## Building Heights and Gravitational Time Dilation

Being all conditions and characteristics equal, to buy a penthouse is more expensive than buying a street-level apartment. Since I like penthouses, I would like lower prices for them. So here's my argument to fight against higher prices for penthouses focusing on one negative externality of living in the top floor of a building: gravitational time dilation.

Albert Einstein taught us that gravity has huge effects on time, attracting and slowing it. Of course, to be aware of this effect we need a point with huge gravitational force: a black hole. The more powerful the gravitational force of the black hole is, the more slow time passes by. This is what is known as "gravitational time dilation".

Albert Einstein taught us that gravity has huge effects on time, attracting and slowing it. Of course, to be aware of this effect we need a point with huge gravitational force: a black hole. The more powerful the gravitational force of the black hole is, the more slow time passes by. This is what is known as "gravitational time dilation".

*Source: Wikipedia (https://en.wikipedia.org/wiki/Gravitational_time_dilation)*

Imagine we have a friend who happens to be the best spaceship pilot in the world (how cool is that?). Now imagine that he is piloting a spaceship toward a black hole and we look at him from distance. By doing so, we would see him as increasingly slower the closer he gets to the black hole. Not only that, we would see our friend aging (biologically) more slowly than us. That is, depending on the size of the black hole and the proximity of the spaceship, a couple of hours flying close to the black hole are equivalent to about 50 years for the observers. Put it in another way, our friend would have traveled into the future.

Various experiments have confirmed this effect and this has enabled better calibration for satellites:

Various experiments have confirmed this effect and this has enabled better calibration for satellites:

*P. Fraundorf (2012) "A fun intro to 1D kinematics", arXiv:1206.2877*

Although the resulting errors in satellite timing are measured in nanoseconds, light speed is 30 centimeters per nanosecond so that the combined effects can result in GPS errors as large as 15 meters if not taken into account.

Following this statement, the effect of the gravitational force also occurs in small magnitudes. So small that they are imperceptible for us. However, the effect of gravity on time remains: the more gravitational pull, the more time slows down.

This means that those living in a penthouse are exposed to a lower gravitational force (by its distance from earth) than those living in a street-level apartment. Time, forced by physical laws, must pass more quickly in a penthouse than in a street-level apartment. The person who lives in the penthouse will age faster (at a proportional rate determined by Einstein's equations) just as we aged 50 years more than our cool cosmonaut friend.

But how much older will be a person who lives in a penthouse compared to another living in a street-level apartment?

If we follow the example proposed in this Wikipedia article, considered over the total lifetime of the earth (4.6 giga-annums), a clock set at the peak of Mount Everest would be about 39 hours ahead of a clock set at sea level. This means that the gravitational time dilation effect in the Everest is 0.0000305217391304348 seconds per year. If we round the height of Everest at 8,000 meters we know that the effect is 0.00000000381521739130435 per year and every meter we move away from the surface of the Earth. Therefore, one who lives in a penthouse located on the tenth floor of a building (about 30 meters) would age 0.00000011445652173913 seconds more per year than his neighbor at the street level apartment.

Following this statement, the effect of the gravitational force also occurs in small magnitudes. So small that they are imperceptible for us. However, the effect of gravity on time remains: the more gravitational pull, the more time slows down.

This means that those living in a penthouse are exposed to a lower gravitational force (by its distance from earth) than those living in a street-level apartment. Time, forced by physical laws, must pass more quickly in a penthouse than in a street-level apartment. The person who lives in the penthouse will age faster (at a proportional rate determined by Einstein's equations) just as we aged 50 years more than our cool cosmonaut friend.

But how much older will be a person who lives in a penthouse compared to another living in a street-level apartment?

If we follow the example proposed in this Wikipedia article, considered over the total lifetime of the earth (4.6 giga-annums), a clock set at the peak of Mount Everest would be about 39 hours ahead of a clock set at sea level. This means that the gravitational time dilation effect in the Everest is 0.0000305217391304348 seconds per year. If we round the height of Everest at 8,000 meters we know that the effect is 0.00000000381521739130435 per year and every meter we move away from the surface of the Earth. Therefore, one who lives in a penthouse located on the tenth floor of a building (about 30 meters) would age 0.00000011445652173913 seconds more per year than his neighbor at the street level apartment.

*Source: Me, myself and my impressive artistic skills.*

If both neighbors have 30 years, they have lived 946,100,000 seconds. Now, after a year of living in the penthouse, the neighbor number 1 will not have the same

In conclusion, if you value your time and your life, buy an apartment at a lower floor of the building. Forget about the penthouse and hopefully I will get a lower price for it.

*exact*age as neighbor 2: he will have aged faster. In fact, his age will be 977,600,000.00000011445652173913 seconds, while the downstairs neighbor will have a lower age of "only" 977,600,000 seconds.

In conclusion, if you value your time and your life, buy an apartment at a lower floor of the building. Forget about the penthouse and hopefully I will get a lower price for it.