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Understanding how General Relativity describes space-time in a given region of space is part of understanding how space and time are being warped. The Schwarzschild metric does a good job of describing space-time around a single spherical body but going beyond that gets complicated if you try to use it for more complicated systems. The best solution to this problem is a generalized metric that gives the full picture.
The relationship between dr and ds as it relates the bending of space in a gravitational field.
Figure 1 illustrates the relationship between dr and ds as it relates the bending of space in a gravitational field. It also shows that when the gravitational acceleration goes to zero the space part of the equation flattens out showing the need for the new metric. This new metric uses the total potential for the time dilation part but replaces it with the gravitational acceleration times the distance in the spacial direction (w) resulting in the new metric shown in formula 1.
The new metric can be expanded for w2 = x2 + y2 + z2 produces the following a general rectangular coordinate format shown in formula 2.
Translating Formula 2 to spherical coordinate format results in in formula 3.
Plugging in the conditions for the Schwarzschild metric where and producing formula 4.
This in turn simplifies to the Schwarzschild metric in formula 5.
Thus this new metric reproduces the Schwarzschild metric by simply plugging in the conditions of an outside gravitational field for which the Schwarzschild metric is intended to describe. Further more the fact that g r = vo2 where vo is orbital velocity at distance r relates the curvature of space to the circular orbital be velocity at that distance from the center of attraction.
Now if we include the Special Relativity relationships in our new metric we get a general metric for both moving and non moving objects in formula 6.
This results in the following general equations for both space (formula 8) and time. (formula 9)
Formula 8. Formula 9.
As in the case of Schwarzschild metric proper length (dl) and proper time (dτ) are real if v < c and if Φ/c2 > -0.5. However if g(w) =0 and v =0 and then dl is is real and dτ is imaginary if Φ/c2 < -0.5. As a result an achronous region (a region where time is stopped) exists with real values for all three spacial dimensions. Furthermore dl is real for any case where v < c and g(w) w < 0.5 c2 and so the achronous region holds for these cases as well.
The generalized metric easily describes the space time between two or more masses. A good example of this is the equal gravitational point between the Earth and the Moon. According to this metric the area should experience time dilation but be essentially flat spatially, have only the curvature produced by the sun. This means that this metric makes predictions about local time dilation and spacial distortions do to gravity for any circumstances. The fact that it makes predictions about space time at any location indicates that this metric is testable.
Illustration of the affect of generalized metric in a bounded universe F is the gravitational potential and Fr is gravitational potential at radius r from the center.
Fr is lower at the center resulting in a higher level of time dilation at the center than towards the edge. However at the edge there is spatially curved while the center is spatially flat. The end result is that the earth is slightly time dilated compared to distant parts of the Universe. This means that distant galaxies would be slightly blue shifted compared to the expected cosmic expansion red shift. This is exactly what is observed in Type Ia supernovae data, suggesting that a spatially bounded universe could explain the Type Ia supernovae data rather that hypothetical dark energy causing accelerating expansion of the universe.
This means that the accelerating expansion of universe interpretation of the Type Ia supernovae data, is a result of assuming a spatially unbounded universe. If on the other hand one assumes a spatially bounded universe with the Earth near the center then gravitational time dilation provides a possible explanation for the Type Ia supernovae data without resorting to hypothetical dark energy. This indicates that dark energy is nothing but a contrivance to allow the Big Bang to fit reality.
In conclusion this generalized metric not only provides a general description of space-time that make testable predictions but when applied to a spatially bounded universe it explains the Type Ia supernovae data, without an accelerating expansion of universe nor hypothetical dark energy. It turns out that an accelerating expansion of universe and dark energy are only necessary to explain the Type Ia supernovae data in a spatially unbounded universe. The simple fact is that the Type Ia supernovae data is a major predictive failure of the Big Bang Cosmology, with dark energy being nothing more than a contrivance to make the Big Bang fit reality.