General relativity
Back to Advanced Higher hubFrames of reference and the equivalence principle
Special relativity only deals with inertial frames. General relativity is what you need once a frame is accelerating.
An inertial frame of reference moves at constant velocity — it is not accelerating. A non-inertial frame is one that is accelerating with respect to an inertial frame.
Throw a ball across a cabin moving at constant velocity and it travels in a straight line. Throw the same ball across a cabin that is accelerating and, to someone inside, it curves — even though no force acts on it sideways. The curve is the frame accelerating out from under the ball.
Now seal the cabin. Inside, that curved path is exactly what you would see if the cabin were sitting still in a gravitational field. There is no measurement you can make inside to tell the two apart.
The equivalence principle: “It is not possible to distinguish between the effects on a body of uniform acceleration and a uniform gravitational field.”
Watch out: the marking instructions say “effects must be implied”. An answer that says acceleration and gravity are the same thing does not get the mark — it is their effects that cannot be told apart.
And the other definition: a non-inertial frame of reference is “a frame of reference that is accelerating with respect to an inertial frame”. That one is asked directly too.
Explain why a pupil sealed inside a windowless cabin, who drops a ball and watches it fall to the floor at 9.8 m s−2, cannot tell whether the cabin is resting on the surface of the Earth or accelerating through deep space. 2 marks
Answer
Spacetime, world lines and geodesics
Mass tells spacetime how to curve; the curve of spacetime tells matter how to move.
Spacetime is the three dimensions of space and one of time taken together as a single four-dimensional structure. An object’s path through it is its world line.
- An object at rest has a vertical straight world line — time passes, but its position does not change.
- An object at constant velocity has a straight sloping world line.
- An accelerating object has a curved world line, because its velocity is changing.
Plot spacetime diagrams the way the exam does: time up the vertical axis, position across the horizontal axis.
A mass curves the spacetime around it. The more mass, and the more concentrated it is, the greater the curvature.
A geodesic is the shortest path between two points in that curved spacetime — and it is the path a freely moving object follows. So a planet is not being pulled off a straight line by a force; it is following the straightest path available through spacetime that the Sun’s mass has curved.
Two pieces of evidence you should be able to name: the precession of the perihelion of Mercury (its orbit slowly rotates, by more than Newtonian gravity predicts), and gravitational lensing (light from a distant galaxy is bent as it passes a massive object, so we see it stretched into arcs or rings).
Describe the world line of an object that stays at rest, and explain how the world line of an object moving at constant velocity differs from that of an accelerating object. Time is on the vertical axis and position on the horizontal axis. 3 marks
Answer
Black holes and the Schwarzschild radius
Every mass has a Schwarzschild radius — the Earth included. What makes a black hole is being squeezed inside your own.
The Schwarzschild radius is the distance from the centre of a mass such that the escape velocity at that distance equals the speed of light. Equivalently, it is the distance from the centre of a mass to the event horizon. Either wording gets the mark.
The event horizon is the boundary at that radius. Nothing — not even light — can escape from inside it, so no information ever reaches us from within. At the very centre lies the singularity, where the mass is concentrated.
A mass becomes a black hole only when its physical radius shrinks to less than its own Schwarzschild radius. The Sun’s Schwarzschild radius is about 3 km and its actual radius is about 700 000 km, so it is nowhere near one — and an answer that says so must compare the two radii to score.
| Symbol | Quantity | Unit |
|---|---|---|
| rS | Schwarzschild radius | m |
| G | gravitational constant, 6.67 × 10−11 | m3 kg−1 s−2 |
| M | mass of the object | kg |
| c | speed of light, 3.00 × 108 | m s−1 |
Calculate the Schwarzschild radius of a star of mass 15 solar masses. (1 solar mass = 2.0 × 1030 kg) 3 marks
Answer
Clocks run at different rates in different gravitational fields: the weaker the field, the faster a clock runs. Three separate marking instructions punish the same habit here.
- Say it as a rate. “Time passes faster” on its own is not accepted — it must refer to the rate at which time passes.
- Always justify, and justify in terms of general relativity, because the GR effect dominates the SR one in these situations.
- A correction stated as a bare negative number is not accepted — say what it is: “reduced by 39 µs per day”, not “−39 µs”.
Glossary
Every word on this page that the exam can ask you to define, with its picture.
Inertial frame of reference
A frame moving at constant velocity — it is not accelerating.
Non-inertial frame of reference
A frame that is accelerating with respect to an inertial frame.
Equivalence principle
It is not possible to distinguish between the effects on a body of uniform acceleration and a uniform gravitational field.
World line
The path an object takes through spacetime. Straight means constant velocity; curved means accelerating.
Geodesic
The shortest path between two points in curved spacetime — and the path a freely moving object follows.
Event horizon
The boundary at the Schwarzschild radius. Nothing, not even light, escapes from inside it.
Schwarzschild radius, rS
The distance from the centre of a mass at which the escape velocity equals the speed of light — the distance from the centre out to the event horizon.
Singularity
The point at the centre of a black hole where the mass is concentrated.
Check yourself
Sources & credits: notes and diagrams written and drawn by Mr Stewart. The linked class slides reproduce past-paper figures, questions and marking-instruction wording for educational use, © Qualifications Scotland (SQA), and name Institute of Physics Scotland and Perimeter Institute as the authors of material used in class. R Stewart - September 2026.