Gravity/Curved Spacetimes effect on Time

[TheIsland24]

I have really been trying to wrap my mind around the effects of gravity and curved space on time. Einstein said that those moving near a very massive object would experience slower time. Heres my explanation for this.


+ THe mass of the object curves and wraps spacetime, obviously causing more time to pass between two separate points compared to a straight line between these points. This is why time seems slower.


This a basic understanding, but is it true? Also, is there something more to the effects of massive objects and gravity on time?

 

[Widdekind]

I understand, that Gravitational Time Dilation is related to the Gravitational Redshift Effect.

A photon moving up & out of a Gravity Well is Redshifted, so its frequency slows down.

But, imagine a single frequency signal sent up said Gravity Well. Over any given interval, the higher observer measures a slower frequency for said signal. But, both observers count the same number of cycles.

This is only possible if the higher observer's clock runs faster, so that over the interval, the higher observer has more seconds of clock time to accumulate the cycles from the lower frequency signal:

\nu \; t = N = \nu^{'} \; t^{'}​

 

[A.T.]

+ THe mass of the object curves and wraps spacetime, obviously causing more time to pass between two separate points compared to a straight line between these points. This is why time seems slower.

The time dimension is stretched more near the mass. This causes both:
- Gravity, because free falling objects advance straight in space-time
- Gravitational time dilation, because all objects advance at the same rate in space time.

This picture explains it best:
http://www.physics.ucla.edu/demoweb...alence_and_general_relativity/curved_time.gif

 

[Widdekind]

This picture explains it best:
http://www.physics.ucla.edu/demoweb...alence_and_general_relativity/curved_time.gif

That is an excellent link !According to Rudolf v.B. Rucker's Geometry, Relativity, and the Fourth Dimension (pp. 107-112), matter causes Space to stretch, and Time to shrink. This causes rulers to shrink, and clocks to slow.

Note that rulers don't physically contract -- they just don't reach as far across stretched space. And, clocks don't slow -- their ticks just "span" across more of shrunken time.

http://www.freeimagehosting.net/uploads/b086a6f81d.jpg

 

[Widdekind]

Thus, mass causes Space to stretch, and Time to shrink. This makes Space-Time appear to be "hyperbolically" curved -- Space curves "upward" ( V ), and Time curves "backward" ( ^ ):

http://www.freeimagehosting.net/uploads/878b16376e.jpg

 

[Widdekind]

This "hyperbolic" curvature of Space-Time can cause particles to orbit about the central mass. Such Time-Like Geodesics maximize the Proper Time of those particles. In the 1+1D Space-Time of "Lineland":

In Lineland, it turns out that a Time-Like Geodesic of a particle P, moving near a massive segment M, would look like Figure 130 (if the particle were free to move through M). This World-Line is, of course, that of a particle oscillating back and forth. By staying near M, the World Line gets "more time" (since the Time scale is shrunken near M), and "less space" (since the Space scale is stretched near M), and thus the Interval (= \sqrt{time^{2}-space^{2}}) is maximized (ibid., pg. 110).​

http://www.freeimagehosting.net/uploads/e36189dccd.jpg

 

[Widdekind]

This "backwards bending" of Time near masses might be more clearly depicted by the curvature, caused by a gradually collapsing Black Hole, in Lineland's 1+1D Space-Time:

http://www.freeimagehosting.net/uploads/62f3a600fc.jpg

 

[Widdekind]

Using the prescription, described by Lewis Carroll Epstein's Relativity Visualized, cited in the hyperlink above, of approximating smoothly curved Space-Time with simpler, angular polygons, we can simply visualize the essential structure, of curved Space-Time near masses, in Lineland's 1+1D Space-Time:

http://www.freeimagehosting.net/uploads/b19a4c260e.jpg

Note that, in the Schwarzschild Metric, Space is stretched by the factor ( 1 - \frac{r_{s}}{r} )^{- \frac{1}{2}}, while Time is shrunken by the factor ( 1 - \frac{r_{s}}{r} )^{+ \frac{1}{2}}. Thus, the curvature of Space-Time, caused by massive objects, preserves the "volume" of Space-Time (stretched Space x shrunken Time).
WILD SPECULATION: Naively extending Lewis Carroll Epstein's approximation scheme, seems to suggest, that Schwarzschild Space-Time is "cylindrical" -- as Time continuously curves "backwards" more & more, making Space-Time "wrap back" upon itself, eventually, in a vaguely "cylindrical" structure. If so, Schwarzschild Space-Time would keep "looping back" upon itself, in a never-ending, and unchanging, cycle. Would this not create a completely static, Time-independent, Space-Time -- as per the assumptions underlying the Schwarzschild solution ?

 

[A.T.]

Using the prescription, described by Lewis Carroll Epstein's Relativity Visualized, cited in the hyperlink above, of approximating smoothly curved Space-Time with simpler, angular polygons, we can simply visualize the essential structure, of curved Space-Time near masses, in Lineland's 1+1D Space-Time:

http://www.freeimagehosting.net/uploads/b19a4c260e.jpg

Actually Epstein uses space-proper_time diagrams, where the proper_time-dimension is stretched by mass as shown here:
http://www.adamtoons.de/physics/gravitation.swf

Your picture is that of curved Minkowski space-coordinate_time, where the coordinate_time-dimension is shrunk by mass, as shown here:
http://www.relativitet.se/spacetime2.html

Both visualization methods are described here:
http://www.relativitet.se/Webtheses/lic.pdf

WILD SPECULATION:Naively extending Lewis Carroll Epstein's approximation scheme, seems to suggest, that Schwarzschild Space-Time is "cylindrical"

No, forget it. It is just an embedding. Rolling the space-time diagram into a cylinder has no physical meaning, and nothing to do with the Schwarzschild metric as such. You can do this with a flat space time as well and the radius of the cylinder is arbitrary. When you go one time around the cylinder you arrive on a new "layer" in this visualizations. It is not a cylinder, but roll of space-time if you want.

 

[Widdekind]

Thanks for the links !

This http://www.ligo-la.caltech.edu/archive%20of%20stuff%20not%20used%20anymore/surftalks/LLO.pdf generates an entirely different 1+1D embedding diagram:

http://www.freeimagehosting.net/uploads/277031d316.jpg

 

[Widdekind]

It would seem simplest to rotate the Flamm Paraboloid about the r-axis, to produce these Surfaces of Revolution. But, the Flamm Paraboloid does not produce the correct t-axis scaling -- you would need to divide by \sqrt{r}, so that the "ends" of the "cylinder" asymptote to perfect flatness.

If you can create an accurate Embedding Diagram in 2+0D -- to wit, the Flamm Paraboloid -- you must be able to create an accurate Embedding Diagram in 1+1D...

 

[atyy]

This http://www.ligo-la.caltech.edu/archive%20of%20stuff%20not%20used%20anymore/surftalks/LLO.pdf generates an entirely different 1+1D embedding diagram:

The link says: Warren G. Anderson, The University of Texas at Brownsville

 

[A.T.]

This http://www.ligo-la.caltech.edu/archive%20of%20stuff%20not%20used%20anymore/surftalks/LLO.pdf generates an entirely different 1+1D embedding diagram:
http://www.freeimagehosting.net/

This one uses the spherical coordinate r which is always positive no matter in which direction you go from the mass. Therefore in the diagram you have two space-time points with the same (r,t) coordinates.

The diagrams I linked above use a Cartesian space coordinate x, which goes from negative to positive along a straight line trough the center of the mass. I find this easier to connect with the observed reality.