A student asked me last week if I could explain the difference between time dilation in Special Relativity (SR) and that in General Relativity (GR), so here is my attempt at doing so. Time dilation in SR comes about when something travels near the speed of light, and is due to the Lorentz transformations which ensure that experiments in any inertial frame are indistinguishable from each other.
I have already derived the Lorentz Transformations from first principles in this blog, and these equations are at the heart of SR, and show why time dilation occurs when one travels near the speed of light. In this blog here, I worked through some examples of time dilation in SR. But, what about time dilation in GR?
How does time dilation come about in GR?
As I have already explained in this blog here, Einstein’s principle of equivalence tells us that whatever is true for acceleration is true for a gravitational field. So, to see how gravity affects time the easiest way is to consider how time would be affected in an accelerating rocket.
We will consider a rocket in empty space, away from any gravitational fields, which is accelerating with an acceleration . We will have two people in the rocket, Alice and Ben. Alice is at the top end of the rocket, the nose end. Bob is at the bottom end of the rocket, where the tail is. Alice sends two pulses of light, one at time , and the second one at a time later. They are received at the back of the rocket by Bob; the first pulse is received when the time is , and the second one when the time is , where is the time interval between flashes as measured by Bob.
This is illustrated in the figure below.
We will set it up so that Bob’s position at time when the first flash is emitted by Alice is , and so his position at any other time is given by
(this just comes from Newton’s 2nd equation of motion , see my blog here which derives those equations).
The position of Alice will just be Bob’s position plus the distance between them, which we will call (the height of the rocket), so
We will assume that the first pulse takes a time of to travel from Alice to Bob. The second pulse is emitted by Alice at a time after the first pulse, this is the time interval between each light pulse that Alice sends. This second pulse is received by Bob at a time of , where is the time interval between pulses as measured by Bob using a clock next to him.
When the first pulse leaves Alice her position is , which from equation (2) is h, as she is at the top of the rocket. When Bob receives the pulse at time his position will be which, from equation (1) is . So, the distance travelled by the pulse is going to be
as the speed of light is and it travels for seconds. Because the rocket is accelerating, the distance travelled by the second pulse will not be same (as it would be if the rocket were moving with a constant velocity). The distance travelled by the second pulse will be less, and is given by
We can use Equations (1) and (2), which give expressions for as a function of , to put in the values that would have when for and for respectively.
Substituting from Equations (1) and (2) into Equation (3) we have
which makes equation (3) become
Doing the same kind of substitution into equation (4) we have
assuming that we can ignore terms in
Substituting these expressions into equation (4) gives
We now subtract equation (6) from (5) to give
Re-arranging equation (5) as and using the quadratic formula to find we can write that
(we can ignore the negative solution because the time is always positive). We will next use the binomial expansion to write
(where we have ignored terms in and higher in the Binomial expansion), and so we can write for
Substituting this expression for into equation (7) we now have
We can cancel the in each term and bringing the terms in onto one side and the term in on the other side we have
and using the binomial expansion for (and ignoring terms in and higher), we can finally write
Because is always positive, this means that is always less than , or to put it another way the time interval as measured by Bob at the back-end of the rocket will always be less than the time interval measured by Alice where the light pulses were sent. This means that Bob will measure time to be going at a slower rate than Alice, Bob’s time will be dilated compared to Alice.
From the principle of equivalence, whatever is true for acceleration is true for gravity, so if we now imagine the rocket stationary on the Earth’s surface, with the top end in a weaker gravitational field than the bottom end, we can see that a gravitational field will also lead to pulses arriving at Bob being measured closer together than where they were emitted by Alice. So, gravity slows clocks down!
A very important difference between time dilation in SR and time dilation in GR is that the time dilation in GR is not symmetrical. In SR, both observers in their respective inertial frames think it is the other person’s clock which is running slow. In GR, both Alice and Bob will agree that it is Bob’s clock which is running slower than Alice’s clock.
In a future blog I will do some calculations on this effect in different situations, but as you can see from Equation (8), the size of the dilation depends on the acceleration and the difference in height between . I will also discuss whether it is time dilation due to GR or time dilation due to SR which affect the satellites which give us GPS the more, as both effects have to be taken into account to get the accuracy we seek in the GPS position.