Figure: Time dilation for circular orbits aroung the Earth due to special and general relativity

Caption: A plot illustrating time dilation for circular orbits around the Earth due to both special relativistic time dilation (called kinematic time dilation on the plot) and gravitational time dilation (which a general relativistic time dilation).

The red curve (which increases from about -30 microseconds to about zero over the horizontal range) is the special relativistic time dilation curve. The blue curve (which increases from about zero to about 55 microseconds over the horizontal range) is the gravitational time dilation curve. The purple curve (which icreases from about -30 microseconds to about 55 microseconds over the horizontal range) is the net time dilation curve.

The plot is actually a broken-axis plot. We discuss this aspect below.

Further explication:

1. The plot is tricky to read.

2. The vertical axis of plot is decrease clock time in microseconds per day due to time dilation relative to a fiducial clock at sea level at rest in the inertial frame of the Earth's center: i.e., a reference frame with the Earth center as the origin and unrotating relative to the observable universe.

   Note well, the positive part of the scale gives decrease and the negative part of the scale gives increase.

   So the positive region is for a clock running fast compared to the fiducial clock and the negative region is for a clock running slow compared to the fiducial clock.

3. The horizontal axis is a logarithmic scale for orbit altitude in kilometers relative to sea level. You add about 6731 kilometers to get the orbital radius relative to the Earth's center. Recall, the Earth equatorial radii R_eq_⊕ = 6378.1370 km.

   Note, the plot is actually a broken-axis plot with a broken region at altitude at 1,500,000 km which is approximately the same as distance from the Earth's center. This distance is about 250 Earth radii or about 0.01 astronomical units. At this distance corotating with the Earth around the Sun and aligned with the Earth-Sun line is the L2 point which is an unstable equilibrium point for bodies orbiting the Sun, but that corotate with the Earth around the Sun because of the combination of the Earth and Sun's gravitational effect.

4. As an example of reading the plot, the gravitational time dilation at altitude 100,000 km (obital radius about 106,731 km) is positive 55 microseconds, and so a clock day here and at rest compared Earth center of mass frame completes 55 microseconds before the fiducial clock day.

   Both clocks are at rest in inertial frame of the Earth's center: i.e., a reference frame with the Earth center as the origin and unrotating relative to the observable universe. The deeper you are in a gravitational well, the slower your clock runs relative to the outside of the gravitational well.

5. As another example of reading the plot, the special relativistic time dilation at 100 km above sea level for an orbiting clock (which requires a speed of about 8 km/s) is about -30 microseconds, and so the clock day in orbit here completes 30 microseconds after the fiducial clock at rest in the inertial frame of the Earth's center. This agrees with the general rule accelerated clocks relative to an inertial frame run slow compared to unaccelerated clocks relative to the same inertial frame.

6. The special relativistic and gravitational time dilation effects are verified every moment of the day since the Global Positioning System (GPS, 1978--) would not work accurately if they were not accounted for.

7. The plot shows the altitudes of various spacecraft, and therefore what their time dilation is relative to the fiducial clock: e.g.,
   1. International Space Station (ISS, 1998--): about -25 microseconds per day time dilation.
   2. Deep Space Atomic Clock (DSAC, 2019--2021): about -20 microseconds per day time dilation.
   3. Gravity Probe A (GP-A, 1976): about 37 microseconds per day gravitational time dilation. This was a 1 hour 55 minute mission and the spacecraft was not in orbit and was used to experimentally test the equivalence principle of general relativity.
   4. The Global Positioning System (GPS, 1978--) satellites: about 46 microseconds per day time dilation.
   5. Gaia spacecraft (mission 2013--2025): about 60 microseconds per day gravitational time dilation. This spacecraft was at the L2 point, and so corotated with the Earth around the Sun.