{"id":200,"date":"2025-06-28T20:49:37","date_gmt":"2025-06-28T17:49:37","guid":{"rendered":"https:\/\/www.makarov.od.ua\/?p=200"},"modified":"2025-06-28T21:44:52","modified_gmt":"2025-06-28T18:44:52","slug":"curvature-of-space-by-time","status":"publish","type":"post","link":"https:\/\/www.makarov.od.ua\/?p=200","title":{"rendered":"Curvature of space by time."},"content":{"rendered":"\n

Everyone knows that the proper time of a material object flying relative to a resting frame of reference differs from the time of the resting frame of reference and is calculated by the formula.<\/p>\n\n\n\n

\"\"<\/figure>
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<\/p>\n<\/div><\/div>\n\n\n\n

Where t0 <\/sub><\/strong>is the time of the reference frame, V <\/strong>is the velocity of relative motion, C <\/strong>is the velocity of light.<\/p>\n\n\n\n

Suppose that two spherical bodies of diameters D1 <\/sub>and D2<\/sub>,<\/p>\n\n\n\n

where D1 <\/sub>= D2 <\/sub>= D and masses m1 <\/sub>and m2<\/sub>, where m1 <\/sub>= m2 <\/sub>fly towards each other with velocity V1 <\/sub>and V2<\/sub>, where V1 <\/sub>= V2<\/sub>, as well as proper time t1 <\/sub>and t2, <\/sub>where t1 <\/sub>= t2 <\/sub>and fly relative to each other at distance R with relative velocity VREL<\/sub> Figure 1.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

If we consider this example from the reference frame of body 1, body 2 is flying relative to body 1 with velocity V21<\/sub>.<\/p>\n\n\n\n

\"\"<\/figure>
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<\/p>\n<\/div><\/div>\n\n\n\n

From the General Theory of Relativity we know that the expression for the time t21 <\/sub>of body 2 flying relative to the frame of reference of body 1 will look like this.<\/p>\n\n\n\n

\"\"<\/figure>
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<\/p>\n<\/div><\/div>\n\n\n\n

Just as if we take body 2 as the reference frame, the expressions for time and velocity are identical.<\/p>\n\n\n\n

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<\/p>\n<\/div><\/div>\n\n\n\n

\"\"<\/figure>
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<\/p>\n<\/div><\/div>\n\n\n\n

Where<\/p>\n\n\n\n

\"\"<\/figure>
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<\/p>\n<\/div><\/div>\n\n\n\n

Then the time of body 2 t21 <\/sub>(2) flying with velocity V21 <\/sub>relative to the frame of reference of body 1 with time t1<\/sub>, is less than time t1 <\/sub>by the value \u2206t.  <\/p>\n\n\n\n

\"\"<\/figure>
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<\/p>\n<\/div><\/div>\n\n\n\n

Just as for body 1 with time t12 <\/sub>(3) flying with velocity V12 <\/sub>relative to the frame of reference of body 2 with time t2<\/sub>, is less than time t2 <\/sub>by the value \u2206t.<\/p>\n\n\n\n

\"\"<\/figure>
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<\/p>\n<\/div><\/div>\n\n\n\n

Based on my assumption, expressed in the article “Relativity of the principle of relativity” that the increment of time \u2206t extends to the distance R from the center of mass of the body moving relative to the center of mass of the reference frame with the regularity \u2206t\/R2<\/sup>.<\/p>\n\n\n\n

We obtain that the change of the time course \u2206tR <\/sub>in points of space at a distance R from a material body of mass m2 <\/sub>and velocity V2 <\/sub>moving relative to a stationary frame of reference, is equal to \u2206tR<\/sub> = \u2206 t\/R2 <\/sup>where<\/p>\n\n\n\n

\"\"<\/figure>
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<\/p>\n<\/div><\/div>\n\n\n\n

That is, in the general case.<\/p>\n\n\n\n

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<\/p>\n<\/div><\/div>\n\n\n\n

The time course of an object moving relative to the reference frame slows down. Correspondingly, when moving away from such a body, time accelerates its course towards t0<\/sub>.  Thus, the time course at a distance R from such an object  tR <\/sub>= t0 <\/sub>– \u2206tR<\/sub>   <\/p>\n\n\n\n

\"\"<\/figure>
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<\/p>\n<\/div><\/div>\n\n\n\n

\"\"<\/figure>
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<\/p>\n<\/div><\/div>\n\n\n\n

We obtain that the time course at a distance R from the center of body 2 flying relative to body 1 taken as a reference frame is tR2 <\/sub>and the time course at a distance  R from the center of body 1 will also be tR1  <\/sub> where  tR1<\/sub> = tR2 <\/sub>= tR  <\/sub><\/p>\n\n\n\n

Hence the distance L traveled by points  a and b of bodies 1 and 2 flying relative to each other Fig. 2 with velocity V1 <\/sub>= V2 <\/sub>= V will be equal to L = V tR<\/sub>.<\/p>\n\n\n\n

\"\"<\/figure>
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<\/p>\n<\/div><\/div>\n\n\n\n

From the condition of equality of velocities, masses and sizes of bodies we obtain that, <\/p>\n\n\n\n

La1<\/sub> = La2<\/sub><\/p>\n\n\n\n

 Lb1 <\/sub>= Lb2<\/sub><\/p>\n\n\n\n

 La1 <\/sub>< Lb1<\/sub><\/p>\n\n\n\n

La2 <\/sub>< Lb2<\/sub><\/p>\n\n\n\n

Fig. 2<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Where Ra1 <\/sub>= Ra2 <\/sub>= Ra<\/sub>, Rb1<\/sub> = Rb2 <\/sub>= Rb<\/sub>, Ra <\/sub>= R – 1\/2D, Rb <\/sub>= R + 1\/2D, Rb <\/sub>= Ra <\/sub>+ D,<\/p>\n\n\n\n

Ra <\/sub>= Rb <\/sub>– D.<\/p>\n\n\n\n

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<\/p>\n<\/div><\/div>\n\n\n\n

\"\"<\/figure>
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<\/p>\n<\/div><\/div>\n\n\n\n

From which it follows that the bodies will describe some arc around each other, Fig. 3.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Let’s try to calculate the radius of the described arc.<\/p>\n\n\n\n

The angles \u03d5 for the arc described by point a and point b of the body will be equal to \u03d5a = <\/sub>\u03d5b <\/sub>and from the formula for arc length L = \u03c0R \u03d5\/180 , where \u03d5 is the angle of the described sector per unit time, R is the radius of the arc.<\/p>\n\n\n\n

From the formula for arc length, \u03d5 is equal to.<\/p>\n\n\n\n

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<\/p>\n<\/div><\/div>\n\n\n\n

Where Ra <\/sub>= Rb <\/sub>– D<\/p>\n\n\n\n

From here we can find out the radius described by a body relative to another body.<\/p>\n\n\n\n

\"\"<\/figure>
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<\/p>\n<\/div><\/div>\n\n\n\n

Or<\/p>\n\n\n\n

Rb <\/sub>= Ra <\/sub>+ D<\/p>\n\n\n\n

\"\"<\/figure>
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<\/p>\n<\/div><\/div>\n\n\n\n

The question arises at what velocity of relative motion <\/p>\n\n\n\n

VREL <\/sub>= V12 <\/sub>= V21 <\/sub>bodies 1 and 2 will rotate around each other while flying uniformly straight and without experiencing any centrifugal forces.<\/p>\n\n\n\n

 Let us solve this problem by fitting values of V, R and D, so that the result of calculation of the radius described by the body Ra <\/sub>and Rb <\/sub>is equal to the substituted value.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

The results of the calculations are recorded in the table.<\/p>\n\n\n\n

 <\/td>R<\/strong><\/td>      1\u00b710-3 <\/sup>m.<\/td>        1m.<\/td>       1.8 m.<\/td><\/tr>
D\/R<\/strong><\/td> <\/td> <\/td> <\/td> <\/td><\/tr>
1\/2<\/td> <\/td>D= 0.5\u00b710-3 <\/sup>m. V=0.0011365\u00b7108<\/sup>m\/s.<\/td>D=0.5 m. V=1.227\u00b7108<\/sup>m\/s.<\/td>D=0.9 m. V=2.7971\u00b7108<\/sup>m\/s.<\/td><\/tr>
1\/5<\/td> <\/td>D=0.2\u00b710-3 <\/sup>m. V=0.001211\u00b7108<\/sup>m\/s.<\/td>D=0.2 m. V=1.32295\u00b7108<\/sup>m\/s.<\/td> <\/td><\/tr>
1\/10<\/td> <\/td>D=0.1\u00b710-3 <\/sup>m. V=0.001221\u00b7108<\/sup>m\/s.<\/td>D=0.1 m. V=1.3358\u00b7108<\/sup>m\/s.<\/td> <\/td><\/tr>
1\/100<\/td> <\/td>D=0.01\u00b710-3 <\/sup>m. V=0.001225\u00b7108<\/sup>m\/s.<\/td>D=0.01 m. V=1.3415\u00b7108<\/sup>m\/s.<\/td> <\/td><\/tr>
1\/1000<\/td> <\/td>D=0.001\u00b710-3 <\/sup>m. V=0.0012247\u00b7108<\/sup>m\/s.<\/td>D=0.001 m. V=1.34187\u00b7108<\/sup>m\/s.<\/td> <\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n