Do we have two sets of co-ordinate systems when space-time is bent

[San K]

my knowledge of time-space is limited, so my question might be poorly/wrongly constructed/verbalized:

Do we have two sets of co-ordinate systems when space-time is bent (by say, mass)?

in one system the circle becomes, say, an ellipsoid
while in other it remains a circle?

in one system the photon continues to move in a straight line
while in other photon takes a non-straight path?

I mean there must be a "reference" system to compare and say the other one is bent

can such co-ordinate systems be called as frames of references?

 

[WannabeNewton]

General coordinate systems are not equivalent to reference frames. When a manifold is curved, there will be potentially many charts needed to cover it, not just two. An example of a curved space that only requires two is the 2-sphere. However you can't make curvature vanish with a coordinate transformation so just because you are in a different coordinate system doesn't mean you suddenly see vanishing curvature if it wasn't vanishing before. You don't need a special coordinate system to quantify curvature (if we did then curvature would be meaningless). The curvature endomorphism is defined in a coordinate independent way in terms of the connection, lie bracket, and vector fields.

 

[Nugatory]

Do we have two sets of co-ordinate systems when space-time is bent (by say, mass)?

A coordinate system is just a rule for assigning coordinates to points, so you can have as many or as few coordinate systems as you please, whether the spacetime is flat or curved. For example, in a flat two dimensional space I can have (x,y) cartesian coordinates, (r,θ) polar coordinates, or hyperbolic coordinates, or oddballs like (u,v) where u=(x+y) and v=(x-y), or ... As many as I want.

I mean there must be a "reference" system to compare and say the other one is bent.

Not necessarily. For example, I can create a triangle by hammering three stakes into the ground and stretching ropes between them; no preferred "reference" system is needed to decide what's straight and what's not straight. Mathematically, we use the geodesic equation to calculate the path of a stretched-straight rope in a particular coordinate system; and it works just fine using only that coordinate system.

Now if I measure the three inner angles of the triangle and they add up to something other than 180 degrees, then I know that I'm working in a curved space even though I never had anything flat to compare against. (For example: My three points are the north pole and the two points on the equator at 0 degrees and 90 degrees longitude. The ropes will follow the equator and the lines of longitude, forming a right angle at each corner, and I can conclude that the surface of the Earth is curved).

 

[PeterDonis]

I mean there must be a "reference" system to compare and say the other one is bent

No, this is wrong. You can detect curvature in a manifold without having any "reference" outside the manifold to compare to. For example, you can detect the curvature of the Earth's surface purely by taking measurements within the surface, without having any other "reference" to compare it to. The same goes for spacetime itself.

 

[Agerhell]

No, this is wrong. You can detect curvature in a manifold without having any "reference" outside the manifold to compare to. For example, you can detect the curvature of the Earth's surface purely by taking measurements within the surface, without having any other "reference" to compare it to. The same goes for spacetime itself.

Tell me how you measure spacetime curvature. If we imagine the Earth as a perfect sphere and ignore all other bodies in the universe, how do we measure the space-time curvature around it?

 

[WannabeNewton]

Tell me how you measure spacetime curvature. If we imagine the Earth as a perfect sphere and ignore all other bodies in the universe, how do we measure the space-time curvature around it?

The Riemann curvature and Gauss curvature are intrinsic properties of the 2-sphere and can be measured without any external references (the fact that this is true for Gauss curvature is the statement of Gauss's Theorema Egregium). Nugatory gave one way of doing it above, using triangles. You can also use an infinitesimal closed parallelogram constructed on the 2-sphere and parallel transport a vector around this loop to see how the vector has changed when it comes back, relative to the initial conditions. This will let you measure Riemann curvature.

 

[PeterDonis]

Tell me how you measure spacetime curvature. If we imagine the Earth as a perfect sphere and ignore all other bodies in the universe, how do we measure the space-time curvature around it?

You measure tidal gravity, since tidal gravity is what physically corresponds to spacetime curvature in the math of GR. Measuring tidal gravity is easy: just start two free-falling objects at rest with respect to each other but spatially separated by some small amount, and look to see how their relative motion changes with time.