculating the entire Riemann tensor.
espen180
I'm just getting a taste of computational GR, and I have a question regarding the metric for the single point mass solution for the einstein equation.

The metric in spherical coordinates for a point mass at $$r=0$$ is

$$\eta=\left(\begin{matrix}-\left(1-\frac{2GM}{c^2r}\right) & 0 & 0 & 0 \\ 0 & \frac{1}{1-\frac{2GM}{c^2r}} & 0 & 0 \\ 0 & 0 & r^2 & 0 \\ 0 & 0 & 0 & r^2\sin^2\theta \end{matrix}\right)$$

As expected, it is symmetrical wrt angle, and I recognize $$\eta_{00}$$ (time entry) as the negativa square of the gravitational time dilation constant.

The spatial $$\eta_{11}$$ entry for radius goes to infinity as r approaches the swartzschild radius. At this radius, photons can orbit the point mass.

Is the a way to compute the curvature of space and of space-time a radius r away from the sphere without computing the entire Riemann tensor?

espen180 said:
At this radius, photons can orbit the point mass.

Not really, though. At r=2M, photons directed radially outward stay at constant radial coordinate, but I wouldn't call this an orbit. The photon sphere, where photons truly do move in circular orbits, is at r=3M.

Yes, there are ways to compute the curvature at a specific radius without computing the entire Riemann tensor. One way is to use the Ricci scalar, which is a scalar quantity that describes the overall curvature of a space or spacetime. It can be calculated using the metric components and their derivatives at a specific radius. Another method is to use the Kretschmann scalar, which is a scalar quantity that measures the strength of the gravitational field at a specific point. It can also be calculated using the metric components at a specific radius. Additionally, there are other curvature invariants that can be calculated at a specific radius, such as the Weyl tensor. These methods can provide a simpler way to understand the overall curvature at a specific radius without having to compute the entire Riemann tensor.

## 1. What is the Einstein equation point mass solution?

The Einstein equation point mass solution is a solution to the Einstein field equations, which describe the relationship between the curvature of spacetime and the distribution of matter and energy. It is a theoretical concept proposed by Albert Einstein in his theory of general relativity.

## 2. How does the Einstein equation point mass solution relate to black holes?

The Einstein equation point mass solution is used to describe the spacetime curvature around a point mass, such as a black hole. It predicts that the spacetime curvature becomes infinitely strong at the center of a black hole, known as the singularity.

## 3. What is the significance of the Einstein equation point mass solution in physics?

The Einstein equation point mass solution is a cornerstone of modern physics, as it provides a framework for understanding the behavior of spacetime and gravity. It has been confirmed through numerous experiments and observations, and is essential for many technological advancements, such as GPS systems.

## 4. Can the Einstein equation point mass solution be applied to objects other than black holes?

Yes, the Einstein equation point mass solution can be applied to any object with a significant amount of mass. It is commonly used to describe the behavior of large celestial bodies, such as planets, stars, and galaxies.

## 5. How does the Einstein equation point mass solution differ from Newton's law of gravitation?

The Einstein equation point mass solution is a more accurate and comprehensive theory of gravity compared to Newton's law of gravitation. It takes into account the curvature of spacetime, while Newton's law is based on the concept of a force acting between two masses. The Einstein equation point mass solution also predicts effects such as gravitational time dilation and gravitational lensing, which are not explained by Newton's law.

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