# Coordinates in general relativity

• GarageDweller
I like the consistency, but I certainly don't claim that it is a better or more natural way to do things. I just find it works better for me.In summary, the conversation discusses the issue of coordinates in curved space-time and the challenges of determining the flatness of a metric when the dimensions of the coordinates are not revealed. It is noted that the convention for assigning dimensions to coordinates and components of tensors may vary, but as long as the requirement of ##ds^2## having units of length squared is met, the choice of convention is not critical. The use of arbitrary coordinate transformations and the preference for coordinates to be unitless are also mentioned.
GarageDweller
Hello fellow PF go-ers
I am having trouble with coordinates in curved space time lately, allow me to demonstrate my issue.
Take the metric of flat space in spherical coordinates for example, a diagonal metric with values 1,r^2 and r^2sinΘ. It appears to me that only when we know that the Θ and ψ variables are dimensionless, can we transform this back into the usual flat spacetime in cartesian coordinates.
However what if I made an abitrary transformation of coordinates and did not reveal to you the dimensions of the various coordinates, how would one know whether the metric is flat or not?

The problem seems to be that the cartesian system of coordinates is held on a pedestal, and I think this should not be.

GarageDweller said:
Hello fellow PF go-ers
I am having trouble with coordinates in curved space time lately, allow me to demonstrate my issue.
Take the metric of flat space in spherical coordinates for example, a diagonal metric with values 1,r^2 and r^2sinΘ. It appears to me that only when we know that the Θ and ψ variables are dimensionless, can we transform this back into the usual flat spacetime in cartesian coordinates.
However what if I made an abitrary transformation of coordinates and did not reveal to you the dimensions of the various coordinates, how would one know whether the metric is flat or not?

The problem seems to be that the cartesian system of coordinates is held on a pedestal, and I think this should not be.
Hi GarageDweller,

I think that there are a few different conventions possible. The bottom line requirement that everything has to meet is that ##ds^2=g_{\mu\nu}dx^{\mu}dx^{\nu}## must have units of length squared. However, how you split that up into dx or g doesn't really matter, and you can do it separately for each coordinate or over the whole tensor. My personal preference is to think of dx as always being unitless and g as always having units of length squared so that everything is consistent from coordinate system to coordinate system and from component to component, but I know it is not a universal preference.

As long as you meet the requirement you are OK.

As far as flatness goes, I don't think that the units are an issue. If the components of the curvature tensor are 0 then they will be 0 regardless of the units.

GarageDweller said:
Take the metric of flat space in spherical coordinates for example, a diagonal metric with values 1,r^2 and r^2sinΘ. It appears to me that only when we know that the Θ and ψ variables are dimensionless, can we transform this back into the usual flat spacetime in cartesian coordinates.
Tensor analysis does not know or care about dimensions. The metric gμν transforms as a tensor. In the new coordinates, always

gμ'ν' = ∂xμ/∂xμ' ∂xν/∂xν' gμν.

Dimensions don't play a role in this at all. The new coordinates can be ANY functions of the old ones, even functions that seem to be dimensionally inconsistent, such as x = r + Θ.

GarageDweller said:
However what if I made an abitrary transformation of coordinates and did not reveal to you the dimensions of the various coordinates, how would one know whether the metric is flat or not?
I would calculate the Riemann tensor.

Bill_K said:
The new coordinates can be ANY functions of the old ones, even functions that seem to be dimensionally inconsistent, such as x = r + Θ.
This actually is one of the reasons that I like to think of coordinates as just numbers. Then there is no hesitation to use functions like that.

DaleSpam said:
I think that there are a few different conventions possible. The bottom line requirement that everything has to meet is that ##ds^2=g_{\mu\nu}dx^{\mu}dx^{\nu}## must have units of length squared. However, how you split that up into dx or g doesn't really matter, and you can do it separately for each coordinate or over the whole tensor. My personal preference is to think of dx as always being unitless and g as always having units of length squared so that everything is consistent from coordinate system to coordinate system and from component to component, but I know it is not a universal preference.

As long as you meet the requirement you are OK.

Yeah, I prefer the opposite convention, so r has units of distance, and Θ is unitless. This has the advantage that quantities like r and Θ have the units you expect, but the disadvantage that different components of the same tensor can have different units. But as DaleSpam says, this is just a matter of personal preference.

bcrowell said:
This has the advantage that quantities like r and Θ have the units you expect, but the disadvantage that different components of the same tensor can have different units.
But even with that disadvantage, it still works out that ##ds^2## has units of length squared. The different components of ##dx## and the different components of ##g## combine together to give you the right units.

## 1. What are coordinates in general relativity?

Coordinates in general relativity refer to the mathematical quantities used to describe the position and motion of objects in the fabric of spacetime. They are used to measure distances, angles, and time intervals in the four-dimensional spacetime continuum.

## 2. How are coordinates different in general relativity compared to classical physics?

In classical physics, coordinates are fixed and absolute, meaning they do not change based on an observer's frame of reference. In general relativity, coordinates are relative and can change depending on the observer's frame of reference and the curvature of spacetime.

## 3. What is the significance of choosing different coordinate systems in general relativity?

Choosing different coordinate systems can simplify the mathematical equations used to describe the behavior of objects in spacetime. It can also reveal different perspectives and insights into the same physical phenomenon.

## 4. Can any coordinate system be used in general relativity?

Yes, any coordinate system can be used in general relativity as long as it is consistent and follows the principles of differential geometry. However, some coordinate systems may be more useful or appropriate for certain situations than others.

## 5. How do coordinates affect the measurement of physical quantities in general relativity?

In general relativity, physical quantities are not invariant and can vary depending on the coordinates used. This is due to the fact that coordinates are relative and can change based on the observer's frame of reference and the curvature of spacetime.

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