# Number of unknowns - Coordinate Transforms

• thehangedman
In summary, the number of unknown functions involved in a generic coordinate transform in general relativity is 16. This is based on the two indexes in the transformation matrix, each running over 4 values. However, there may be symmetries that eliminate elements, reducing the number of independent components to 10. This is in contrast to special relativity, where the Lorentz transformation only involves 4 unknown functions.
thehangedman
In general relativity, what are the total number of unknowns for a generic coordinate transform? Is it just 4 * 4 = 16? Is there a way to break those down into combinations of types, such as boosts, rotations, reflections (parity?), etc, or is it just left wide open from an interpretive standpoint? My feeling is the answer is in fact 16 unknown functions of space-time and that the actual interpretation can't really be broken out like we do in SR (Lorentz)...

It is an infinite number of unknowns. Even if you only had a 1D manifold, there are an infinite number of degrees of freedom.

DaleSpam said:
It is an infinite number of unknowns. Even if you only had a 1D manifold, there are an infinite number of degrees of freedom.

I made a mistake in my question. I know that functions have an unlimited number of degrees of freedom, I meant to ask how many functions are involved in a generic coordinate transform in R^4. My guess is 16, since there are two indexes in the transformation matrix, each running over 4 values, but wanted clarity since there might be symmetries that eliminate elements (though I'm guessing not).

I think it is only 4 unknown functions. You can always write it as:
$x'_0=f_0(x_0,x_1,x_2,x_3)$
$x'_1=f_1(x_0,x_1,x_2,x_3)$
$x'_2=f_2(x_0,x_1,x_2,x_3)$
$x'_3=f_3(x_0,x_1,x_2,x_3)$

May be you are talking about gμv which seems to have 16 components.But it is symmetric,which will eliminate 6 so there will be only 10 independent components.

## 1. What is the meaning of "number of unknowns" in coordinate transforms?

The "number of unknowns" in coordinate transforms refers to the number of variables or parameters that need to be solved for in order to determine the transformation between two coordinate systems. These unknowns could include translation, rotation, and scaling factors.

## 2. How do you determine the appropriate number of unknowns for a coordinate transform?

The appropriate number of unknowns for a coordinate transform depends on the complexity of the transformation and the number of dimensions involved. For example, a simple 2D translation would only require 2 unknowns (x and y), while a 3D rotation might require 3 unknowns (x, y, and z rotations). In general, the number of unknowns should be equal to the number of degrees of freedom in the transformation.

## 3. What is the role of the number of unknowns in the accuracy of a coordinate transform?

The number of unknowns plays a crucial role in the accuracy of a coordinate transform. Generally, the more unknowns that are solved for, the more accurate the transformation will be. However, increasing the number of unknowns also increases the complexity and computational cost of the transformation.

## 4. Can the number of unknowns be reduced in a coordinate transform?

Yes, the number of unknowns can be reduced in a coordinate transform by simplifying the transformation or by using additional constraints. For example, if the translation between two coordinate systems is always in a fixed direction, the number of unknowns can be reduced from 2 to 1. However, reducing the number of unknowns may also result in a less accurate transformation.

## 5. How does the number of unknowns impact the computational efficiency of a coordinate transform?

The number of unknowns has a direct impact on the computational efficiency of a coordinate transform. Generally, the more unknowns there are, the longer it will take to solve for them and determine the transformation. This is especially important in real-time applications where efficiency is crucial.

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