time and gravity

Can you feel the flow of time? If you enter this question in your favourite search engine, you get many hits about how we perceive a certain amount of time, say two weeks. There is a branch of research in the psychological sciences on the perception of time. This is not what I am going to discuss here. Can you feel the wind on your face or water running over your hands? Can you feel the fire? Indeed, it even can hurt you. Can you physically sense the flow of time? Can the flow of time hurt you? I am going to write about the latter two questions.

Twin Paradox

From my previous blogs, you can convince yourself that the constancy of the vacuum speed of light implies that the flow of time is personal: time itself slows down if you move with speed compared with the time of an observer at rest. Let us think this through a bit more, and we will discover that things might not add up. We seem to encounter an inconsistency that puzzled the greatest minds of our time for decades - the twin paradox.

In my previous blogs, we built a clock with light bouncing between mirrors to measure time flow. If we look at the clock in a moving train, we see that the time in the train flows slower than on the platform. This is a rock-solid conclusion, but not without some apparent issues. If the train's speed is perfectly constant, the traveller in the train does not feel any force from accelerations. We don't feel any force either by just standing at the platform. From the traveller's perspective, the platform moves in the other direction while they rest on the train. If the traveller looks at our clock on the platform, they see that time flows slower there.

Let's take this to the extreme. Consider two twins: one stays at Earth and the other flies away in a spacecraft at high velocity. In some distant future, the spacefarer turns around and flies home at the same high speed. When both twins look at the clocks of each other, both will claim that the time of the other sibling flows slower. When the spacefaring twin disembarks the spacecraft after their return and stands next to their sibling, a crucial question remains: who of the twins is younger?

Acceleration

Before we resolve this conundrum, a quick word about force and acceleration. Acceleration quantifies how much the velocity changes per second. If it does not change, the acceleration is zero. Mechanics tells us that acceleration is related to force: if a car accelerates, we feel a force pushing us into the seat. The gravitational acceleration on Earth is about 9.81 m/s/s, and the force coming with it keeps our feet on the surface of the Earth.

There is a profound postulate by Albert Einstein, which must not necessarily be true, but has been verified in many observations: you cannot distinguish between the force from an accelerating car and the gravitational force. This is the cornerstone of Albert Einstein's Theory of General Relativity.

I won't delve into this matter any deeper here but point out that this solves an issue for interstellar space travellers. Assume that we would have the technology for a space rocket to sustain an acceleration of 9.81 m/s/s over the years. Our spacefarers onboard would feel very much at home with normal gravity and without the degradation of bones and other issues that come with the absence of sustained gravity.

Let us now return to the twin paradox: the apparent issue is that the situation looks symmetric: the twin on Earth sees their sibling move away to the right, return and come to a halt. The twin on the spacecraft sees the guy at home move out to the left, return and come to a halt. Both claim the other is younger.

The resolution is that the situation is not symmetric. Only if both twins stand next to each other, the flow of time is the same, and their clocks are in sync. The spacefarer has then to accelerate to get to the high travel speed. The traveller feels a force during the acceleration process. The twin at home sees their twin fly away with increasing velocity but does not feel any force at all. It is the acceleration that tips the balance of the scale: the twin who travels is younger.

With Einstein's postulate that there is no difference between gravitational and acceleration forces, this observation has the exquisite consequence that time flow is slower nearer to Earth than further away. You can think of it in this way: the flow of time at your feet is slower than at your head (since your feet are closer to the centre of gravity). This tiny difference in the flow of time manifests itself as quite significant gravitational acceleration - gravity in layman’s terms or also called sometimes (not very accurately) the weight of our body.

Back to the beginning: Can you feel the flow of time physically? You can feel if the flow of time changes: you do it all the time on Earth as the gravitational force.

Sat Nav

Is this all relevant for our lives on Earth? Yes - you might use these effects every day.

A hand-held Sat Nav (e.g., GPS) receives time signals from as many as 24 satellites. It then calculates your position with an accuracy of 5-10 metres. Note, however, that time flows slower for you on Earth. This effect is partially compensated by the high speed of the satellites, which makes their clock going slower. The net result is that the time flows faster for the satellites by about 0.000,000,038 seconds per day.

This is slow enough not to mess with our everyday life experience, namely, a universal flow of time. But to achieve position accuracy, the time measurements must be very precise. The details are (see the Ohio State website) that your Sat Nav would be wrong by about 10 kilometres at the end of a day if it would not correct for the differences in the flow of time.