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the big one
2024-08-27 21:48:20 -05:00
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<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:19.4958%;top:8.68014%;width:61.0084%;height:2.14031%>Testing relativity of simultaneity using GPS satellites
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:36.9748%;top:10.7015%;width:26.0504%;height:1.54578%>Kuan Peng 彭宽 titang78@gmail.com
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:43.5294%;top:12.2473%;width:12.9412%;height:1.30797%>25 October 2019
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:22.6891%;top:15.22%;width:50.7563%;height:1.30797%>Abstract: Relativity of simultaneity can be measured with clocks of GPS satellites.
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:14.7899%;top:17.8359%;width:25.3782%;height:1.54578%>1. Time-slide in GPS system
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:20.8086%;width:73.2773%;height:1.54578%>In Special Relativity relativity of simultaneity is the fact that 2 simultaneous events occurring in a
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:22.3543%;width:72.7731%;height:1.54578%>stationary frame does not appear simultaneous in a moving frame. For example, in Einstein's train
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:23.7812%;width:76.4706%;height:1.54578%>thought experiment 2 simultaneous flashes of light on the platform do not appear simultaneous for the
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:25.327%;width:76.9748%;height:1.54578%>observer in the train. But relativity of simultaneity has never been tested with real simultaneous events.
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:28.2996%;width:76.6387%;height:1.54578%>For testing relativity of simultaneity we need 2 synchronized clocks moving at high speed and we will
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:29.8454%;width:70.9244%;height:1.54578%>read them in a stationary frame. Fortunately, we have at hand many GPS satellites which carry
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:31.2723%;width:70.084%;height:1.66468%>precision clocks and broadcast their time, with which we can check relativity of simultaneity.
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:34.3639%;width:55.6303%;height:1.54578%>Figure 1 shows an example of 8 GPS satellites in a circular orbit. F1 is the
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:35.7907%;width:54.6219%;height:1.54578%>frame moving with the satellite 1, its x axis passes through the satellites 1
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:37.3365%;width:56.9748%;height:1.54578%>and 2. In a frame that coincides with F1 at time 0 and stationary with respect
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:38.8823%;width:54.6219%;height:1.54578%>to the Earth, the positions of the satellites 1 and 2 are x1 and x2. Suppose
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:40.3092%;width:56.3025%;height:1.54578%>that 2 simultaneous events occur at x1 and x2 in the stationary frame. In the
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:41.8549%;width:56.1345%;height:1.54578%>moving frame F1 these same events will occur at the times t'1 and t'2 on the
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:43.4007%;width:57.1429%;height:1.42687%>satellites 1 and 2 respectively. t'1 and t'2 are determined by the time equation
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:44.7087%;width:51.5966%;height:1.54578%>of the Lorentz transformations which is equation (1) with v being the
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:46.2545%;width:54.2857%;height:1.54578%>velocity of the satellite 1. We call t'2 - t'1 the time-slide of the event at x2
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:47.6813%;width:56.9748%;height:1.66468%>with respect to that at x1. The time-slide of the satellite 2 with respect to the
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:49.2271%;width:49.7479%;height:1.54578%>satellites 1 is approximately expressed by equation (2) because the
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:50.7729%;width:37.1429%;height:1.54578%>velocities of the satellites 1 and 2 are not parallel.
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:53.7455%;width:54.4538%;height:1.54578%>In order to get a precise expression of the time-slide of the satellite 2, let
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:55.2913%;width:52.1008%;height:1.54578%>us imagine that there are n satellites between the satellites 1 and 2. In
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:56.7182%;width:13.2773%;height:1.30797%>the frame of the i
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:24.3697%;top:56.4804%;width:39.1597%;height:1.54578%>th satellite the time-slide between the satellites i and
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:58.3829%;width:52.9412%;height:1.54578%>i+1 is 'i which is expressed by equation (3), where i is the distance
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:59.9287%;width:50.2521%;height:1.54578%>between these 2 satellites. Equation (3) is sufficiently precise if the
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:61.3555%;width:49.7479%;height:1.30797%>satellites i and i+1 are so close that their velocities can be taken as
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:63.0202%;width:51.7647%;height:2.97265%>parallel. By summing all i from i=1 to n, we obtain t'2 - t'1 the time-
slides of the satellite 2 which is expressed in equation (4).
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:67.5386%;width:53.4454%;height:1.66468%>When n is very big, the sum of all the distance i equals p2, the length
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:69.0844%;width:53.9496%;height:1.54578%>of the arc between the satellites 1 and 2, see Figure 2. So, the time-slide
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:70.6302%;width:45.8824%;height:1.54578%>of the satellite 2 is precisely expressed with p2 in equation (5).
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:73.6029%;width:40.1681%;height:1.54578%>For all the satellites in Figure 1, the time-slide of the j
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:51.5966%;top:73.365%;width:11.7647%;height:1.54578%>th satellite with
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:75.1486%;width:53.2773%;height:1.54578%>respect to the satellites 1 is t'j- t'1 and is expressed by equation (6) with
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:76.3377%;width:53.2773%;height:1.9025%>pj being the arc from the first satellite to the jth satellite and j being any
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:77.8835%;width:53.2773%;height:1.54578%>number from 1 to 9. The 9th satellite is in fact the first satellite because
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:79.6671%;width:14.6218%;height:1.30797%>the orbit is a circle.
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:82.4019%;width:52.1008%;height:1.66468%>The time-slide of the 9th satellite is t'9 - t'1 and is non-zero. So, due to
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:83.8288%;width:51.5966%;height:1.9025%>relativity of simultaneity the time of the 9th satellite is different from
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:85.6124%;width:54.1177%;height:1.54578%>that of the first satellite. But this is impossible in real world because the
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:86.9203%;width:22.8571%;height:1.54578%>9th satellite is the first satellite.
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:90.1308%;width:52.605%;height:1.54578%>On the other hand, the synchronization of the satellites is not obvious.
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:76.1345%;top:47.5624%;width:6.38656%;height:1.54578%>Figure 1
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:68.0672%;top:51.2485%;width:2.35294%;height:1.18906%>′ =
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:70.7563%;top:50.5351%;width:2.35294%;height:0.713436%>−
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:73.1092%;top:50.8918%;width:0.504202%;height:0.356718%>
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:71.7647%;top:52.9132%;width:2.52101%;height:1.07015%>1 −
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:86.2185%;top:51.6052%;width:2.18487%;height:1.30797%>(1)
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:68.2353%;top:56.2426%;width:0.672269%;height:0.713436%>′
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:68.5714%;top:56.7182%;width:2.52101%;height:0.951249%>−
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:71.5966%;top:56.2426%;width:0.840336%;height:0.713436%>′
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:72.1008%;top:56.5993%;width:4.03361%;height:1.07015%>≈ −
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:76.8067%;top:55.1724%;width:2.01681%;height:1.42687%>(
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:79.1597%;top:55.648%;width:2.52101%;height:0.951249%>−
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:83.0252%;top:55.1724%;width:1.0084%;height:1.30797%>)
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:78.3193%;top:57.9073%;width:2.68908%;height:1.18906%>1 −
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:86.2185%;top:56.5993%;width:2.18487%;height:1.42687%>(2)
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:67.563%;top:61.3555%;width:1.34454%;height:1.07015%>∆
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:69.2437%;top:61.2366%;width:0.672269%;height:0.713436%>′
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:69.5798%;top:61.7122%;width:3.86555%;height:0.951249%>= −
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:74.958%;top:60.2854%;width:2.18487%;height:1.42687%>∆
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:74.6219%;top:63.0202%;width:2.52101%;height:1.07015%>1 −
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:86.2185%;top:61.7122%;width:2.18487%;height:1.30797%>(3)
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:68.7395%;top:66.5874%;width:2.35294%;height:0.832342%>−
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:71.7647%;top:66.1118%;width:0.840336%;height:0.713436%>′
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:72.1008%;top:66.4685%;width:2.68908%;height:0.951249%>=
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:76.9748%;top:66.2307%;width:1.34454%;height:1.07015%>∆
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:67.563%;top:70.2735%;width:3.02521%;height:0.713436%>= −
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:71.7647%;top:71.5815%;width:2.68908%;height:1.07015%>1 −
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:78.6555%;top:69.9168%;width:1.34454%;height:1.07015%>∆
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:86.2185%;top:68.4899%;width:2.18487%;height:1.30797%>(4)
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:72.7731%;top:79.6671%;width:6.72269%;height:1.66468%>Figure 2
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:68.2353%;top:82.8775%;width:0.672269%;height:0.713436%>′
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:68.5714%;top:83.3532%;width:2.52101%;height:0.951249%>−
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:71.5966%;top:82.8775%;width:0.840336%;height:0.713436%>′
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:72.1008%;top:83.2342%;width:4.03361%;height:1.07015%>= −
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:77.3109%;top:84.5422%;width:2.68908%;height:1.07015%>1 −
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:82.8572%;top:83.2342%;width:5.54622%;height:1.30797%>(5)
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:68.2353%;top:87.8716%;width:0.672269%;height:0.713436%>′
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:68.4034%;top:88.3472%;width:2.35294%;height:0.951249%>−
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:71.4286%;top:87.8716%;width:0.840336%;height:0.713436%>′
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:71.7647%;top:88.2283%;width:4.20168%;height:1.07015%>= −
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:77.1429%;top:89.6552%;width:2.52101%;height:1.07015%>1 −
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:82.521%;top:88.3472%;width:5.88235%;height:1.30797%>(6)
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:69.4118%;top:75.0297%;width:1.51261%;height:1.07015%>
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:78.8235%;top:77.8835%;width:1.51261%;height:1.07015%>
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:75.9664%;top:75.2675%;width:2.18487%;height:0.832342%>=
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:79.6639%;top:75.0297%;width:1.17647%;height:0.951249%>∆
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:76.4706%;top:40.9037%;width:2.68908%;height:0.951249%>Earth
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:74.1176%;top:46.0166%;width:7.89916%;height:1.07015%>GPS satellites
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:76.9748%;top:35.3151%;width:1.51261%;height:1.07015%>F1
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:82.1849%;top:36.6231%;width:3.19328%;height:1.18906%>x2, t'2
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:73.1092%;top:37.8121%;width:1.34454%;height:1.07015%>
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:77.3109%;top:36.6231%;width:1.51261%;height:1.07015%>
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:83.1933%;top:40.7848%;width:1.34454%;height:1.07015%>
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:72.9412%;top:43.7574%;width:1.51261%;height:0.832342%>
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:71.2605%;top:40.7848%;width:1.51261%;height:1.07015%>
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:81.5126%;top:37.931%;width:1.34454%;height:1.07015%>
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:77.3109%;top:44.7087%;width:1.51261%;height:1.07015%>
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:81.3445%;top:43.6385%;width:1.51261%;height:0.951249%>
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:76.3025%;top:37.6932%;width:3.36134%;height:1.18906%>x1, t'1
</p></div><div class=ndfHFb-c4YZDc-cYSp0e-wxLEad-sn54Q style=display:none></div><div class="ndfHFb-c4YZDc-vWsuo-fmcmS-IDNFyf ndfHFb-c4YZDc-vWsuo-fmcmS-gvZm2b"></div><div class="ndfHFb-c4YZDc-vWsuo-fmcmS-IDNFyf ndfHFb-c4YZDc-vWsuo-fmcmS-G0jgYd"></div><div class="ndfHFb-c4YZDc-vWsuo-fmcmS-IDNFyf ndfHFb-c4YZDc-vWsuo-fmcmS-G0jgYd"></div><div class="ndfHFb-c4YZDc-vWsuo-fmcmS-IDNFyf ndfHFb-c4YZDc-cYSp0e-oYxtQd-gvZm2b"></div><div class=ndfHFb-c4YZDc-dZssN-qkEl5b></div><div class=ndfHFb-c4YZDc-dZssN-G0KQoc></div><img src=data:, class=ndfHFb-c4YZDc-cYSp0e-DARUcf-RJLb9c alt="Page 1 of 4" aria-hidden=true style=opacity:1></div><div class=ndfHFb-c4YZDc-cYSp0e-DARUcf style=padding-bottom:141.375%><div class=ndfHFb-c4YZDc-cYSp0e-DARUcf-PLDbbf><a href="https://en.wikipedia.org/w/index.php?title=Global_Positioning_System&amp;oldid=922144667" target=_blank class=ndfHFb-c4YZDc-cYSp0e-DARUcf-hSRGPd tabindex=0 role=link aria-label="https://en.wikipedia.org/w/index.php?title=Global_Positioning_System&amp;oldid=922144667" data-saferedirecturl="https://www.google.com/url?q=https://en.wikipedia.org/w/index.php?title%3DGlobal_Positioning_System%26oldid%3D922144667&amp;sa=D&amp;source=apps-viewer-frontend&amp;ust=1717124493546278&amp;usg=AOvVaw3oLN0AmZJjqinDF4t_52wn&amp;hl=en" rel=noreferrer style=left:36.6387%;top:8.44233%;width:14.6218%;height:1.54578%></a><a href=https://en.wikipedia.org/wiki/Global_Positioning_System target=_blank class=ndfHFb-c4YZDc-cYSp0e-DARUcf-hSRGPd tabindex=0 role=link aria-label=https://en.wikipedia.org/wiki/Global_Positioning_System data-saferedirecturl="https://www.google.com/url?q=https://en.wikipedia.org/wiki/Global_Positioning_System&amp;sa=D&amp;source=apps-viewer-frontend&amp;ust=1717124493546336&amp;usg=AOvVaw2g3z6FYTRNVmRQhX6-qn3v&amp;hl=en" rel=noreferrer style=left:23.6975%;top:23.5434%;width:25.3782%;height:1.54578%></a><a href=https://en.wikipedia.org/wiki/Ladder_paradox target=_blank class=ndfHFb-c4YZDc-cYSp0e-DARUcf-hSRGPd tabindex=0 role=link aria-label=https://en.wikipedia.org/wiki/Ladder_paradox data-saferedirecturl="https://www.google.com/url?q=https://en.wikipedia.org/wiki/Ladder_paradox&amp;sa=D&amp;source=apps-viewer-frontend&amp;ust=1717124493546348&amp;usg=AOvVaw2hr_H4LHb5GGe2tAhgchOk&amp;hl=en" rel=noreferrer style=left:43.1933%;top:56.4804%;width:5.21008%;height:1.66468%></a><a href=https://en.wikipedia.org/wiki/Ladder_paradox target=_blank class=ndfHFb-c4YZDc-cYSp0e-DARUcf-hSRGPd tabindex=0 role=link aria-label=https://en.wikipedia.org/wiki/Ladder_paradox data-saferedirecturl="https://www.google.com/url?q=https://en.wikipedia.org/wiki/Ladder_paradox&amp;sa=D&amp;source=apps-viewer-frontend&amp;ust=1717124493546359&amp;usg=AOvVaw0ygTh5FkU7c5XV5-7exec4&amp;hl=en" rel=noreferrer style=left:11.7647%;top:58.0262%;width:5.54622%;height:1.54578%></a></div><div class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-bN97Pc-haAclf><h2 class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-tJHJj tabindex=0>Page 2 of 4</h2><p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:87.0588%;top:92.8656%;width:1.17647%;height:1.30797%>2
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:8.56124%;width:72.1008%;height:1.54578%>According to the Wikipedia page Basic concept of GPS: "The satellites carry very stable atomic
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:9.98811%;width:74.2857%;height:1.54578%>clocks that are synchronized with one another and with the ground clocks." So, the satellites 1 and 2
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:11.5339%;width:75.9664%;height:1.54578%>for example must be synchronized with one clock on Earth, that is, the event "time of the satellite 1 is
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:13.0797%;width:75.7983%;height:1.54578%>t0" and the event "time of the satellite 2 is t0" occur simultaneously on Earth. But, as the satellite 2 has
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:14.5065%;width:73.4454%;height:1.54578%>a non-zero time-slide with respect to the satellite 1, the satellite 2 cannot be synchronized with the
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:16.0523%;width:76.6387%;height:1.54578%>satellite 1 in orbit. Conversely, if the satellite 2 is synchronized with the satellite 1 in orbit, the reading
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:17.5981%;width:73.1092%;height:1.54578%>of their time on Earth would be different. So, due to relativity of simultaneity the satellites 1 and 2
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:19.025%;width:64.3698%;height:1.54578%>cannot be synchronized with the clock on Earth and with one another at the same time.
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:21.9976%;width:74.4538%;height:1.66468%>Nevertheless, let us compute the value of t'9 - t'1. The radius of the GPS orbit is 26 600 km (see the
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:23.5434%;width:76.8067%;height:1.54578%>Wikipedia page Structure of the orbit of GPS satellites), the circumference of this orbit is 167 133 km,
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:25.0892%;width:75.4622%;height:1.54578%>that is, p9 = 167 133 km. The velocity of the satellites is 3.87 km/s, the speed of light is 299792 km/s.
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:26.2782%;width:74.6219%;height:1.9025%>Then, the time-slide of the 9th satellite is t'9 - t'1 = - 7204 ns. If this time-slide really exists but is not
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:28.0618%;width:57.479%;height:1.54578%>correctly dealt with, the GPS system would give wrong coordinates on Earth.
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:31.0345%;width:74.4538%;height:1.66468%>The coordinates computed by GPS devices on Earth using the time of satellites are actually correct,
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:32.5803%;width:75.7983%;height:1.54578%>which proves that the clocks of the satellites are really synchronized with the clock on Earth and also
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:34.126%;width:64.2017%;height:1.54578%>with one another. This is impossible if relativity of simultaneity affects GPS satellites.
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:14.7899%;top:37.2176%;width:32.437%;height:1.54578%>2. Time-slide and length contraction
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:40.1903%;width:58.4874%;height:1.54578%>What is the consequence if relativity of simultaneity were not true? Relativity
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:41.6171%;width:57.1429%;height:1.66468%>of simultaneity is given by the time equations of the Lorentz transformations
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:43.1629%;width:60%;height:1.42687%>which are derived from the space equations that are the equations (7) and (8). By
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:44.5898%;width:58.6555%;height:1.54578%>substituting equation (7) for x' in equation (8), we obtain equation (9) which we
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:46.0166%;width:57.9832%;height:1.66468%>transform into equation (10). By substituting equation (8) for x in equation (7),
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:47.5624%;width:57.3109%;height:1.54578%>we obtain equation (11) which we transform into equation (12). Equations (10)
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:49.1082%;width:24.5378%;height:1.54578%>and (12) are the 2 time equations.
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:52.0809%;width:36.6387%;height:1.54578%>So, the system of the 2 time equations is a linear
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:53.6266%;width:32.7731%;height:1.54578%>rearrangement of the system of the 2 space
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:55.0535%;width:33.7815%;height:1.54578%>equations and these 2 systems are equivalent.
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:56.5993%;width:37.6471%;height:1.54578%>This is why the length-contraction-caused Ladder
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:58.1451%;width:33.4454%;height:1.54578%>paradox can be explained using relativity of
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:59.5719%;width:32.7731%;height:1.54578%>simultaneity. But also, a contradiction with
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:61.1177%;width:36.9748%;height:1.54578%>relativity of simultaneity leads to a contradiction
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:62.5446%;width:17.9832%;height:1.66468%>with length contraction.
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:14.6218%;top:65.6362%;width:26.2185%;height:1.66468%>3. Orbital length contraction
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:68.7277%;width:75.1261%;height:1.54578%>Does the said contradiction with length contraction exist? Let us show it with geostationary satellites
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:70.1546%;width:69.916%;height:1.54578%>which are particularly appropriate for this purpose because they are stationary with respect to
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:71.7004%;width:60.1681%;height:1.54578%>observers on Earth, they all move at the same velocity and their orbit is circular.
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:74.673%;width:76.9748%;height:1.54578%>A satellite is located with respect to the center of the Earth by a vector called position vector which we
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:76.2188%;width:57.1429%;height:1.54578%>can precisely determine with radars on Earth. Once the position vectors of 2
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:77.7646%;width:59.4958%;height:1.42687%>satellites are determined, we can derive the distance between them and check if
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:79.1914%;width:29.916%;height:1.54578%>this distance verifies length contraction.
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:82.1641%;width:59.8319%;height:1.66468%>In the example shown in Figure 3, the 8 geostationary satellites are immobile in
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:83.7099%;width:58.6555%;height:1.54578%>the sky for an observer on Earth, that is, they are immobile in the frame of the
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:85.2556%;width:59.3277%;height:1.54578%>observer which we denote by A. In a frame in space which does not rotate with
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:86.6825%;width:57.6471%;height:1.66468%>the Earth, these satellites move around the Earth. This frame is denoted by B.
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:89.7741%;width:57.1429%;height:1.54578%>The position vectors of the satellites 1 and 2 are R1 and R2 in the frame A at
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:74.1176%;top:40.6659%;width:2.35294%;height:1.18906%>′ =
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:77.1429%;top:40.3092%;width:2.52101%;height:0.713436%>−
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:77.6471%;top:42.3306%;width:2.52101%;height:1.18906%>1 −
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:86.2185%;top:41.4982%;width:2.18487%;height:1.30797%>(7)
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:73.2773%;top:45.4221%;width:2.85714%;height:0.713436%>=
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:77.1429%;top:44.2331%;width:2.18487%;height:1.18906%>′ +
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:80.8403%;top:44.2331%;width:0.840336%;height:0.59453%>′
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:77.3109%;top:46.7301%;width:2.52101%;height:1.07015%>1 −
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:86.2185%;top:45.6599%;width:2.18487%;height:1.42687%>(8)
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:53.2773%;top:53.6266%;width:2.52101%;height:1.07015%>1 −
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:56.6387%;top:53.8645%;width:2.68908%;height:1.18906%>=
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:60%;top:53.151%;width:2.68908%;height:0.713436%>−
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:60.5042%;top:55.1724%;width:2.68908%;height:1.18906%>1 −
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:65.2101%;top:53.6266%;width:1.34454%;height:1.07015%>+
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:68.0672%;top:53.8645%;width:5.21008%;height:1.30797%>(9)
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:75.7983%;top:53.3888%;width:2.18487%;height:1.18906%>′ =
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:78.3193%;top:52.7943%;width:2.35294%;height:0.713436%>−
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:80.8403%;top:53.151%;width:0.504202%;height:0.356718%>
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:79.3277%;top:55.1724%;width:2.52101%;height:1.07015%>1 −
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:85.3782%;top:53.8645%;width:3.02521%;height:1.30797%>(10)
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:52.1008%;top:58.7396%;width:0.672269%;height:0.59453%>′
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:53.6135%;top:58.7396%;width:2.68908%;height:1.18906%>1 −
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:57.1429%;top:59.0963%;width:2.68908%;height:1.18906%>=
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:61.0084%;top:58.0262%;width:2.18487%;height:1.07015%>+
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:64.7059%;top:57.9073%;width:0.672269%;height:0.713436%>′
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:61.0084%;top:60.4043%;width:2.52101%;height:1.07015%>1 −
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:65.5462%;top:59.2152%;width:1.51261%;height:0.713436%>−
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:67.8992%;top:58.9774%;width:5.88235%;height:1.42687%>(11)
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:75.1261%;top:58.9774%;width:2.52101%;height:0.832342%>=
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:78.4874%;top:57.6694%;width:2.18487%;height:1.07015%>+
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:81.5126%;top:57.5505%;width:0.840336%;height:1.18906%>′
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:79.1597%;top:60.2854%;width:2.52101%;height:1.18906%>1 −
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:85.3782%;top:58.9774%;width:3.02521%;height:1.42687%>(12)
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:77.479%;top:90.1308%;width:6.55462%;height:1.54578%>Figure 3
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:76.8067%;top:87.396%;width:8.40336%;height:1.18906%>Geostationary
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:78.3193%;top:88.4661%;width:5.04202%;height:1.07015%>satellites
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:78.9916%;top:78.9536%;width:1.84874%;height:1.07015%>R1
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:81.1765%;top:80.7372%;width:1.17647%;height:0.832342%>
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:78.9916%;top:82.7586%;width:4.03361%;height:0.951249%>Earth's
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:79.1597%;top:83.8288%;width:3.19328%;height:0.832342%>center
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:83.8656%;top:80.4994%;width:1.84874%;height:1.07015%>R2
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:75.9664%;top:78.9536%;width:1.51261%;height:1.07015%>
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:80.1681%;top:77.7646%;width:1.51261%;height:1.07015%>
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:86.0504%;top:81.9263%;width:1.51261%;height:1.07015%>
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:75.9664%;top:84.8989%;width:1.34454%;height:1.07015%>
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:74.2857%;top:81.9263%;width:1.51261%;height:1.07015%>
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:84.3698%;top:79.0725%;width:1.51261%;height:1.07015%>
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:80.1681%;top:86.088%;width:1.51261%;height:0.951249%>
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:84.3698%;top:84.78%;width:1.51261%;height:1.07015%>
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:82.1849%;top:78.8347%;width:0.840336%;height:1.07015%>l
</p></div><div class=ndfHFb-c4YZDc-cYSp0e-wxLEad-sn54Q style=display:none></div><div class="ndfHFb-c4YZDc-vWsuo-fmcmS-IDNFyf ndfHFb-c4YZDc-vWsuo-fmcmS-gvZm2b"></div><div class="ndfHFb-c4YZDc-vWsuo-fmcmS-IDNFyf ndfHFb-c4YZDc-vWsuo-fmcmS-G0jgYd"></div><div class="ndfHFb-c4YZDc-vWsuo-fmcmS-IDNFyf ndfHFb-c4YZDc-vWsuo-fmcmS-G0jgYd"></div><div class="ndfHFb-c4YZDc-vWsuo-fmcmS-IDNFyf ndfHFb-c4YZDc-cYSp0e-oYxtQd-gvZm2b"></div><div class=ndfHFb-c4YZDc-dZssN-qkEl5b></div><div class=ndfHFb-c4YZDc-dZssN-G0KQoc></div><img src=data:, class=ndfHFb-c4YZDc-cYSp0e-DARUcf-RJLb9c alt="Page 2 of 4" aria-hidden=true></div><div class=ndfHFb-c4YZDc-cYSp0e-DARUcf style=padding-bottom:141.375%><div class=ndfHFb-c4YZDc-cYSp0e-DARUcf-PLDbbf><a href=https://en.wikipedia.org/wiki/Geostationary_orbit target=_blank class=ndfHFb-c4YZDc-cYSp0e-DARUcf-hSRGPd tabindex=0 role=link aria-label=https://en.wikipedia.org/wiki/Geostationary_orbit data-saferedirecturl="https://www.google.com/url?q=https://en.wikipedia.org/wiki/Geostationary_orbit&amp;sa=D&amp;source=apps-viewer-frontend&amp;ust=1717124493558824&amp;usg=AOvVaw2orzk4R8DqrfHLA7WDXRdE&amp;hl=en" rel=noreferrer style=left:69.7479%;top:45.1843%;width:12.605%;height:1.54578%></a><a href=https://pengkuanonphysics.blogspot.com/2019/06/how-to-test-length-contraction-by.html target=_blank class=ndfHFb-c4YZDc-cYSp0e-DARUcf-hSRGPd tabindex=0 role=link aria-label=https://pengkuanonphysics.blogspot.com/2019/06/how-to-test-length-contraction-by.html data-saferedirecturl="https://www.google.com/url?q=https://pengkuanonphysics.blogspot.com/2019/06/how-to-test-length-contraction-by.html&amp;sa=D&amp;source=apps-viewer-frontend&amp;ust=1717124493558862&amp;usg=AOvVaw0Loa6-ZxtfKcA9-muaNfhQ&amp;hl=en" rel=noreferrer style=left:14.958%;top:57.3127%;width:12.2689%;height:1.54578%></a><a href=https://www.academia.edu/39584663/How_to_test_length_contraction_by_experiment target=_blank class=ndfHFb-c4YZDc-cYSp0e-DARUcf-hSRGPd tabindex=0 role=link aria-label=https://www.academia.edu/39584663/How_to_test_length_contraction_by_experiment data-saferedirecturl="https://www.google.com/url?q=https://www.academia.edu/39584663/How_to_test_length_contraction_by_experiment&amp;sa=D&amp;source=apps-viewer-frontend&amp;ust=1717124493558877&amp;usg=AOvVaw2aP0mUed-cn0r0aO7Ww-yl&amp;hl=en" rel=noreferrer style=left:29.0756%;top:57.3127%;width:17.479%;height:1.54578%></a><a href=https://pengkuanonphysics.blogspot.com/2019/08/astrophysical-jet-and-length-contraction.html target=_blank class=ndfHFb-c4YZDc-cYSp0e-DARUcf-hSRGPd tabindex=0 role=link aria-label=https://pengkuanonphysics.blogspot.com/2019/08/astrophysical-jet-and-length-contraction.html data-saferedirecturl="https://www.google.com/url?q=https://pengkuanonphysics.blogspot.com/2019/08/astrophysical-jet-and-length-contraction.html&amp;sa=D&amp;source=apps-viewer-frontend&amp;ust=1717124493558956&amp;usg=AOvVaw0SWEJ5yH-6DAbLzDh0LBzK&amp;hl=en" rel=noreferrer style=left:46.7227%;top:87.5149%;width:11.4286%;height:1.54578%></a><a href=https://www.academia.edu/40066246/Astrophysical_jet_and_length_contraction target=_blank class=ndfHFb-c4YZDc-cYSp0e-DARUcf-hSRGPd tabindex=0 role=link aria-label=https://www.academia.edu/40066246/Astrophysical_jet_and_length_contraction data-saferedirecturl="https://www.google.com/url?q=https://www.academia.edu/40066246/Astrophysical_jet_and_length_contraction&amp;sa=D&amp;source=apps-viewer-frontend&amp;ust=1717124493558972&amp;usg=AOvVaw20BqTVBA94gGWfCFVmcNzJ&amp;hl=en" rel=noreferrer style=left:11.7647%;top:88.9417%;width:11.9328%;height:1.66468%></a></div><div class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-bN97Pc-haAclf><h2 class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-tJHJj tabindex=0>Page 3 of 4</h2><p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:87.0588%;top:92.8656%;width:1.17647%;height:1.30797%>3
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:8.56124%;width:74.6219%;height:1.66468%>time 0. The angle made by the vectors R1 and R2 is , which is the angular position of the satellite 2
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:10.107%;width:72.1008%;height:1.54578%>with respect to the satellite 1. For n satellites that are equally spaced in orbit, the angle from one
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:11.7717%;width:47.395%;height:1.54578%>satellite to the next is always  and the angular position of the i
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:58.6555%;top:11.415%;width:26.8908%;height:1.9025%>th satellite is the angle (i-1) in the
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:13.1986%;width:6.89076%;height:1.30797%>frame A.
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:16.2901%;width:76.6387%;height:1.54578%>The frame B is such that the position vectors of the satellite 1 in the frame A and B coincides at time 0
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:17.717%;width:75.7983%;height:1.54578%>and equal R1. In the frame B the position vector of the satellite 2 is R'2 and the angle made by R1 and
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:19.3817%;width:69.916%;height:1.54578%>R'2 is . Since the angle from one satellite to the next is always , the angular position of the i
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:81.3445%;top:19.1439%;width:1.34454%;height:0.832342%>th
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:20.9275%;width:20.6723%;height:1.54578%>satellite is the angle (i-1).
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:23.9001%;width:60.8403%;height:1.30797%>Let l be the distance between the satellites 1 and 2 in the frame A. In the frame B
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:25.4459%;width:57.9832%;height:1.54578%>the satellites being moving, the length l undergoes length contraction and the
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:26.9917%;width:57.3109%;height:1.54578%>distance between these 2 satellites is l'. The ratio of length contraction is l'/l,
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:28.4186%;width:61.0084%;height:1.66468%>which is given by equation (13). Because l' &lt; l , R'2 is slightly closer to R1 than R2
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:30.0832%;width:13.6134%;height:1.54578%>and we have &lt;.
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:32.937%;width:71.7647%;height:1.78359%>The angular position of the n+1th satellite is n=2 in the frame A and n in the frame B. n is
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:34.4828%;width:75.4622%;height:1.9025%>smaller than 2 because &lt;. But, the geostationary orbit being a circle, the n+1th satellite is the first
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:36.1474%;width:76.9748%;height:1.78359%>satellite and it is impossible that the n+1th satellite is not at the angular position 2 while being the first
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:37.931%;width:6.55462%;height:1.18906%>satellite.
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:40.6659%;width:72.9412%;height:1.78359%>So, length contraction in orbit creates a gap between the n+1th satellite and the first satellite in the
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:42.4495%;width:75.7983%;height:1.42687%>frame B, which is the same type of contradiction than the contradiction with relativity of simultaneity.
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:45.3032%;width:76.1345%;height:1.54578%>The value of this gap is computed by equation (14) using the parameters of the geostationary orbit: the
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:46.849%;width:75.7983%;height:1.54578%>orbital speed is 3.0746 km/s, the radius of the orbit is 42 164 km and the circumference of the orbit is
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:48.2759%;width:58.8235%;height:1.54578%>nl = 264 924 km. The value of the gap is then 14 mm, which is too small to be
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:49.8216%;width:60%;height:1.54578%>measured. But for particles traveling in a circular accelerator at a fraction of the
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:51.2485%;width:31.7647%;height:1.66468%>speed of light, this gap is significantly big.
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:14.7899%;top:54.3401%;width:21.3445%;height:1.42687%>4. Circular accelerator
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:57.4316%;width:54.6219%;height:1.54578%>In « How to test length contraction by experiment? », I have proposed to
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:58.8585%;width:48.9076%;height:1.54578%>test length contraction using n fast moving electrons in a circular
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:60.4043%;width:52.7731%;height:1.54578%>accelerator. The electrons are equally spaced in the accelerator tube as
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:61.9501%;width:53.4454%;height:1.54578%>Figure 4 shows. For an observer situated at the center of the accelerator
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:63.3769%;width:54.958%;height:1.54578%>and rotating with the electrons, the electrons are immobile and the length
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:64.9227%;width:49.5798%;height:1.54578%>of the chain of electrons from number 1 to number n+1 equals the
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:66.1118%;width:52.9412%;height:1.9025%>circumference of the accelerator tube, the n+1th electron being the first
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:67.8954%;width:6.72269%;height:1.30797%>electron.
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:70.868%;width:51.2605%;height:1.66468%>At the velocity of 0.865 c, the ratio of length contraction equals 0.5,
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:72.4138%;width:53.7815%;height:1.54578%>which implies that, for an observer who measures the moving electrons
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:73.9596%;width:53.6135%;height:1.54578%>using the detectors in the laboratory, the length of the chain of electrons
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:75.3864%;width:55.1261%;height:1.54578%>equals half the circumference of the accelerator tube and he would see all
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:76.9322%;width:55.9664%;height:1.54578%>the n electrons squeezed into one half of the accelerator tube and the other
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.5966%;top:78.2402%;width:55.7983%;height:1.78359%>half is empty, as shown in Figure 5. It is impossible that the n+1th electron
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:79.9049%;width:54.2857%;height:1.54578%>is not at the place of the first electron because they are the same electron.
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:14.7899%;top:82.9964%;width:12.605%;height:1.42687%>5. Comments
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:85.9691%;width:54.1177%;height:1.66468%>In section 1, I have shown that GPS satellites cannot be synchronized in
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:87.5149%;width:51.7647%;height:1.54578%>accordance with relativity of simultaneity. In « Astrophysical jet and
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:11.7647%;top:89.0606%;width:52.9412%;height:1.54578%>length contraction » I have shown that relativistic jets ejected by black
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:75.4622%;top:25.8026%;width:2.68908%;height:1.42687%>=
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:79.3277%;top:25.4459%;width:2.68908%;height:1.07015%>1 −
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:76.6387%;top:27.824%;width:5.37815%;height:1.9025%>≈ 1 − 1
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:80.8403%;top:29.2509%;width:1.17647%;height:1.18906%>2
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:85.3782%;top:26.5161%;width:3.02521%;height:1.30797%>(13)
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:73.9496%;top:49.7027%;width:1.0084%;height:1.30797%>(
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:74.6219%;top:50.1784%;width:2.35294%;height:0.713436%>−
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:77.479%;top:49.7027%;width:2.85714%;height:1.30797%>′) ≈
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:80.6723%;top:50.5351%;width:1.17647%;height:1.07015%>2
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:82.8572%;top:49.5838%;width:5.54622%;height:1.66468%>(14)
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:75.2941%;top:70.6302%;width:6.72269%;height:1.54578%>Figure 4
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:75.2941%;top:89.7741%;width:6.55462%;height:1.54578%>Figure 5
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:76.6387%;top:83.1153%;width:3.19328%;height:1.18906%>n+1th
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:75.7983%;top:84.4233%;width:4.70588%;height:0.951249%>electron
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:73.4454%;top:73.7218%;width:8.90756%;height:0.951249%>Accelerator's tube
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:77.479%;top:88.1094%;width:8.23529%;height:1.18906%> Lab observer
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:76.8067%;top:76.0999%;width:3.36134%;height:1.07015%>First
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:75.9664%;top:77.2889%;width:4.70588%;height:1.07015%>electron
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:82.0168%;top:55.648%;width:5.54622%;height:0.951249%>Electrons
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:75.7983%;top:69.2033%;width:4.87395%;height:0.951249%>Detectors
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:73.4454%;top:54.459%;width:8.23529%;height:0.951249%>Accelerator tube
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:77.3109%;top:60.9988%;width:1.68067%;height:1.18906%>
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:75.4622%;top:62.3068%;width:4.87395%;height:1.07015%>Rotating
<p class=ndfHFb-c4YZDc-cYSp0e-DARUcf-Df1ZY-eEGnhe style=left:75.4622%;top:63.258%;width:4.53782%;height:0.951249%>observer
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