Equivalence Principle: Locally Flat Spacetimes Explained

[WWCY]

Hi all, I have ran into some confusion about the equivalence principle; perhaps I should state what I understand and then proceed to ask questions.

It is my understanding that the equivalence principle states that spacetimes are locally Minkowski, and so the rules of SR apply in that locality. Mathematically this means that though we may solve for $g_{\mu \nu}$ using an arbitrarily chosen coordinate system $\theta ^{\mu}$, for every $\theta ^{\mu}$ there exists a Jacobian $J$ such that $J^{T} g J = \eta$. This also means that at every $\theta ^{\mu}$, there us some coordinate change one can apply such that we obtain rectangular coordinates $(t,\vec{x})$.

Question: Doesn't this mean that we can always conspire (via coordinate transformations at every $\theta ^{\mu}$) to describe events using flat spacetimes? How then, does this differ from SR? I can't quite put my finger on the difference.

Thanks in advance!

 

[PeroK]

Hi all, I have ran into some confusion about the equivalence principle; perhaps I should state what I understand and then proceed to ask questions.

It is my understanding that the equivalence principle states that spacetimes are locally Minkowski, and so the rules of SR apply in that locality. Mathematically this means that though we may solve for $g_{\mu \nu}$ using an arbitrarily chosen coordinate system $\theta ^{\mu}$, for every $\theta ^{\mu}$ there exists a Jacobian $J$ such that $J^{T} g J = \eta$. This also means that at every $\theta ^{\mu}$, there us some coordinate change one can apply such that we obtain rectangular coordinates $(t,\vec{x})$.

Question: Doesn't this mean that we can always conspire (via coordinate transformations at every $\theta ^{\mu}$) to describe events using flat spacetimes? How then, does this differ from SR? I can't quite put my finger on the difference.

Thanks in advance!

A local inertial frame only describes a small part of spacetime - approximately well enough for practical purposes. As you extend the local frame the approximation breaks at some point. For example:

If you are on the ISS (space station), then you have your local inertial frame. The approximation is that all local objects orbit in parallel paths. Or, to put it better, Newton's first law applies and two objects at rest with respect to each other remain at rest with respect to each other inside the ISS.

But, if you tried to imagine a huge space station, extending out into a much greater orbit, then this approximation would break down. The outer part of the space station would need enormous force to keep up with the angular speed of the inner part. In addition, if you took two objects at different heights in the ISS, then gradually the outer one would drift behind the inner one. The local inertial frame does not extend forever.

In conclusion, locally we have SR, but not globally.

 

[PeroK]

$J^{T} g J = \eta$.

PS note that, more precisely, you have:

$J^{T} g J (x_P) = \eta (x_P)$

only at a single point $x_P$ and

$J^{T} g J (x) \approx \eta (x)$

for $x$ sufficiently close to $x_P$.

 

[Orodruin]

PS note that, more precisely, you have:

$J^{T} g J (x_P) = \eta (x_P)$

only at a single point $x_P$ and

$J^{T} g J (x) \approx \eta (x)$

for $x$ sufficiently close to $x_P$.

Just adding to this, the curvature in such coordinates at $x_P$ are going to show up as second order corrections in the latter equation.

 

[Dale]

Doesn't this mean that we can always conspire (via coordinate transformations at every θμθμ\theta ^{\mu}) to describe events using flat spacetimes? How then, does this differ from SR? I can't quite put my finger on the difference.

Those local coordinates are only good to first order. The difference shows up as soon as you go far enough away from your reference event that you notice the second order effects.

 

[Pencilvester]

Question: Doesn't this mean that we can always conspire (via coordinate transformations at every θμθμ\theta ^{\mu}) to describe events using flat spacetimes? How then, does this differ from SR?

It might be beneficial for you to understand the distinction between the tangent space at a point on a manifold and the tangent bundle for a manifold. The tangent bundle is the set of tangent spaces at every point in the manifold.

In Minkowski space, there is not much point in making this distinction because any given tangent space can perfectly represent the entire manifold and therefore moving between tangent spaces on this manifold is trivial.

On a curved manifold, however, tangent spaces vary from point to point, so you need a connection that tells you how to move objects in one tangent space to another. And even then, the resulting transported object will depend on which path was taken to get there.

So while we can make a set out of all the tangent spaces at every point on a manifold (i.e. the tangent bundle), we can’t meaningfully combine them to make a single Minkowski space.

 

[WWCY]

Thanks everyone, for the responses. I believe I'm starting to understand the issue better now. Cheers!