Gravity and Wavelength

Einstein’s general theory of relativity describes gravity as the curvature of spacetime caused by mass and energy. One consequence is that light climbing out of a gravitational well loses energy and its wavelength increases. This phenomenon is called gravitational redshift. Observing it provides evidence for relativity and helps measure the properties of stars, galaxies, and even black holes. Gravitational redshift also affects precision instruments such as atomic clocks in satellites.

The Redshift Formula

For a non-rotating, spherically symmetric body, the gravitational redshift $z$ of light emitted at radius $r$ is

$z={\left(1-\frac{2}{G}\right)}^{-1/2}-1$

Here $G$ is the gravitational constant, $M$ is the mass of the body, and $c$ is the speed of light. The formula assumes the emitting source is stationary relative to the body and that we observe the light far away where gravity is negligible.

Redshift in Astrophysics

Gravitational redshift is pronounced near compact objects. For example, light emitted from the surface of a white dwarf or neutron star is redshifted by several percent. Around black holes, the effect becomes extreme as the radius approaches the Schwarzschild limit. Measuring the redshift in spectral lines can reveal the mass-to-radius ratio of these dense bodies, providing clues about their composition and internal structure.

Worked Example

Consider a star with mass 1.4 solar masses and radius 10 km, similar to a typical neutron star. Converting mass to kilograms and radius to meters yields $M=1.4\times 1.989\times {10}^{30}\mathrm{kg}$ and $r=10\times 1000m$. Plugging into the equation gives $z\approx 0.2$, meaning photons leaving the surface lose about twenty percent of their original energy. Observers at infinity would see spectral lines shifted to longer wavelengths accordingly.

Using the Calculator

Enter the mass of the object in solar masses and its radius in kilometers. The script converts these to SI units, evaluates the redshift formula, and outputs $z$. For small values, it also converts the shift to a change in wavelength using $\mathrm{\backslash Delta\backslash lambda}=z\mathrm{\backslash lambda\_0}$, where $\mathrm{\backslash lambda\_0}$ is the emitted wavelength. This feature helps astronomers compare predicted shifts with actual observations of spectral lines.

Gravitational Time Dilation

Closely related to redshift is the slowing of time in strong gravity. A clock on the surface of a massive object runs slower than a distant clock by the same factor that redshifts the light. Global Positioning System satellites must correct for this effect to maintain accuracy. Our calculator focuses on light, but understanding the time dilation connection deepens appreciation of how gravity affects both matter and radiation.

Experimental Tests

The gravitational redshift has been verified in numerous experiments. The Pound-Rebka experiment in 1959 measured the shift in gamma rays over a 22-meter tower using the Mössbauer effect. More recently, spacecraft have observed redshift in signals sent from deep-space probes as they climb out of the Sun’s gravitational well. These observations match the predictions of general relativity to high precision, strengthening our confidence in the theory.

Beyond Newtonian Gravity

In Newtonian physics, gravity acts as a force between masses and does not affect light directly. Einstein’s general relativity changed that picture by showing that mass curves spacetime, altering the paths of both particles and light. The gravitational redshift is one of many relativistic phenomena that have no analog in classical mechanics. It underscores the idea that energy and mass are intimately connected, as photons must expend energy to escape gravity’s pull.

Limitations of the Formula

The redshift equation used here assumes a static, spherically symmetric field and neglects rotation. If the object rotates rapidly, the metric changes and additional effects like frame dragging come into play. For extremely compact objects or those inside strong fields, more sophisticated models from general relativity may be needed. Still, the simple formula provides accurate estimates for many real-world scenarios.

Historical Context

The notion that gravity affects light predates Einstein, with early speculations by John Michell and Pierre-Simon Laplace in the eighteenth century. However, it was Einstein’s theory that supplied the mathematical framework and quantitative predictions. The first astronomical measurements of gravitational redshift came in the 1920s with studies of white dwarfs. Ever since, researchers have used redshift to probe the extreme physics of dense stars and black holes.

Redshift in Cosmology

While this calculator focuses on local gravitational redshift, cosmologists also study redshifts caused by the expansion of the universe. The two effects are distinct: gravitational redshift stems from curvature due to mass, whereas cosmological redshift arises from stretching of spacetime itself. Nonetheless, the basic concept—light changing wavelength due to the geometry of spacetime—connects these phenomena, making redshift a key tool throughout astrophysics.

Conclusion

Gravitational redshift reveals how strongly gravity influences light and time. By entering mass and radius into this calculator, you can estimate the shift photons experience when escaping a gravitational field. Whether analyzing emission lines from a white dwarf or studying the intense gravity near a black hole, understanding redshift deepens our grasp of the universe’s most extreme environments.