F: net force in newtons; m: constant mass in kilograms; a: acceleration in metres per second squared.

Read it as: the net force equals the mass multiplied by the acceleration. Acceleration describes a change of velocity, not velocity itself.

A net force of 10 N acting on a mass of 2 kg gives an acceleration of 5 m/s². Doubling the force doubles the acceleration if the mass is unchanged.

For constant mass in an inertial frame, within classical mechanics. Force and acceleration are vectors; this scalar form can describe one chosen direction.

G: gravitational constant; m₁ and m₂: the two masses; r: distance between their centres for spherical bodies.

The force grows with each mass and decreases with the square of separation. The formula gives the magnitude of an attractive force.

Keep both masses fixed and double r. The denominator becomes 4r², so the force becomes one quarter of its previous value.

Exact for ideal point masses in Newtonian gravity; also applies to separated spherically symmetric bodies. It is a useful approximation in weak gravitational fields with slow motion.

Δτ: proper time measured by a clock between events on its path; Δt: time interval in an inertial frame; v: constant clock speed in that frame; c: speed of light in vacuum.

An inertial observer assigns a longer time interval than the elapsed time on a clock moving at constant speed relative to that observer. The moving clock measures its own proper time normally.

At v = 0.6c, γ = 1/√0.64 = 1.25. If the moving clock records 8 seconds between two events, the inertial frame assigns 10 seconds.

This calculation assumes constant relative speed and neglects gravity. Comparing twins who reunite requires analysing their complete paths, not simply applying this formula symmetrically.

E₀: rest energy in joules; m: invariant mass in kilograms; c = 299,792,458 m/s in vacuum.

A system has rest energy associated with its mass. This is not the same as saying all of that energy is available for extraction.

For m = 1 kg, E₀ is approximately 8.99 × 10¹⁶ J. Squaring c includes squaring its units: kg × m²/s² is a joule.

E₀ is rest energy. More generally E² = p²c² + m²c⁴, where p is momentum magnitude. A photon has zero rest mass and can still carry energy.

Gμν: Einstein tensor, built from spacetime curvature; gμν: metric tensor; Λ: cosmological constant; Tμν: stress–energy tensor; G without indices: gravitational constant.

This is a set of tensor equations relating spacetime geometry to energy, momentum and stresses. The indices label spacetime components: the equation is not a single ordinary multiplication exercise.

Begin by separating the two sides: the left describes geometry, while the right describes physical sources. Then study how a metric defines distances and time intervals before attempting a full calculation.

General relativity is a classical theory. It is not a complete quantum theory of gravity. The equation requires a matter model and initial or boundary conditions; notation and sign conventions must be stated.

v: recession speed inferred in the nearby-universe approximation; d: distance; H₀: present expansion rate.

For nearby galaxies following the large-scale expansion, recession speed is approximately proportional to distance. H₀ is the present value of an expansion rate that evolves with cosmic time.

Using 70 km/s/Mpc purely as an illustrative value: at 10 Mpc, the expansion contribution is approximately 700 km/s. Local peculiar motions can also affect observations.

This nearby relation is not a universal conversion of any redshift into speed. At large distances, cosmological distances and redshifts require an expansion model. The illustrative H₀ value is not a claim of a settled measurement.