Does Time Dilation Affect Synchronized Clocks in Different Frames?

[chef99]

Homework Statement

Two atomic clocks are synchronized. One is placed on a satellite, which orbits around the Earth at a high speed for a whole year. The other is placed in a lab and remains at rest, with respect to the earth. You may assume that both clocks can measure time accurately to many significant digits.



a) Will the two clocks still be synchronized after one year? Explain your reasoning.



b) Imagine that the speed of light was much slower than its actual value. How would the results of this experiment change if the speed of light was only twice the average speed of the satellite? Explain your reasoning, using a calculation.

Homework Equations

Δt_m = Δt_s / √ 1- v² / c2[/SUP] )

The Attempt at a Solution

a) No, after a year, the two atomic clocks will not be synchronized. The clock on the satellite will be slightly behind the one in the lab, as the amount of time measured in the satellite's frame of reference will be slightly less than that measured from the earth’s frame of reference (the clock in the lab), because the orbiting clock is stationary with respect to the satellite’s frame of reference, whereas it is moving in relation to the earth’s frame of reference. The time dilation would be slight, however, as the satellite will likely not be moving at nearly the speed of light (the question does not specify the satellite’s speed). The faster the satellite moves, the greater the time difference.

b) If the speed of light was much slower, the results of this experiment would be dramatically different, as time dilation increases the closer an object is traveling to the speed of light. If the satellite traveled at half the speed of light, as asked in the question above, the result would be as follows:

Δt_m = Δt_s / √ 1- v² / c² )

Δt_m = 3.154 x10⁷s / √ 1- 0.5c² / c²

Δt_m = 3.154 x10⁷s / √ 1- 0.5

Δt_m = 44604295.76

Δt_m = 4.5 x10⁷sThe same advent will take 4.5 x10⁷s. This is a noticeable difference.I think I have the ideas right, I'm just not sure if I'm explaining it properly. If someone could help clarify this that would be most appreciated.[/SUB][/SUB][/SUB]

 

[PeroK]

Your answer to part a) is probably what is expected. The question, however, is very poor as it ignores the rotation of the Earth and gravity.

In reality, astronauts on the space station age slightly more than those on Earth.

Search for the Hafele-Keating experiment if you are interested.

For b), how long is $4.5 \times 10^7 s$?

 

[chef99]

Your answer to part a) is probably what is expected. The question, however, is very poor as it ignores the rotation of the Earth and gravity.

In reality, astronauts on the space station age slightly more than those on Earth.

Search for the Hafele-Keating experiment if you are interested.

For b), how long is $4.5 \times 10^7 s$?

4.5×10⁷s is 12390.1 hours, which is almost one and a half years (1.41). That would be considered a huge time difference Id imagine?

Thank you for the research, very interesting. I didn't even think of how the direction the object is traveling would affect the time dilation, but that makes perfect sense as the launch direction of rockets is all important to launch with the Earth's rotation [east]. So my answers while perhaps inaccurate in realaity, they are correct for the purpose of the question?

 

[PeroK]

4.5×10⁷s is 12390.1 hours, which is almost one and a half years (1.41). That would be considered a huge time difference Id imagine?

Thank you for the research, very interesting. I didn't even think of how the direction the object is traveling would affect the time dilation, but that makes perfect sense as the launch direction of rockets is all important to launch with the Earth's rotation [east]. So my answers while perhaps inaccurate in realaity, they are correct for the purpose of the question?

The problem is also poor in that it didn't say explicitly in whose reference frame the ship orbits for a year.

Which have you assumed in your answer? What does the year and a half represent?

 

[chef99]

The problem is also poor in that it didn't say explicitly in whose reference frame the ship orbits for a year.

Which have you assumed in your answer? What does the year and a half represent?

I calculated the satellite orbiting the Earth from the Earth's frame of reference. But as you said, the Earth is not truly an inertial frame of reference when its rotation is factored so the equation I guess doesn't really work in reality.

 

[PeroK]

I calculated the satellite orbiting the Earth from the Earth's frame of reference. But as you said, the Earth is not truly an inertial frame of reference when its rotation is factored so the equation I guess doesn't really work in reality.

Let me interpret your answer. The ship orbits for a year in the Earth frame. At the end of an Earth year the ship's clock differs by a year and a half?

 

[chef99]

Let me interpret your answer. The ship orbits for a year in the Earth frame. At the end of an Earth year the ship's clock differs by a year and a half?

Yes, that is correct.

 

[PeroK]

Yes, that is correct.

Does the ship clock read 2.5 years or -0.5 years?

 

[chef99]

Does the ship clock read 2.5 years or -0.5 years?

I believe it reads -0.5, as it is the clock moving from the Earth's frame of reference, but I could be mistaken.

 

[PeroK]

I believe it reads -0.5, as it is the clock moving from the Earth's frame of reference, but I could be mistaken.

The ship clock can't be moving backwards.

What you have calculated is the time on an Earth clock (about 1.5 years) after a year on the ship clock.

 

[chef99]

The ship clock can't be moving backwards.

What you have calculated is the time on an Earth clock (about 1.5 years) after a year on the ship clock.

Ok, so the ship's clock reads 1 year, and the Earth clock reads 1.5? Is this is because time can't go backward?