# Why Are Coordinates Independent in GR? - Exploring the Motivation

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• TomServo
In summary: But do you have an answer for the question? Why do we require independent coordinates? Is it because if we had dependent coordinates we could make the metric flat by rescaling just some of the dependent coordinates?Yes, your hypothesis is correct. It is just that the notation ##u=t+x## is sometimes used as the definition of the new set of coordinates ##(u,\tilde{x})## and sometimes it is used to denote the dependence between the two sets of coordinates.I don't know the answer. I just wanted to point out the abuse of notations.In summary, the requirement for independent coordinates in a space is motivated by the fact that if we allowed dependent coordinates, we could make the metric flat by res
TomServo
TL;DR Summary
Looking for fundamental reason why we only look at independent coordinates in GR
I can see that by the tensor transformation law of the Kronecker delta that

##\frac{\partial x^a}{\partial x^b}=\delta^a_b##

And thus coordinates must be independent of each other.

But is there a more straightforward and fundamental reason why we don’t consider dependent coordinates? Is it just built-in to the transformation law?

Obviously if we could have dependent coordinates, then for a metric like this plane wave one:

##ds^2=-2dudv+f(u)g(u)dy^2+f(u)/g(u)dz^2##

...I could rescale my y and z coordinates by:

##dy’^2=f(u)g(u)dy^2## and ##dz’^2=f(u)/g(u)dz^2##

And have

##ds^2=-2dudv+dy’^2+dz’^2##

And thus turn the metric (or any metric by such a rescaling) to flat spacetime. So I see why it’s good to have linearly independent coordinates to forbid transformations like the above.

But why? Is this requirement the very motivation for the tensor transformation laws? What’s the underlying reason?

Thanks.

In general, the number of independent coordinates is determined by the dimension of the space. If, for example, in the 2D plane you have ##y## dependent on ##x##, then you have a 1D curve.

You could, I guess, have more than 2 coordinates for the x-y plane, but one of them would be redundant.

If you have the minimum number of coordinates then they must be independent.

Dale and Cryo
PeroK said:
In general, the number of independent coordinates is determined by the dimension of the space. If, for example, in the 2D plane you have ##y## dependent on ##x##, then you have a 1D curve.

You could, I guess, have more than 2 coordinates for the x-y plane, but one of them would be redundant.

If you have the minimum number of coordinates then they must be independent.

TomServo said:

Independent does not mean orthogonal.

PeroK said:
Independent does not mean orthogonal.
That's true, but perhaps you can help me to see something. Let's say I have the 1+1 Minkowski space, with coordinates x and t with basis vectors ##\hat{x}=(0,1)## and ##\hat{t}=(1,0)##. Now I define a new coordinate u=t+x to replace t, and so my coordinates are u and x with basis vectors ##\hat{u}=(1,1)## and ##\hat{x}=(0,1)##.

These are independent basis vectors, so in that sense they are independent, but based on the definition of u it seems that ##\frac{\partial u}{\partial u}=1## and ##\frac{\partial u}{\partial x}=1\neq 0##, in conflict with the notion that ##\frac{\partial x^a}{\partial x^b}=\delta^a_b##, so it seems they aren't independent after all.

Where have I gone wrong?

Is it that we cannot introduce just one new coordinate, but when we say we are introducing a new coordinate we are implicitly introducing D new coordinates, but the new special coordinate and D-1 coordinates such that ##x'^a=x^a##? And thus what I should have said is that I'm introducing new coordinates u and x', such that u=t+x and x'=x, and thus ##\frac{\partial u}{\partial u}=1## but ##\frac{\partial u}{\partial x'}=0##?

Sorry, will be offline for a bit. You're really differentiating a curve there. There's a subtle difference between that and a new coordinate system.

Hopefully someone else can explain in full.

TomServo
TomServo said:
That's true, but perhaps you can help me to see something. Let's say I have the 1+1 Minkowski space, with coordinates x and t with basis vectors ##\hat{x}=(0,1)## and ##\hat{t}=(1,0)##. Now I define a new coordinate u=t+x to replace t, and so my coordinates are u and x with basis vectors ##\hat{u}=(1,1)## and ##\hat{x}=(0,1)##.

These are independent basis vectors, so in that sense they are independent, but based on the definition of u it seems that ##\frac{\partial u}{\partial u}=1## and ##\frac{\partial u}{\partial x}=1\neq 0##, in conflict with the notion that ##\frac{\partial x^a}{\partial x^b}=\delta^a_b##, so it seems they aren't independent after all.

Where have I gone wrong?

Is it that we cannot introduce just one new coordinate, but when we say we are introducing a new coordinate we are implicitly introducing D new coordinates, but the new special coordinate and D-1 coordinates such that ##x'^a=x^a##? And thus what I should have said is that I'm introducing new coordinates u and x', such that u=t+x and x'=x, and thus ##\frac{\partial u}{\partial u}=1## but ##\frac{\partial u}{\partial x'}=0##?
This is because of abuse of notations. When you switch from the coordinates ##(t,x)## to ##(u,x)## by the relation ##u=t+x##, what is really meant is that you are considering new coordiantes ##(u,\tilde{x})## with ##u=f(t,x)=t+x## and ##\tilde{x}=g(t,x)=x##.

TomServo
martinbn said:
This is because of abuse of notations. When you switch from the coordinates ##(t,x)## to ##(u,x)## by the relation ##u=t+x##, what is really meant is that you are considering new coordiantes ##(u,\tilde{x})## with ##u=f(t,x)=t+x## and ##\tilde{x}=g(t,x)=x##.
Okay, so my hypothesis at the end was correct?

I don't like abuse of notations because of the confusion they cause, I'm grateful to you for point it out.

## 1. Why are coordinates independent in General Relativity?

Coordinates are independent in General Relativity (GR) because GR is a theory of gravity that is based on the idea of curved spacetime. In this theory, the geometry of spacetime is determined by the distribution of matter and energy, rather than by a fixed coordinate system. This means that the laws of physics should be the same regardless of the coordinates used to describe them.

## 2. How does the concept of curved spacetime relate to independent coordinates in GR?

The concept of curved spacetime is crucial to understanding why coordinates are independent in GR. In this theory, gravity is not a force between masses, but rather a consequence of the curvature of spacetime caused by the presence of mass and energy. This curvature affects the paths of objects, including light, and therefore the way we measure distances and time intervals. The independence of coordinates arises because the geometry of spacetime is determined by the distribution of mass and energy, not by a fixed coordinate system.

## 3. Can you provide an example of how coordinates are independent in GR?

One example of how coordinates are independent in GR is the phenomenon of gravitational lensing. This occurs when light from a distant object is bent by the curvature of spacetime caused by a massive object, such as a galaxy. The path of the light is affected by the distribution of mass and energy, rather than by a specific coordinate system. This means that different observers, using different coordinates, will measure the same bending of light, demonstrating the independence of coordinates in GR.

## 4. How does the independence of coordinates in GR differ from Newtonian mechanics?

In Newtonian mechanics, the laws of physics are described in terms of an absolute space and time. This means that the laws of physics are the same for all observers, regardless of their coordinates. However, in GR, the laws of physics are described in terms of the curvature of spacetime, which is determined by the distribution of mass and energy. This means that the laws of physics can appear different to different observers, depending on their coordinates. This is a fundamental difference between the two theories.

## 5. Are there any practical implications of coordinates being independent in GR?

Yes, there are practical implications of coordinates being independent in GR. One example is the Global Positioning System (GPS), which relies on the principles of GR to accurately determine the position and time on Earth. The GPS satellites use atomic clocks, which are affected by the curvature of spacetime due to the Earth's mass, to calculate the position of a receiver on Earth. This demonstrates how the independence of coordinates in GR has real-world applications.

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