Gravity can really slow you down

While the future’s all abuzz about the World Time Machine Project, it’s timely to talk about what gravity has to do with it. With time, that is.

It’s all thanks to general relativity, Albert Einstein’s extension to his special theory of relativity in order to take into account both acceleration and gravity (which Einstein actually showed were equivalent).

General relativity is both famously elegant – among physicists and mathematicians, at least – and horrifically difficult to understand. When the astrophysicist Sir Arthur Eddington was told about a rumour that only three people in the world really understood it, he responded “I’m trying to think who the third person is.”

But today you can be the fourth person on that list, as we explore gravitational redshift, one of the simplest demonstrations of how gravity slows down time.

Diagram of gravitational red-shifting
Image via Wikimedia Commons

WARNING: Equations ahead

So, general relativity tells us that time runs slower in a gravitational field, like on Earth, than it does out in space. But how can we tell? How do we compare how fast time is running in two different locations?

One way is by sending a light beam between the two locations. Light is an electromagnetic wave, which means that it’s created by oscillations in the electromagnetic field. So if someone out in space sees time on Earth running slow relative to themselves, that means they will see light emitted from the Earth oscillating slower than an observer on the ground.

Slower oscillations mean a lower frequency. And red light has a lower frequency than blue light, so that means it will be shifted towards the red end of the spectrum, i.e. gravitational redshift.

We can prove that the frequency will be lower by looking at how gravity affects the energy of the light. But to do that we need to first look at how gravity affects matter.

Imagine there’s a billiard ball floating out in space, far, far away from the Earth, with mass m. Now everyone knows from Einstein’s theory of special relativity that its mass is equivalent to energy, given by the formula E=mc².

If this ball falls towards Earth, it will pick up speed. In doing so, it gains energy of motion, also known as kinetic energy. In classical physics, this kinetic energy is ½mv² (where v is the ball’s velocity). So now the ball’s total energy is:

E’mc² + ½mv²

But what is the amount of this kinetic energy? Well, once more using the classical formulae – which are accurate enough for our purposes when we’re dealing with things the size of billiard balls and the Earth – the kinetic energy gained by the ball is equal to the difference in gravitational potential energy.

Good old Sir Isaac Newton, who gave us the classical theory of gravitation, taught us that gravitational potential energy depends on the mass m of the object, the mass M of the object generating the gravitational field (i.e. the mass of the Earth, which is 5.97 x 1024 kg), the distance R from the centre of the Earth, and the gravitational constant G (6.67 x 10-11 m3 kg-1 s-2). The actual formula is:

G Mm / R

Because it’s inversely proportional to R, as you go far out in space and R gets bigger, the potential energy approaches zero. So let’s just say that we started far enough out that the kinetic energy is:

½mv² = GMm/R

Where R here is the radius of the Earth (6.371 x 106 m, or 6731 km). So the total energy of the ball is in fact:

E’mc² + GMm/R

Because this is a thought experiment, we can do whatever we want and work out the consequences. So let’s use the fact that energy can never be created or destroyed, but only converted into other forms, and turn this billiard ball into pure energy, in the form of light.

Because energy must be conserved, this light now must have the full energy E’mc² + GMm/R. So it contains the original rest energy of the ball, plus the kinetic energy, which remember was the same as the gravitational potential energy.

Now, let’s direct this light out into space, back to the original starting point. When we get there, we can of course reverse our trick and turn the light back into matter. But how much matter?

Remember, energy must be conserved – it can never be created or destroyed. So we can’t end up with any more energy, or mass, than the original E=mc². Somehow, we’ve lost the extra energy!

But how? If it was a billiard ball, then by going into space gravity would have slowed it down. But this is light, and the speed of light is constant (represented by c, equal to 3.00 x 108 m s-1). So how can it have lost energy?

Fortunately, we also have quantum physics – which Einstein also had a hand in – that tells us the energy of light is given by the formula

E = ,

where h is Planck’s constant (6.63 x 10-34 J s) and ν is the frequency of the light (represented by the Greek letter nu).

So, the energy can be reduced if the frequency is also reduced. And remember, a lower frequency means the light is redder, so therefore we have… gravitational redshift. Ta da!

How much? Well, because energy is proportional to frequency (and vice versa), the ratio of the difference between the two is the same. That ratio is, which we shall call the letter z, is:

z = Δν/ν = ΔE/E = (GMm/R) / mc² = GM/Rc²

Okay, so this was just an approximation using classical physics. The actual formula from general relativity is:

z=\frac{1}{\sqrt{1-\left(\frac{2GM}{rc^2}\right)}}-1

As mentioned before, if you plug in values for Earths and billiard balls, this formula gives a result pretty close to the rough, classical approximation we worked out.

In fact, both formulas tell us that light emitted from the surface of the Earth should drop its frequency by the proportion z = 6.96 x 10-10 . That’s a factor of about 1 in 1.5 billion.

And of course, we get a bigger effect the stronger the gravity. So from Jupiter we get a gravitational redshift of 1 in 50 million; from the Sun it would be 1 in 500 thousand. And from a black hole the redshift is infinite.

In fact, because the frequency change is also the same as the amount time has slowed – what we call time dilation – this formula tells us that, at the event horizon of a black hole, time actually stops. This is why not even light can escape from a black hole; and why we call it the event horizon.

But of course on Earth, as we’ve seen, it’s much less, and clocks on the surface of the Earth only run a little bit slower than those out in space. It would take over 45 years to get a time difference of 1 second.

That may not sound like much, but we can actually measure it. GPS, or the global positioning system, relies on triangulation from a network of satellites to find your position. It does that by calculating how far you are from each of 4 satellites, and you measure those distances by calculating the time it takes for their signal to reach you (for an explanation, see this story on ABC’s Catalyst program).

Accurate time measurement is so important to making GPS work that we have to take into account the effects of relativity. At the height of satellite orbits, time is running faster than it is for us to the extent that they get ahead about about 45 microseconds every day.

However, it’s further complicated by the fact that special relativity also says that time slows down when you’re moving. The satellites are zooming around the Earth so fast that their clocks slow down by about 7 microseconds per day.

The net effect is a time difference of 45 – 7 = 38 microseconds per day. That’s 38 millionths of a second. Which doesn’t sound like much, but that measly 38 microseconds adds up to your GPS being wrong by a distance of about 10 km.

So there you go. Gravity really does slow down time, and it’s proved routinely every day by people using their GPS. Just think, all your life you’ve been living in a time warp and you didn’t know it. Suddenly time travel doesn’t seem so far-fetched, does it?

All the more reason to join in the World Time Machine Project! Go on, check it out, spread the word, and help us solve the secrets of time travel!

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