Gravitational Time Dilation Problem

[J_M_R]

Homework Statement

A person lived 75 years in a city located 3.1km above sea level. How much longer could they have lived at sea level? (Times are measured by an observer at infinite distance).

Homework Equations

tr/t∞ = {1 - [ (2GM) / (r(c^2)) ]}^(1/2)

Rc (Radius at city) = Rearth + 3.1km

∴ t(sea-level)/t∞ = {1 - [ (2GMe) / (Rearth(c^2)) ]}^(1/2)

and ∴ t(city)/t∞ = {1 - [ (2GMe) / (Rc(c^2)) ]}^(1/2)

The Attempt at a Solution

t(sea-level) / {1 - [ (2GMe) / (Rearth(c^2)) ]}^(1/2) ≈ t(city) / {1 - [ (2GMe) / (Rc(c^2)) ]}^(1/2)

Having made the the two t∞ equal to each other.

Knowing t(city) = 75 years this gave t(sea-level) as 74.99 years.

Where have I gone wrong as shouldn't t(sea-level) be longer?

 

[PeroK]

Where have I gone wrong as shouldn't t(sea-level) be longer?

What gives you that idea?

 

[J_M_R]

What gives you that idea?

I was just using the fact that the question says "How much longer could they have lived at sea level?" so assumed t(sea-level) should therefore be longer?

 

[Bandersnatch]

I was just using the fact that the question says "How much longer could they have lived at sea level?" so assumed t(sea-level) should therefore be longer?

Ask yourself, which of the two has longer until his 76th birthday according to your calculations.

 

[J_M_R]

Ask yourself, which of the two has longer until his 76th birthday according to your calculations.

Ah, so the t(sea-level) has longer until his 76th birthday so the person could have lived 0.01 years longer at sea level according to my calculations?

 

[Bandersnatch]

You got it.

 

[J_M_R]

You got it.

Thanks!