As many of you have probably known there was a famous Hafele–Keating experiment, which was about a plane traveling the globe in order to prove predictions of special relativity and general relativity.

After the plane has landed a time difference was observed. It is widely accepted that clock in the frame of reference of moving object goes slower than stationary clock (additional effect comes from gravity).

Hence clock isn't some magical device, but rather a device made of moving parts (even if the parts are atoms or elementary particles) you must assume that all processes on the plane have slowed down. But this means that to the observers on the ground plane consumes less fuel than it should to travel certain distance.

How would you explain this?

In real Hafele–Keating experiment amount of gained fuel would be extremely small. But you can imagine super-advanced planes flying at relativistic velocities or planes flying for years around the globe. In edge-case scenario - plane traveling near the light speed - it could travel the world on a drop of a fuel. This seems to break energy conservation laws, because fuel can release only certain amount of energy. How would it be possible to travel such distance, against tremendous forces, on a drop of a fuel?

Yet another look at the experiment. In original experiment it was stated that plane traveling eastwards had lost some nanoseconds, while plane traveling westwards gained some. This would mean that energy is not conserved in respect to direction of travel, because $$W = \int \vec{F}d\vec{S}$$ and the distance $$s$$ is obviously the same. Fuel burns slower at one direction than the other.

One more example. Imagine two helicopters traveling the globe at velocities $$\vec{V}$$ and $$2\vec{V}$$. They are built in such manner that they have two independent engines - one to keep them up and the second one to move them horizontally. The one that travels faster uses less fuel and will remain in the air longer than one that goes slower. They took the same amount of fuel for their vertical engines. The question is: where does the energy to keep faster helicopter in the air for longer time come from, if it burns fuel slower?

• Shouldn't the time also catch up in the end due to acceleration? You can make a clock that measures time by counting how much fuel particles are being burnt.
– doc
Sep 15 '20 at 18:05
• Have you made sure to take into account how the Force each observer measures will not be the same? Sep 15 '20 at 21:21
• @BioPhysicist Use ground as a frame of reference.
– doc
Sep 15 '20 at 21:42
• I thought this was an idealized situation (without friction) ! Do we need to keep track of heat dissipation as well? This question has been edited so much as to make some answers moot. Sep 16 '20 at 1:06
• In line with @ZeroTheHero, I think all of the additional scenarios have really made your post lose focus here. Sep 16 '20 at 1:33

I think it will help to first consider a simple scenario involving a straight flight from A to B, at a speed high enough to make the Lorentz factor equal to 2. We suppose A and B are on the ground (of some huge planet), at rest relative to one another.

Suppose observers in the aircraft find that their clocks advance by $$1$$ minute on the journey from A to B, and that $$1$$ mole of fuel molecules were burned.

Then observers on the ground find that the plane arrives at B at a time $$2$$ minutes after it left A (according to clocks at A and B), and $$1$$ mole of fuel molecules were burned. Thus the time dilation effect means that observers on the ground consider the rate at which the plane burns fuel (and does everything else) is slow. Things like "number of molecules which combined with oxygen between two given events" are the same no matter how distance and time are measured; they are called Lorentz invariant.

You might ask, why did the plane have to burn any fuel at all? It is because it has to push against air resistance. In the reference frame of the plane there is less distance to travel, but the air is denser. So it burns fuel at a higher rate (compared to what ground observers measure for the plane as they observe it flying), but ends up using the same amount of fuel as the ground observers measure.

I hope this allows you to settle your questions.

So how is it possible, that second plane uses less fuel, if amount of work $$W$$ has to be the same for both planes.

It doesn't use less fuel.

If the plane's engine and its fuel pumps and everything else are time dilated (relative to a family of clocks with zero airspeed), that in no way relieves them of the requirement to generate enough thrust to counter air resistance. To keep the plane's speed up, the engine and fuel pumps and everything else have to run more quickly (relative to their internal clocks / proper time) than they would if the relativistic slowdown effect didn't exist. If they can't handle that higher rate, then they'll simply fail to maintain the speed.

• what about no friction and acceleration/deceleration only at the beginning and end of the trip?
– user65081
Sep 15 '20 at 23:59
• @Wolphramjonny I think that's a completely different question. This one is about working against friction. Sep 16 '20 at 0:02
• well the trip is contracted so the fuel burn rate is no different in the frame of the plane but the net fuel burn is less since the trip is shorter. Sep 16 '20 at 1:05

From the point of view of a stationary observer, the plane's clock ran slower because it was moving at relativistic speeds. Any speed is relativistic if the difference in clock speed matters to you. It is a question of how sensitive your experiment is.

At relativistic speeds, the engines run slower, the fuel burns slower, the fuel gauge drops slower.

From the point of view of the pilot, distance contracted, so the trip was shorter.

For general relativity, time runs slower when you are deeper in a gravitational well.

It might be easier to think of two helicopters, one on a mountain and the other in a valley. They take off together, hover at different altitudes, and land together. The lower helicopter burned less fuel while hovering because it spent less time hovering. (Ignore such things as different air density.)

• Imagine a plane flying near-light speed. Its turbine is almost stopped, it burns almost no fuel, yet it goes faster than F22 raptor... Doesn't something seem to contradict here in your mind :)
– doc
Sep 15 '20 at 15:02
• @doc I think that you, as others have noted, also should take into account the amount of fuel necessary to reach such velocities. Sep 15 '20 at 18:25
• @doc - Counter-intuitive, yes. Contradictory, no. It burns almost no fuel, no. You can see the beginnings of it when you compare a regular plane and a Raptor, both of which fly every day at relativistic speeds. The raptor flies faster and uses more fuel. There is a relativistic effect that slows their clocks and makes them consume less fuel that classical physics would suggest. But even with the effect, the faster plane burns more fuel. Continuing up near the speed of light, the same pattern would continue if you ignore the planes melting, etc. Sep 15 '20 at 21:12
• @mmesser314 I have updated my question with another view at this, which addresses your observation. Take a look.
– doc
Sep 15 '20 at 21:14
• The pilot would see the plane burning fuel at an immense rate over a short distance. This would still be a lot of fuel. More if you go faster. A stationary observer would see the plane burning fuel at a slower rate than the pilot sees because of time dilation. But it would still be an immense rate that consumed a lot of fuel. Sep 15 '20 at 21:20

Inside both planes clocks run normally according to pilots, small paddle wheels on fuel pipes measuring fuel consumption rotate normally according to pilots, and small paddle wheels outside the plane measuring airspeed rotate normally according to pilots.

So, if plane's clocks slow down, then plane's fuel pumps slow down, and plane's airspeed slows down.

A slower fuel pump runs for a longer time than a faster fuel pump, both pumps pump the same amount of fuel during one round around the earth.

• According to pilots everything runs normally, except space being contracted. But on the ground they observe plane hanging in the air for longer time than that allowed by the fuel, which has finite energy. Take a look at example with helicopters as it is more picturesque.
– doc
Sep 16 '20 at 12:47
• The above answer is a good answer to a question asked. I am not going to answer the helicopter question, it's not a good question. I could answer a question about energy consumption of air conditioning during flight. Sep 16 '20 at 14:00
• why helicopter question isn't a good one?
– doc
Sep 16 '20 at 14:43
• The question is: 1: Too difficult: motion changes gravity and lift. 2: Too easy: Ideal helicopter uses zero energy to hover. Sep 16 '20 at 16:01
• I wish you would focus more on a phenomenon, rather than details how helicopters work. What matters is that fuel is burnt slower. If two "helocopters" are exactly the same and they need to use some fuel to overcome the gravity, then how is it possible that same amount of fuel allows one helicopter to hang in the air longer than the other.
– doc
Sep 16 '20 at 16:09

One more example. Imagine two helicopters traveling the globe at velocities V⃗ and 2V⃗ . They are built in such manner that they have two independent engines - one to keep them up and the second one to move them horizontally. The one that travels faster uses less fuel and will remain in the air longer than one that goes slower. They took the same amount of fuel for their vertical engines. The question is: where does the energy to keep faster helicopter in the air for longer time come from, if it burns fuel slower?

The faster helicopter crashes on the ground because of its extra slow vertical motor.

As the fuel tank of the vertical motor of the faster helicopter has extra kinetic energy, the crater is extra large.

If the kinetic energy is included, then there was enough energy in the fuel tank of the vertical motor of the faster helicopter to keep the faster helicopter up as long as the other helicopter, but the vertical motor was just too slow.

• But in the frame of reference of a pilot motor isn't slow at all. So why would helicopter fall down?
– doc
Sep 17 '20 at 10:45
• @doc Earth has trillions of gigatons of gravitating kinetic energy, according to the pilot. Or in one second large number of momentum units are absorbed by the copter according to the pilot, because pilot is time dilated. (why does kg*m/s not have a name) (maybe this question is not so bad as I thought earlier) Sep 17 '20 at 13:36

It took atomic clocks to measure time differences of $$100$$s of nanoseconds. No fuel gauge on an airplane could measure the difference in fuel burned in so little time.

Edit: Since you've clarified to indicate this is about airplanes flyings for days or at relativistic speeds (what then is the connection with the Hafele-Keating experiment?): yes the processes slow down in the moving frame so such a plane would burn less fuel than it would if you did not account for special relativity.

• Yep, but it is a thought experiment. You can imagine plane flying for years around the globe or super-advanced plane traveling at relativistic velocities.
– doc
Sep 15 '20 at 13:13
• @doc with due respect your question says nothing of planes flying for years or planes traveling at relativistic velocities. Sep 15 '20 at 13:54
• i appeal to your intelligence to deduce it, especially that the question is tagged as "thought-experiment". But even with literal interpretation it has physical meaning, so you should respond to it in scientific manner, rather than telling me that "no fuel gauge can measure" this...
– doc
Sep 15 '20 at 14:00
• @doc Just edit the question to be more clear. Other readers cannot look into your mind and see exactly what you meant. If another user interprets your question a certain way that was not intended, it just means it is not 100% clear. It is easy to edit your question to fix this. Sep 15 '20 at 14:25