If you were to test gravitational time dilation from the surface of the Earth through the barycenter (center of mass between the Earth and another celestial body, such as the Moon) and back to the surface, the resulting time curve would show variations in the passage of time due to the difference in gravitational potential.
Gravitational time dilation arises due to the influence of gravity on the flow of time. According to general relativity, clocks in regions with stronger gravitational fields run slower compared to clocks in regions with weaker gravitational fields. In the case of the Earth, the gravitational field is stronger near the surface and weaker at higher altitudes.
When you start from the surface of the Earth and move towards the barycenter, which is closer to the Moon than the surface of the Earth, you would experience a decrease in gravitational potential. As a result, time would pass slightly faster for you compared to an observer who remains stationary at the Earth's surface. This is because you are moving to a region with a weaker gravitational field.
Upon reaching the barycenter and moving back towards the Earth's surface, you would experience an increase in gravitational potential. This would cause time to pass slightly slower for you compared to the stationary observer on the Earth's surface. As you approach the Earth, which has a stronger gravitational field, the time dilation effects would become more pronounced.
The resulting time curve would show a gradual decrease in time dilation as you move away from the Earth's surface towards the barycenter, followed by a gradual increase in time dilation as you move back towards the surface. The exact shape of the curve would depend on factors such as the distance between the Earth and the Moon and the specific trajectory taken. However, it would generally exhibit a symmetric pattern with respect to the barycenter, reflecting the changing gravitational potential and the corresponding effects on the flow of time.