physics

Theoretical motivation for general relativity

Theoretical motivation for general relativity is a physics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Theoretical motivation for general relativity rather than just read about it. In short: A theoretical motivation for general relativity, including the motivation for the geodesic equation and the Einstein field equation, can be obtained from special relativity by examining the dynamics of particles in circular orbits about the Earth. A key advantage in examining circular orbits is that it is possible to know the solution of the Einstein Field Equation a priori.

Key takeaways

Theoretical motivation for general relativity belongs to physics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Theoretical motivation for general relativity to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Theoretical motivation for general relativity from memory before moving on to harder problems.

Reference excerpt

A theoretical motivation for general relativity, including the motivation for the geodesic equation and the Einstein field equation, can be obtained from special relativity by examining the dynamics of particles in circular orbits about the Earth. A key advantage in examining circular orbits is that it is possible to know the solution of the Einstein Field Equation a priori. This provides a means to inform and verify the formalism. General relativity addresses two questions:

How does the curvature of spacetime affect the motion of matter? How does the presence of matter affect the curvature of spacetime? The former question is answered with the geodesic equation. The second question is answered with the Einstein field equation. The geodesic equation and the field equation are related through a principle of least action. The motivation for the geodesic equation is provided in the section Geodesic equation for circular orbits. The motivation for the Einstein field equation is provided in the section Stress–energy tensor.

Geodesic equation for circular orbits

Kinetics of circular orbits

For definiteness consider a circular Earth orbit (helical world line) of a particle. The particle travels with speed v. An observer on Earth sees that length is contracted in the frame of the particle. A measuring stick traveling with the particle appears shorter to the Earth observer. Therefore, the circumference of the orbit, which is in the direction of motion appears longer than π {\displaystyle \pi } times the diameter of the orbit. In special relativity the 4-proper-velocity of the particle in the inertial (non-accelerating) frame of the earth is

u = ( γ , γ v c ) {\displaystyle u=\left(\gamma ,\gamma {\mathbf {v} \over c}\right)}

where c is the speed of light, v {\displaystyle \mathbf {v} } is the 3-velocity, and γ {\displaystyle \gamma } is

γ = 1 1 − v ⋅ v c 2 {\displaystyle \gamma ={1 \over {\sqrt {1-{{\mathbf {v} \cdot \mathbf {v} } \over c^{2}}}}}} . The magnitude of the 4-velocity vector is always constant

u α u α = − 1 {\displaystyle u_{\alpha }u^{\alpha }=-1}

where we are using a Minkowski metric

η μ ν = η μ ν = ( − 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ) {\displaystyle \eta ^{\mu \nu }=\eta _{\mu \nu }={\begin{pmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}}} . The magnitude of the 4-velocity is therefore a Lorentz scalar. The 4-acceleration in the Earth (non-accelerating) frame is

… excerpt ends here. Continue reading the full article.

Illustrations

Theoretical motivation for general relativity: Circular orbits at the same radius — Circular orbits at the same radius

Theoretical motivation for general relativity: Diagram 1. Changing views of spacetime along the world line of a rapidly accelerating observer. In this animation, the dashed line is the spacetime trajectory ("world line") of a particle. The balls are placed at regular intervals of proper time along the world line. The solid diagonal lines are the light cones for the observer's current event, and intersect at that event. The small dots are other arbitrary events in the spacetime. For the observer's current instantaneous inertial frame of reference, the vertical direction indicates the time and the horizontal direction indicates distance.
The slope of the world line (deviation from being vertical) is the velocity of the particle on that section of the world line. So at a bend in the world line the particle is being accelerated. Note how the view of spacetime changes when the observer accelerates, changing the instantaneous inertial frame of reference. These changes are governed by the Lorentz transformations. Also note that:
* the balls on the world line before/after future/past accelerations are more spaced out due to time dilation.
* events which were simultaneous before an acceleration are at different times afterwards (due to the relativity of simultaneity),
* events pass through the light cone lines due to the progression of proper time, but not due to the change of views caused by the accelerations, and
* the world line always remains within the future and past light cones of the current event. — Diagram 1. Changing views of spacetime along the world line of a rapidly accelerating observer. In this animation, the dashed line is the spacetime trajectory ("world line") of a particle. The balls are placed at regular intervals of proper time along the world line. The solid diagonal lines are the light cones for the observer's current event, and intersect at that event. The small dots are other arbitrary events in the spacetime. For the observer's current instantaneous inertial frame of reference, the vertical direction indicates the time and the horizontal direction indicates distance. The slope of the world line (deviation from being vertical) is the velocity of the particle on that section of the world line. So at a bend in the world line the particle is being accelerated. Note how the view of spacetime changes when the observer accelerates, changing the instantaneous inertial frame of reference. These changes are governed by the Lorentz transformations. Also note that: * the balls on the world line before/after future/past accelerations are more spaced out due to time dilation. * events which were simultaneous before an acceleration are at different times afterwards (due to the relativity of simultaneity), * events pass through the light cone lines due to the progression of proper time, but not due to the change of views caused by the accelerations, and * the world line always remains within the future and past light cones of the current event.

Worked examples

Example 1 — a first encounter with Theoretical motivation for general relativity

Start with the simplest possible case. Write down what Theoretical motivation for general relativity claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In physics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Theoretical motivation for general relativity before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Theoretical motivation for general relativity ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Theoretical motivation for general relativity

In research: Theoretical motivation for general relativity appears in physics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Theoretical motivation for general relativity in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Theoretical motivation for general relativity is common in secondary-school and first-year university syllabi. It links to neighbouring topics General relativity, so understanding it makes those chapters shorter.
In everyday life: Look for Theoretical motivation for general relativity outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Theoretical motivation for general relativity in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Theoretical motivation for general relativity means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Theoretical motivation for general relativity out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Theoretical motivation for general relativity in simple terms?

Why does Theoretical motivation for general relativity matter?

Because it connects several physics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Theoretical motivation for general relativity?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Theoretical motivation for general relativity.