General relativity, will we learn otherwise?

sanctus · 03-14-2008

Quote:

Originally Posted by Natural

I know the feeling, I don't like it either.
My main problem with curved space is that the "models" are all 2 dimensional. (along with the thinking)

Instead of all of the images and drawings that you've ever seen, imagine something that curves from a point outward IN ALL DIRECTIONS. (ALL)

So now describe a path along which something could travel.
Sorry there is no such thing as a path when you curve in all directions.

I know... I know... rotational orbit makes a difference but gravity is still working in ALL directions simultaneously. So which way is the curve?
There in lies my problem with this theory.

Welcome then! No one claims to be able to imagine without the 2D analogy the curvature, but does this make the theory wrong just because you see it as too abstract?

HydrogenBond · 03-14-2008

Maybe I didn't explain myself well enough. If we assume Newtonian gravity, without any GR considerations, such as time dilation, the calculated Newtonian mass-gravitational-pressures-temperatures and density in the sun's core results in a certain nuke reaction rate. This reaction rate will be directly related to the energy output from the core.

If we add GR to the blend, and therefore some time dilation, because the reaction rates have a time element, these reaction rates should dilate. Or the reactions rates should appear to slow in our earth reference relative to the Newtonian predictions. If there is no change there is no time dilation affect coming from the GR relative to the Newtonian prediction.

Let me use a easier example. If one fell into a black hole, from the earth reference they would appear to fall forever. If we had a flashlight with a battery that last 24 hours on earth, it would now appear to last longer near the black hole. Or the energy output from the flashlight slows to trickle.

If we assume GR is valid, the slower nuke reaction rate in stars, should get proportionally slower with increasing star mass, since increasing mass implies increasing GR. If it doesn't, then GR time dilation is not valid or the assumptions of star innards need work.

CraigD · 03-14-2008

Quote:

Originally Posted by HydrogenBond

If we assume GR is valid, the slower nuke reaction rate in stars, should get proportionally slower with increasing star mass, since increasing mass implies increasing GR. If it doesn't, then GR time dilation is not valid or the assumptions of star innards need work.

This is a good thought, but I believe HydrogenBond fails to consider how many and complicated are the factors affecting the rate of stellar fusion, and how gravitational field strength varies within stars and other gravity-contained objects, such as planets.

The primary factors determining rate of stellar fusion is the temperature and pressure in the stellar core (by definition, where fusion occurs), and the abundance of elements there. The fusion rate changes significantly on a long time scale over the life of the star, and on the short term, as stars oscillate and vary (which, observation makes increasingly clear, even “non-variable” stars like Sol do, to some extent).

Like an Earth-like planet’s, the gravitational field strength that determines the gravitational time dilation factor actually decreases at some point as it approaches the center of the star, to nearly zero at the center. So the time dilation factor in the core may be less than in a star’s outer layers, and the time dilation in the core of a more massive star may be less than that of a less massive one. A good and succinct, if necessarily ideal description of this can be read in the “outside and inside a non rotating sphere” section of the wikipedia article “gravitational time dilation”.

Combined, I think these factors result in a situation where the effect on the rate of stellar fusion due to gravitational time dilation is so small relative to the other factors that it is, for all practical purposes, “lost in the noise”. So observing the bolometric magnitude of stars of different masses isn’t a good test of relativity.

From what I’m able to gather, the most precise tests of General Relativity, such as the Pound-Rebka experiment (a redshift, not time dilation, experiment), have been performed using very precise clocks to and very small differences in gravitational field strength.

HydrogenBond · 03-15-2008

Quote:

From what I’m able to gather, the most precise tests of General Relativity, such as the Pound-Rebka experiment (a redshift, not time dilation, experiment), have been performed using very precise clocks to and very small differences in gravitational field strength.

The experiment demonstrates that even subtle gravity differences can exhibit a GR affect. All I was saying was in comparison to Newtonian predictions, with a slower GR time dilated reaction rate relative to the classical Newtonian prediction.

You brought up an interesting point about the center of gravity mathematically causing vector canceling. Although the directional force vectors cancel, the gravity scalar is at a maximum in the center. Does that scalar addition correspond to GR? The reason I ask that is the GR of a planet is often explained, in simple terms, as analogous to a bowling ball making a well in the fabric of space-time. If the vector addition is important than the bottom of that well should contain a peak since the gravity vector addition decreases toward the center. But it is never explained with a peak in the center, but appears to assume scalar addition for only a well.

This leads to two possible stellar reaction rate dilation scenarios. If the GR well is deepest in the center, no vector canceling peak, than the reaction rates are slower in the center and increase toward the perimeter, relative to classical gravity. If there is a peak in the center due to vector canceling, the rate in the center is slowest and increases outward. This can be understood with red or blue shift coming from the center. The first scenario implies blue shift into the perimeter for hotter reaction energy. The second scenario implies red shift into the perimeter for cooler reaction energy, relative to Newtonian.

CraigD · 03-15-2008

Quote:

Originally Posted by HydrogenBond

You brought up an interesting point about the center of gravity mathematically causing vector canceling. Although the directional force vectors cancel, the gravity scalar is at a maximum in the center. Does that scalar addition correspond to GR? The reason I ask that is the GR of a planet is often explained, in simple terms, as analogous to a bowling ball making a well in the fabric of space-time. If the vector addition is important than the bottom of that well should contain a peak since the gravity vector addition decreases toward the center. But it is never explained with a peak in the center, but appears to assume scalar addition for only a well.

The gravitational time dilation predicted by GR for a specific point depends simply of the gravitational potential of the point. As the graph in preceding linked wikipedia section depicts, the gravitational potential for any body single body – Sol, Earth, etc. – is zero at 2 radii from its center of mass, zero and infinity.

Though not a pretty graphic, here’re results of the two gravitational time dilation equations given in the wikipedia article “gravitational time dilation”, from roughtly the orbit of Mercury to the center of Sol.

Code:

r/rS t0/tf
100  .999999978790432605
99   .999999978576194548
98   .999999978357584286
97   .999999978134466596
96   .99999997790670062
95   .999999977674139572
94   .999999977436630415
93   .999999977194013535
92   .999999976946122375
91   .999999976692783058
90   .999999976433813978
89   .999999976169025367
88   .999999975898218834
87   .999999975621186863
86   .999999975337712289
85   .999999975047567723
84   .999999974750514954
83   .999999974446304287
82   .999999974134673848
81   .999999973815348829
80   .999999973488040686
79   .999999973152446259
78   .999999972808246848
77   .999999972455107191
76   .999999972092674386
75   .999999971720576706
74   .999999971338422332
73   .999999970945797975
72   .999999970542267385
71   .999999970127369737
70   .999999969700617869
69   .999999969261496382
68   .999999968809459557
67   .999999968343929095
66   .99999996786429165
65   .999999967369896128
64   .999999966860050747
63   .999999966334019798
62   .999999965791020108
61   .999999965230217149
60   .999999964650720758
59   .999999964051580421
58   .999999963431780072
57   .999999962790232342
56   .999999962125772193
55   .999999961437149856
54   .999999960723022987
53   .999999959981947934
52   .999999959212369994
51   .999999958412612526
50   .99999995758086476
49   .999999956715168103
48   .999999955813400752
47   .999999954873260321
46   .999999953892244219
45   .9999999528676274
44   .999999951796437087
43   .999999950675423968
42   .999999949501029271
41   .999999948269347027
40   .999999946976080668
39   .999999945616492956
38   .999999944185347994
37   .999999942676843842
36   .999999941084533902
35   .99999993940123482
34   .999999937618918141
33   .999999935728582266
32   .999999933720100396
31   .999999931582039045
30   .999999929301440266
29   .999999926863558806
28   .999999924251542951
27   .999999921446044431
26   .999999918424738324
25   .999999915161727719
24   .999999911626799551
23   .999999907784486311
22   .999999903592871851
21   .999999899002055992
20   .999999893952158524
19   .999999888370692872
18   .999999882169064333
17   .99999987523783239
16   .999999867440196398
15   .999999858602875533
14   .999999848503080163
13   .999999836849469994
12   .999999823253591293
11   .999999807185734406
10   .999999787904305802
9    .999999764338114781
8    .999999734880375224
7    .999999697006137375
6    .999999646507151345
5    .999999575808566619
4    .999999469760680159
3    .999999293014177733
2    .999998939521079163
1    .999997879041033708
.99  .999997921248161006
.98  .999997963031094316
.97  .99999800438983369
.96  .99999804532437918
.95  .99999808583473084
.94  .99999812592088872
.93  .999998165582852873
.92  .999998204820623347
.91  .999998243634200193
.9   .999998282023583461
.89  .999998319988773199
.88  .999998357529769456
.87  .999998394646572278
.86  .999998431339181715
.85  .999998467607597812
.84  .999998503451820615
.83  .999998538871850171
.82  .999998573867686523
.81  .999998608439329717
.8   .999998642586779797
.79  .999998676310036806
.78  .999998709609100787
.77  .999998742483971783
.76  .999998774934649835
.75  .999998806961134984
.74  .999998838563427271
.73  .999998869741526738
.72  .999998900495433422
.71  .999998930825147363
.7   .999998960730668601
.69  .999998990211997173
.68  .999999019269133116
.67  .999999047902076468
.66  .999999076110827264
.65  .999999103895385541
.64  .999999131255751334
.63  .999999158191924679
.62  .999999184703905609
.61  .999999210791694158
.6   .999999236455290359
.59  .999999261694694245
.58  .999999286509905848
.57  .999999310900925199
.56  .99999933486775233
.55  .999999358410387271
.54  .999999381528830052
.53  .999999404223080703
.52  .999999426493139253
.51  .999999448339005728
.5   .999999469760680159
.49  .999999490758162571
.48  .999999511331452991
.47  .999999531480551447
.46  .999999551205457962
.45  .999999570506172563
.44  .999999589382695274
.43  .99999960783502612
.42  .999999625863165122
.41  .999999643467112305
.4   .99999966064686769
.39  .999999677402431301
.38  .999999693733803157
.37  .999999709640983281
.36  .999999725123971691
.35  .999999740182768408
.34  .999999754817373451
.33  .999999769027786839
.32  .999999782814008588
.31  .999999796176038719
.3   .999999809113877246
.29  .999999821627524187
.28  .999999833716979558
.27  .999999845382243373
.26  .999999856623315649
.25  .999999867440196398
.24  .999999877832885635
.23  .999999887801383373
.22  .999999897345689626
.21  .999999906465804403
.2   .999999915161727719
.19  .999999923433459583
.18  .999999931281000007
.17  .999999938704348999
.16  .99999994570350657
.15  .999999952278472728
.14  .999999958429247482
.13  .999999964155830839
.12  .999999969458222808
.11  .999999974336423394
.1   .999999978790432605
.09  .999999982820250445
.08  .999999986425876919
.07  .999999989607312033
.06  .99999999236455579
.05  .999999994697608194
.04  .999999996606469247
.03  .999999998091138953
.02  .999999999151617313
.01  .999999999787904329
0    1

Notice that gravitational time dilation peaks at an unspectacular 0.999997879041033708 at Sol’s surface, r/rS=1, and is non-existent at its center, r/rS=0. Also note that these calculations assume constant density and no rotation, so far more approximate than their number of significant digits imply, but are still useful for illustrative purposes.

To get a dramatic, or even infinite (t0/tf=0) gravitational time dilation, a body’s radius must be nearly less than its Schwarzschild radius. With its low density, Sol doesn’t come close to this: its mean radius is about 696,000,000 m, its Schwarzschild radius 2,952 m. Other stars don’t do much better, as even super-giant ones more than 100 times as massive as Sol have even lower densities, with radii several hundred times Sol’s, which their Schwarzchild radii increase only proportional to their masses.

I just can’t imagine any measurement that could detect the effect of a time dilation factor on the order of 0.9999999 in a Star’s fussion, over the many turbulent and only approximately calculable effect that can effect the rate dramatically.

modest · 03-15-2008

Quote:

Originally Posted by CraigD

Like an Earth-like planet’s, the gravitational field strength that determines the gravitational time dilation factor actually decreases at some point as it approaches the center of the star, to nearly zero at the center. So the time dilation factor in the core may be less than in a star’s outer layers, and the time dilation in the core of a more massive star may be less than that of a less massive one. A good and succinct, if necessarily ideal description of this can be read in the “outside and inside a non rotating sphere” section of the wikipedia article “gravitational time dilation”.

Quote:

Originally Posted by CraigD

The gravitational time dilation predicted by GR for a specific point depends simply of the gravitational potential of the point. As the graph in preceding linked wikipedia section depicts, the gravitational potential for any body single body – Sol, Earth, etc. – is zero at 2 radii from its center of mass, zero and infinity.

I believe the wikipedia article: “outside and inside a non rotating sphere” section of the wikipedia article “gravitational time dilation”

Quote:

Originally Posted by wikipedia

If one is inside the sphere, the sphere can be split in two parts: a hollow sphere above and a solid sphere below. One is weightless anywhere in the interior of a uniform hollow sphere. With respect to one's gravitational potential, it is as if the hollow sphere is not there[1][2]. What is left is the solid sphere below, and its mass is:...

The implication is that the gravitational time dilation reaches its maximum at the surface of the non-rotating massive spherically-symmetric object, and that the gravitational time dilation reaches its minimum at the center of the sphere.

is wrong.

They are ignoring the outer shell or hallow sphere part when calculating time dilation which is not correct. It is true that the spacetime in the hallow shpere would be flat or that only the mass beneath you (when you are descending) would contribute toward the curvature or the local force of the gravitational field. However, gravitational time dilation is not a measure of gravitational force. It is rather proportional to gravitational potential phi or it's positive GR counterpart U. For this, the "outer shell" must be considered. The potential in a hollow sphere would be constant - indicating no gravitational force - but it would not be zero. This is a distinction the wiki authors didn't seem to realize.

The gravitational potential outside a spherical body is -GM/r and inside the sphere it is $-.5\times GM/R \times [1+r^2/R^2]$ where r is the distance to center and R is the radius of the body

So, if the gravitational potential is zero at infinity and it's taken to be positive (which is a conversion you have to make for the following equation) then it gets larger as you approach the sphere and continues to get larger inside the sphere approaching the center. Gravitational time dilation is proportional to the potential. Therefore, gravitational time dilation is greatest at the center of a massive object.

EDIT:

Using U, you can calculate the gravitational time dilation inside and outside the sphere:
$T = \frac{T_0}{\sqrt{1-U/c^2}}$

-modest

modest · 03-15-2008

It occurs to me that the equation in my above post for potential in a ball is either wrong or weird or perhaps both. This website:
Gravitational potential due to rigid body
I just found which has the formula:
$\phi = -\frac{GM}{2a^3}(3a^2-r^2)$
where a is the radius of the mass and r is distance from center.

It shows the derivation for this formula and shows the formula for r=0:
$\phi = -\frac{3GM}{2a}$ and says this about it:

Quote:

This may be an unexpected result. The gravitational field strength is zero at the center of a solid sphere, but not the gravitational potential. However, it is entirely possible because gravitational field strength is rate of change in potential, which may be zero as in this case.

which is basically what I was saying above.

In any case, somebody should change the wiki article.

-modest

Erasmus00 · 03-15-2008

Quote:

Originally Posted by modest

I believe the wikipedia article: “outside and inside a non rotating sphere” section of the wikipedia article “gravitational time dilation”

is wrong.

Then lets solve for it. Without proof, I posit that the spherically symmetric static space-time solution in a vacuum is the Schwarzchild metric, which has a t component given by

$ds^2 = \left(1-\frac{C}{r}\right)dt^2 +...$

Now, C is a constant to be set by boundary conditions. Inside a hollow sphere, one condition is that at $r=0$ everything should be finite, so C =0. So inside a hollow sphere, everything is minkowski, and there is no gravitational time dilation. Hence, for gravitational time dilation you need only consider the part of a sphere that is below you.

So, I'm pretty sure wiki got it right.

Edit: On second thought, I take that back. The above argument assumes superposition is valid (inside a sphere= hollowsphere above + solid sphere below), but GR's eqs. are nonlinear.
-Will

CraigD · 03-15-2008

Quote:

Originally Posted by modest

I believe the wikipedia article: “outside and inside a non rotating sphere” section of the wikipedia article “gravitational time dilation”

is wrong.

I believe you’re right. The wikipedia discussion page does, too, but nobody has corrected the main page yet

Quote:

Originally Posted by modest

Using U, you can calculate the gravitational time dilation inside and outside the sphere:…

As I don’t know a formula for the gravitational potential $U = \frac{GM}{r}$ inside a sphere of uniform or other density, I’ve fallen back to a numeric approximation, representing the sphere as a cloud of 26,252 point masses.

I get this data (r/rS is distance in solar radii, t0/tf is time dilation factor, U is gravitational potential, a is acceleration of gravity):

Code:

r/rS    t0/tf                    U                       a
100     .999999989395200034      1906223767.813480345    .02738835683753690541
90      .999999988216884665      2118027173.356373618    .03381282283334155303
80      .999999986743988548      2382781772.589150796    .04279441867358078436
70      .999999984850261461      2723181173.04637664     .0558948742980524389
60      .999999982325284976      3177048304.152280841    .0760793931197597338
50      .999999978790302062      3812465131.93427698     .1095549435136705927
40      .999999973487785805      4765597898.447833819    .1711813736583099953
30      .999999964650117036      6354177963.942459629    .3043292452352190608
20      .99999994697404736       9531469655.77949344     .6847843307467345995
10      .999999893936075265      19065099315.48882054    2.740056558554051093
9       .999999882147080632      21184183073.96256291    3.383131871617272929
8       .999999867409051173      23833358802.36715171    4.282379482834933189
7       .999999848456927781      27240022127.71027448    5.594436683759973575
6       .999999823181128431      31783372494.31814896    7.61693168208359074
5       .999999787781443652      38146501260.77327098    10.97353725131199335
4       .999999734648802308      47697146293.75077       17.15950419178133069
3       .999999646000573906      63631752231.64106418    30.54568677801467849
2       .999999468448974921      95546821916.539406      68.79537428188544505
1       .999998952223784709      188338761261.5377773    236.2785547840037095
.9      .999998860377357958      204848233543.4009131    232.8268030348554124
.8      .999998774380242872      220306285770.2400017    209.8060072617887853
.7      .999998698623743462      233923577798.4557373    180.9041294609779052
.6      .999998634623462806      245427679195.0222742    149.3471828252405965
.5      .999998583165326013      254677319714.0637629    116.2546377940258245
.4      .999998544699015786      261591668889.2030174    82.36314408431174265
.3      .99999851937075551       266144443224.168515     48.59539333978102965
.2      .999998506800302837      268403991752.1595836    17.02233713424564801
.1      .999998505013092641      268725244156.0610172    4.77347197305522623
.09     .999998505215889535      268688791258.7991411    5.64477991360693624
.08     .999998505445717968      268647479421.5523449    6.167159494655981265
.07     .999998505688868986      268603772839.607712     6.33303746024827012
.06     .99999850593166612       268560129868.6037794    6.15193642753531008
.05     .999998506161135347      268518882599.0099152    5.65030422326076242
.04     .999998506365607828      268482128513.5120911    4.868435598279108475
.03     .999998506535173245      268451648999.7996966    3.856129723603217361
.02     .999998506661966524      268428857810.7520283    2.668634036372359829
.01     .999998506740317896      268414774091.3511006    1.363715835633614655
0       .999998506766814506      268410011305.4346802    .0000000000000000105

Instead of a time dilation factor (t0/tf) of zero at the center of Sol, I get a slightly greater dilation (by a factor of about 1.0000005) than at the surface. Note also that, due to the granularity and accumulated calculator errors, this above-surface t0/tfs differs slightly from the pervious, more accurate calculations.

Still, the gravitational time dilation in Sol, or even the most massive stars, remains very small, far below what I think could be detected by observation.

HydrogenBond · 03-16-2008

If you look at the vector addition of gravity in a planetary sphere, the vectors cancel in the center to give zero gravity, based on Newtonian gravity. One thing nobody talks about is where is the energy going? In other words, if we took apart a planet, each piece of matter could generate gravity. If place them all in a sphere we lose gravity in the center. What this implies is potential energy is being lost. If we assume the conservation of energy, then the lost gravity potential needs to go somewhere else. In this case into GR. Or GR appears to be a result of the lost energy within the Newtonian gravity vector addition.

If we plot Newtonian gravity, starting at distant space, relative to an isolated planet, it starts close to zero. As we approach the planet the gravity will get higher and reaches a max at the planet. As we enter the planet the vectors begin to cancel until at the center, we get zero gravity just like at vast distance. So the Newtonian gravity-distance curve starts at zero reaches a max and then decays back to zero.

If you look at this in term of a GR space-time well, the fabric of space-time starts flat at far distance, it begins to form a well that gets deeper as we approach the planet. As we enter the planet the well should start to form a peak as it attempts to reach the top of the fabric at zero gravity, i.e., same mathematical zero value of gravity as distant space in this example. But this peak is not expressed in GR, but is a virtual peak, with the potential energy of that Newtonian peak providing the energy that makes GR possible. The more massive the object, the more vector canceling and lost potential one will get inside the object. This higher amount of lost potential means a larger virtual peak in the GR well, and more space-time affect.

I realize this is unorthodox, but where does the potential energy within the vector canceling go, within the Newtonian center of gravity? This analysis sort of makes gravity the sum of Newtonian plus GR, with GR due to Newtonian vector canceling and the induction of virtual energy.

Some of the first experiments to show the zero potential in a sphere involved a shell with charge on the surface. The vector canceling or the wave canceling implies lost potential energy in the center. The affect is the surface charge becomes dynamic due to energy conservation.

General relativity, will we learn otherwise?

Hypography?

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