General Relativity

[geordief]

Not homework as such but I would appreciate feedback as to whether I am making progress into understanding the main principles of this theory

Can I take Special Relativity as setting the stage with a "clean sheet" of flat spacetime?

As I see it the main understanding in GR is that ,in a spacetime populated by massive objects at every point it is possible to find 3 2D surfaces corresponding to each spacial dimension plus the time dimension

 

As one finds each of the 3 curvatures on these 3 surfaces (caused by the presence of nearby massive  bodies  or energetic effects)  this overall curvature will allow one to draw a geodesic at any point which will indicate the "natural" direction of movement at the point in question

 

Have I made any obvious errors ?

Is that a decent basic understanding?

[Mordred]

On 12/26/2017 at 1:36 PM, geordief said:

 

Can I take Special Relativity as setting the stage with a "clean sheet" of flat spacetime?

 

No a fully flat geometry is Galiliean relativity or Euclidean space.

[geordief]

32 minutes ago, Mordred said:

No a fully flat geometry is Galiliean relativity or Euclidean space.

They are not spacetime geometries are they?

[Mordred]

No there is no time dilation involved in the above so there is no need to model as a spacetime metric.

It is when you need to consider how time dilation causes an interval deviation from the vector addition under Galilean (Euclidean) relativity do you require SR/GR. Think of it this way. All forms of relativity involve vectors and vector addition. It is through the use of vectors that we define the term ( interval).

More formally Galilean relativity is symmetric on all vectors. This includes unit vectors  used to describe the geometry axis. ( all axis are orthogonal) x,y,z. When length contraction/time dilation occurs under Lorentz this geometry no longer holds true and becomes skew assymmetric.

The amount of skew being described by the Lorentz gamma or beta functions. By accounting for the skew we can restore the orthogonal reference frame (at rest) so that we restore the Pythagorous relations under Euclid geometry. This is described as a transformation. (to transform one frame into another)

Orthogonal (all reference frames are 90 degrees from another.) The 90 degrees provides a perpendicular symmetry at the same time (unit vectors = 1 geometric unit ) are also at unity 1 therefore also symmetric. This is your three 2d hyperplanes. (xyz) with unit vectors (ijk) in sequence. To give time a unit of both energy and interval) vector quantity we define the interval as (ct) .(Under specifically Natural units)[math] c=G=\hbar =1) [/math] Changes to ct are modelled via contraction of the x coordinate axis. How this is graphed is by skewing the y axis on both the plus or minus quadrants towards the positive value x quadrant. (will form a V towards higher value x coordinates.

This is the Lorentz frame skew assymetry graphed. The maximum skew angle being 45 degrees. Above that your velocity exceeeds c.

 

 

Edited December 28, 2017 by Mordred

[Mordred]

So visual aid time....

Take a graph x, y. draw a line where x equal 1. Y is the set of all real numbers.  Now if you skew y axis as per the descriptive above. The straight line forms a hyperbolic curve. At 45 degrees v=c.(timelike) (time has units of length) This forms the surface of the lightcone representation. The direction of motion being in the positive x direction. That surface is the null geodesic under graph.

There is a couple of handy derivitave's to know how we give acceleration and velocity dimension of time. (as length in Natural units via ct)

[math]\frac{d^2x}{dt^2}=0[/math] (if constant non zero acceleration [math]\frac{d^2x}{dt^2}=a[/math]

and [math] \frac{dx}{dt}=v[/math] acceleration and velocity respectively. Lorentz transforms works under constant velocity. Acceleration generates a rotation under symmetry group

Edited December 28, 2017 by Mordred

[geordief]

7 hours ago, Mordred said:

So visual aid time....

Take a graph x, y. draw a line where x equal 1. Y is the set of all real numbers.  Now if you skew y axis as per the descriptive above. The straight line forms a hyperbolic curve. At 45 degrees v=c.(timelike) (time has units of length) This forms the surface of the lightcone representation. The direction of motion being in the positive x direction. That surface is the null geodesic under graph.

There is a couple of handy derivitave's to know how we give acceleration and velocity dimension of time. (as length in Natural units via ct)

d2xdt2=0 (if constant non zero acceleration d2xdt2=a

and dxdt=v acceleration and velocity respectively. Lorentz transforms works under constant velocity. Acceleration generates a rotation under symmetry group

Thanks. That will take me  quite a while to (hopefully ) ingest or work around .

 

 I don't know if you have time (worthwhile time) to skim through this Youtube video (not mine of course   )  to pass comment as to whether it seems rigorous.

 

The animations  seem top notch to me  but if the analysis does not match up  then it  might be  dangerous for those of us with "little knowledge".

 

https://www.youtube.com/watch?v=UfThVvBWZxM

 

Edited December 28, 2017 by geordief

[Mordred]

Looks good, if you learn to understand that vid, your doing good.