# Coordinate singularities and coordinate transformations

• jinbaw
In summary: The "horizon" is a coordinate artifact.A test particle would be in a circular orbit, since d\theta and d\phi are still orthogonal to each other.
jinbaw
I have a metric of the form $$ds^2 = (1-r^2)dt^2 -\frac{1}{1-r^2}dr^2-r^2 d\theta^2 - r^2 sin^2\theta d\phi^2$$

A singularity exists at $$r=\pm 1$$. By calculating $$R^{abcd}R_{abcd}$$ i found out that this singularity is a coordinate singularity.

I found the geodesic equations for radial photons and performed the eddington-finkelstein coordinate transformation.

my metric turns out to be:

$$ds^2 = (1-r^2)dudv-r^2 d\theta^2 - r^2 sin^2\theta d\phi^2$$

where u and v are the constants of integration for the outgoing and incoming radial photons.

I also took a further coordinate transformation where T=(u+v)/2 and X = (u-v)/2

and the metric takes the form:

$$ds^2 = (1-r^2)dT^2-(1-r^2)dX^2-r^2 d\theta^2 - r^2 sin^2\theta d\phi^2$$

However for $$r=\pm 1$$ the metric still does not behave properly.

Do you suggest any other coordinate transformation? Thank you.

In what way does it not behave properly? There does not appear to be a singularity. (I didn't check your math)

jinbaw said:
A singularity exists at $$r=\pm 1$$. By calculating $$R^{abcd}R_{abcd}$$ i found out that this singularity is a coordinate singularity.

One is that just because a particular curvature scalar is not singular, that doesn't mean that there's no singularity there. Only the converse is true: if some curvature scalar *does* blow up, then the singularity *isn't* just a coordinate singularity. Also, note that the Kretchmann invariant isn't the only curvature scalar that can be constructed from the Riemann tensor.

The other is that I doubt that it makes sense to refer to r=-1. I'm pretty sure your solution can't be continued to negative r.

Anyway, getting back to your question, one thing I would suggest is that before you start doing coordinate transformations that are analogous to the ones that we know are helpful in the case of the Schwarzschild metric, I would suggest seeing if you can transform to coordinates such that at large r, the metric is flat. When you calculated the Kretchmann invariant, how did you find that it depended on r and t? This would help in figuring out if there is an asymptotically flat part of your spacetime somewhere. If you can find a relation between r and t such that the Kretchmann invariant vanishes, then presumably that is where asymptotically flat infinity is hiding. Note that r is timelike for large r.

Is this a vacuum solution with zero cosmological constant? If so, then I think Birkhoff's theorem says it has to be the Schwarzschild solution written in unusual coordinates, and you can probably locate the Schwarzschild singularity by looking for a relation between r and t such that the Kretchmann invariant blows up.

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1. Actually, yes you're right i can't take r = -1.
2. I do have a cosmological constant + a gauss-bonnet term.
3. To detect the singularities, i look at my metric and check at what values of r it might blow up. Then, i look at the kretchmann invariant to check whether the singularities are real or coordinate. IS there any other way to find the singularities of my metric?

Your metric is de Sitter. There are horizons, but no singularities.

Ich said:
Your metric is de Sitter. There are horizons, but no singularities.

Can you explain more? :)

so what i have is only points at which the r coordinate becomes timelike and the t coordinate becomes spacelike. This is what happens at r=1. But, i do not have real singularities similar to r=0 in schwarzschild. right?

if i want to think of some test particle that exists in my spacetime, it would not have any "odd" behavior?

Last edited by a moderator:

## 1. What is a coordinate singularity?

A coordinate singularity is a point in a coordinate system where the equations that define the system break down and become undefined. This can occur when the equations involve dividing by zero or taking the square root of a negative number.

## 2. How do coordinate transformations help in dealing with singularities?

Coordinate transformations are used to change from one coordinate system to another. This allows us to avoid coordinate singularities by using a different set of equations that do not break down at those points.

## 3. Can coordinate singularities be avoided completely?

No, it is not possible to completely avoid coordinate singularities. They are inherent in the mathematical equations used to define a coordinate system. However, we can minimize their impact by using appropriate coordinate transformations.

## 4. Are coordinate singularities a problem in all coordinate systems?

No, coordinate singularities are not a problem in all coordinate systems. They are most commonly encountered in spherical and polar coordinate systems, but can also occur in other systems such as cylindrical or elliptical coordinates.

## 5. How do coordinate singularities affect scientific calculations?

Coordinate singularities can have a significant impact on scientific calculations, as they can lead to inaccuracies or errors in the results. It is important for scientists to be aware of coordinate singularities and use appropriate coordinate transformations to minimize their impact on calculations.

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