Maths IA · Calculus · topic 24 of 97
Gravitational time dilation: how good is a Maclaurin approximation? · IB Maths IA idea: guide, sample graph and step-by-step plan
a square root and a tiny xGPS satellites carry clocks that gain about 38 microseconds a day compared with clocks on the ground. That tiny gap comes from a square root, and the square root is almost always replaced by a short series. How short can it be?
- ✓ you like seeing a real number come out of a series you built yourself
- ✓ you want to ask when an approximation is safe, and prove your answer with an error bound
- ✓ you are curious why GPS needs Einstein to work
The big idea
In general relativity, a clock at distance r from a mass runs at the rate √(1 − rs/r) compared with a distant one, where rs depends on the mass. For Earth, rs/r is about a billionth. That makes the square root a perfect case for a series: expand it in x = rs/r, keep one or two terms, and the answer is almost exact. But 'almost' needs a number. How big is the error, where does the series stop working, and is the error bound you derive honest? The expansion is routine. The marks come from the error work: bound the remainder, test the bound against exact arithmetic, find the domain where the series converges, and decide how many terms a real satellite clock needs.
Make it yours: pick one twist
Copying a classic won't show your choices. Pick one of these (or invent your own) and your IA becomes yours.
Swipe for more twists →
Differentiate the factor repeatedly at x = 0, build the first few terms, and check them against the binomial theorem for fractional powers.
→ Two derivations of the same coefficients. Show they agree, and say which is easier to extend.Use a remainder term to prove an upper bound on the error of the first-order and second-order approximations.
→ A proof with inequalities. Test your bound against exact values and say how tight it is.Compare the exact factor and each approximation as r gets close to rs, and find the domain where the series converges.
→ A domain and a limit. Explain what the factor and the series do at r = rs, and why Earth is nowhere near that.Use the first-order series for a clock on the ground and one at GPS height to estimate the gain per day, then add the speed effect.
→ A number you can compare with a published value. Say which approximations you used and how they affect the answer.Repeat the work for a clock on the Moon, on Mars or near the Sun, and ask when the first-order series stops being enough.
→ The size of rs/r varies by orders of magnitude. A table of how many terms each body needs for a stated accuracy.Try to compute the exact error with ordinary calculator precision, then with a more careful method.
→ The error is smaller than the calculator's rounding. A rewrite of the difference that avoids subtracting nearly equal numbers.Your Maths IA research question: run the check
how small is the error?Write your own question first. Then make sure it ticks every box:
- Names the quantity you approximate and what x stands for, with the values of the constants you use.
- Says what 'good enough' means: a relative error, a time error in microseconds per day, or both.
- Includes a proof or derivation of an error bound, not only a table of computed errors.
- Says how you test the bound: exact values with enough digits, or a published result.
- Keeps to one or two physical settings, and is clear about which relativity you quote and which you derive.
The model
A ground clock and a satellite clock, one formula for how fast each runs, and a single variable x that is tiny for every real clock. Draw yours with the planet you choose.
How your Maths IA could be laid out
The usual order for a topic like this, what each part needs to show, and roughly how much of your 12 to 20 pages it takes. The words are all yours.
Why GPS clocks matter, what the clock-rate factor is, and what you expect: how many terms will be enough?
Every symbol defined. The series derived two ways, with the first few coefficients checked.
Where the series converges, what happens at the edge, and a graph of the exact factor and approximations.
A remainder or alternating-series argument, a bound for first and second order, and a test against exact values.
The gain per day from gravity, the speed effect and the net, compared with a published figure.
Back to your aim, with numbers: the terms needed for a stated accuracy and what that means for a satellite clock.
The simplified physics, rounding of constants, the orbit treated as a circle. Say which assumption matters most.
Where every constant, formula and published figure came from.
What your Maths IA graph could look like
show where it failsA made-up example to show the form of a strong graph, not the answer: the exact clock-rate factor and a first-order approximation close to the critical radius. Yours will look different. example only
- Exact factor and approximation on one set of axes, so the gap is visible at a glance.
- Both axes are ratios with no units, so say so in the labels.
- Real clocks sit at huge r over rs, so add a second graph of the error on logarithmic axes.
- Plot a few exact values you computed as points to show your own arithmetic agrees.
- Another graph to consider: the error of each approximation against the number of terms kept.
The maths in action
worked through, not sketched!The graph above was a sketch with made-up numbers. This figure works the idea through from the model itself, so you can see what a finished figure looks like. The source line under it says where every number comes from.
Your game plan
Tick steps off as you go. We'll remember on this device.
0 of 9 done
- Read and predictFind out what the factor means and what rs is. Before any calculation, guess how many terms a GPS clock needs.
- Fix the physicsChoose the body and orbit, list the constants with sources, and compute rs and the value of x for each clock.
- Build the seriesDerive the first terms of the series for the factor in two ways, and define every symbol.
- Trial run (1 lesson)Compare each approximation with exact values at a few radii, including some near rs. Fix any mismatch now.
- Find the domainDecide where the series converges and describe what it does close to the boundary.
- Bound the errorDerive an upper bound on the error for each order and test it against exact arithmetic with many digits.
- Go further, with a reasonApply the first-order series to GPS and compare the net daily gain with a published figure, or repeat for another body.
- Reflect as you goAfter every result, write one line on what it means for a real clock and what it assumes. These lines become your evaluation.
- Cut, check, then sendCheck every graph has labelled axes, every symbol is defined once and the conclusion answers your aim. Then send your teacher one draft.
The maths you'll use
- Maclaurin and binomial series – Build from derivatives, and check it with the binomial theorem for index .
- Convergence – Find the values of where the series converges, and describe the behaviour at ().
- Error bounds – Bound the remainder after one or two terms and show it is at most about times a constant for small .
- Orders of magnitude – Work with in standard form, and compare absolute and relative error.
- Applying the first-order result – Show the clock-rate difference between two radii is and convert it to microseconds per day.
Key equations
Syllabus links
- 1.9Binomial theorem
- 1.10Counting principles; binomial theorem for any indexHL
- 1.8Infinite geometric series
- 5.1Limits and the derivative as a rate of change
- 5.6Standard derivatives; chain, product and quotient rules
- 5.13Limits with l'Hopital's rule or Maclaurin seriesHL
- 5.19Maclaurin seriesHL
- 1.1Standard form
- 1.6Approximation, bounds and percentage error
Read up first
Data and sources to start from
Published constants for Earth and the GPS orbit, and the published daily clock offsets of GPS satellites to test the series against.
- NASA Earth Fact Sheet
- NIST: CODATA fundamental physical constants
- GPS.gov: official U.S. government GPS information
References to cite
- Ashby, N. (2003). Relativity in the Global Positioning System. Living Reviews in Relativity, 6, Article 1.
- Hartle, J. B. (2003). Gravity: An introduction to Einstein's general relativity. Addison-Wesley.
- National Institute of Standards and Technology. (n.d.). Fundamental physical constants. https://physics.nist.gov/cuu/Constants/
Cite what you actually use, in your own format, and say where each number came from.
Traps to avoid in this Maths IA
bound it, don't just compute it- Quoting a result from relativity without saying which parts you derive and which you take from a source.
- Calling the approximation 'accurate' without a number or a bound.
- Computing a tiny error with a calculator that rounds, and getting zero or noise.
- Using the series for r close to rs without checking that it still converges.
- Forgetting that a satellite's speed also changes its clock, which pulls the other way.
- Mixing units, such as kilometres for r and metres for rs.
More Maths IA ideas like this
Maclaurin series: how many terms are enough?●●○
How does a phone find itself? Trilateration●●●
Newton's method: how fast, and when does it fail?●●○
Orbits and ellipses: how well does an ellipse fit?●●●
This guide shows how a topic like this is usually approached. It is not an IA. Your question, choices, data, analysis and words must be your own: change something, tell your teacher where the idea came from, and follow your school's rules on outside help.