125d8253   
   From: relativity@paulba.no   
      
   Den 01.02.2026 16:56, skrev Maciej Woźniak:   
   > On 2/1/2026 2:49 PM, Paul B. Andersen wrote:   
   >   
   >> It is remarkable that a person who is posting in a sci.physics   
   >> Usenet group is ignorant of the existence of accelerometers.   
   >>   
   >> Inertial Navigation Systems (INS), also called   
   >> Inertial Reference Systems (IRS), are used in aircrafts,   
   >> ships, submarines, missiles, spacecrafts, robots, drones,   
   >> you name it.   
   >>   
   >> All commercial aircrafts must have one or multiple IRS.   
   >> That is because the IRS is independent of information   
   >> from the outside (GNSS, radio beacons, radar) which   
   >> may not always be available under all conditions.   
   >>   
   >> An IRS consist of three accelerometers and three gyros   
   >> (ring lasers) measuring the acceleration along, and   
   >> the rotation around the aircraft's three axes: Roll Axis,   
   >> Pitch Axis and Yaw Axis.   
   >>   
   >> When the IRS is switched on, the aircraft must stay   
   >> stationary in 20 minutes.   
   >> During that time, the gyros will have found the direction   
   >> of Earth's spin axis and the accelerometers will have found   
   >> the direction of a vertical to the ground (acceleration is upwards).   
   >> The angle between these two axes will give the latitude   
   >> of the aircraft's position, and the angle between Earth's spin axis   
   >> and the Roll axis will give the heading of the aircraft.   
   >> The longitude must be entered manually.   
   >>   
   >> The position is found by integration of the acceleration.   
   >>   
   >>   
   >> So you see, accelerometers measures acceleration, and   
   >> thousands (millions?) are in action right now.   
      
   > Probably - they're just different   
   > devices than the one from those gedanken   
   > delusions of yours and they don't measure   
   > your "proper acceleration" which has nothing   
   > in common with any acceleration and thus   
   > is completely insigniificant for any sane   
   > person.   
      
   You mean the following "gedanken"?   
   | You sit in your car with an accelerometer in your hand.   
   | The accelerometer shows that your acceleration is a = 2 m/s².   
   | According to the accelerometer dv/dt = 2 m/s².   
   | What is v?   
      
   Can you explain why this is not a normal accelerometer?   
      
   In Newtonian Mechanics (NM) an objects acceleration   
   is independent of in which inertial frame of reference   
   the acceleration is measured.   
      
   Given two inertial frames of reference, K and K'.   
   K is moving with the speed w relative to K'   
      
   K: --o------------------> x ->w   
      
   K': --------------------> x'   
      
   In NM the Galilean transform applies:   
     x' = x + wt   
     dx'/dt = dx/dt + w   
   If the object's speed is v in K, then it is (v+W) in K'   
      
     d²x'/dt² = d²x/dt² + dw/dt   
   Since dw/dt = 0, the acceleration is the same in K and K'   
     a = dv/dt = d(v+W)/dt   
      
   ------------------------------------------------------   
   | In NM the acceleration of an object is the same   
   | regardless of in which inertial frame it is measured.   
   -------------------------------------------------------   
      
   We can choose the frames of reference as we wish,   
   so at any instant, we can choose K to be momentarily   
   comoving with the object.   
   At that instant the speed of object in K is v = 0,   
   but we still have a = dv/dt.   
      
   In an accelerometer the apparatus that measures the   
   acceleration is inside a casing.   
   So the only possible reference is the casing.   
   The acceleration is measured in the momentarily comoving   
   inertial frame, or simpler:   
   ---------------------------------------------   
   | In an accelerometer, the acceleration is   
   | measured in the accelerometer's rest frame.   
   ----------------------------------------------   
      
      
   In relativity (SR and GR) it is more complex.   
   In the momentarily comoving frame of reference K   
   the speed is v = dx/dt and the acceleration   
   is a = d²x/dt² = dv/dt   
      
   But the speed in K' is v'= (v+w)/(1+vw/c²)   
      
   and the acceleration is a' = a/(γ³(1+vw/c²)³)   
      
   That is, the acceleration of an object is dependent of   
   in which inertial frame it is measured.   
   But the acceleration in the equation a = F/m is   
   the acceleration measured in the momentarily comoving   
   frame of reference, or:   
   the acceleration measured in the object's rest frame.   
      
   This acceleration is called the object's "proper acceleration".   
      
   So an accelerometer measures its proper acceleration.   
      
   --   
   Paul   
      
   https://paulba.no/   
      
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