Fundamentals of Redshift

Introduction to Redshift What is Redshift? Redshift is a fundamental observation in astronomy: the wavelength of electromagnetic radiation increases (becomes "redder"), which equivalently means its frequency and energy decrease. The opposite effect, called blueshift, represents a decrease in wavelength and an increase in frequency and energy. These shifts arise from different physical causes in the universe. Understanding redshift is essential because it allows astronomers to determine how fast objects are moving, how far away they are, and ultimately, it provides evidence for the expansion of the universe itself. Quantifying Redshift: The Parameter z To measure and compare redshifts across different observations, astronomers use a dimensionless parameter called z, defined as the fractional change in wavelength: $$z = \frac{\lambda{\text{observed}} - \lambda{\text{rest}}}{\lambda{\text{rest}}}$$ This can be rewritten as: $$1 + z = \frac{\lambda{\text{observed}}}{\lambda{\text{rest}}}$$ Key insight: When $z > 0$, we observe a redshift (wavelength increased). When $z < 0$, we observe a blueshift (wavelength decreased). For most distant galaxies, $z$ is positive and quite small (typically $z < 0.1$ for nearby galaxies). The Three Types of Redshift In astronomy, redshift occurs through three distinct physical mechanisms. It's crucial to understand when each applies, because they have different implications for what the redshift tells us. Doppler Redshift Doppler redshift occurs due to relative motion between a source and an observer. If a source moves away from us, the light waves are stretched, creating a redshift. If it moves toward us, the waves are compressed, creating a blueshift. For small velocities (much less than the speed of light), the relationship is remarkably simple: $$z \approx \frac{v{\parallel}}{c}$$ where $v{\parallel}$ is the velocity component along the line of sight (radial velocity), and $c$ is the speed of light. This linear approximation is used frequently because many objects have velocities well below relativistic speeds. Example: If a star moves away from Earth at 1000 km/s, then $z \approx (1000 \text{ km/s}) / (300,000 \text{ km/s}) \approx 0.0033$. Relativistic Doppler Effect When an object moves at velocities comparable to the speed of light, the simple approximation breaks down. Time dilation becomes important, and we must use the relativistic Doppler formula. The key to understanding relativistic effects is the Lorentz factor: $$\gamma = \frac{1}{\sqrt{1 - (v/c)^2}}$$ This factor becomes significantly larger than 1 as velocity approaches the speed of light, and it appears in the relativistic Doppler formula: $$1 + z = \gamma \left(1 + \frac{v{\parallel}}{c}\right)$$ for motion along the line of sight (radial motion). For motion at an arbitrary angle $\theta$ to the line of sight, the full relativistic formula is: $$1 + z = \gamma \left(1 + \frac{v}{c}\cos\theta\right)$$ Important special case—transverse redshift: When $\theta = 90°$ (motion perpendicular to the line of sight), $\cos\theta = 0$ and: $$1 + z = \gamma = \frac{1}{\sqrt{1 - (v/c)^2}}$$ This is remarkable: even though the source isn't moving toward or away from us, we still observe a redshift due to time dilation. For small velocities, this transverse redshift is: $$z \approx \frac{1}{2}\left(\frac{v{\perp}}{c}\right)^2$$ Cosmological Redshift Cosmological redshift is the most important type for understanding the large-scale universe. It doesn't arise from motion through space, but rather from the expansion of space itself. As the universe expands, the wavelengths of light traveling through space are stretched proportionally. In a homogeneous and isotropic (uniform in all directions) expanding universe, redshift is related to the cosmic scale factor $a(t)$: $$1 + z = \frac{a0}{a{\text{emission})}$$ where $a0$ is the scale factor at the time of observation and $a{\text{emission}}$ is the scale factor at the time the light was emitted. Since the universe expands with time, $a$ increases, making $z$ positive for distant objects that emitted light long ago. Key distinction: A cosmological redshift doesn't mean the galaxy is moving away from us through space in the usual sense. Rather, the space between us and the galaxy is expanding. This subtle but crucial difference means that cosmological redshifts can be much larger than Doppler redshifts without violating relativity—there's no limit to how fast space can expand. For small redshifts, cosmological redshift connects to the recession velocity through Hubble's law: $$v \approx H0 D$$ where $H0$ is the Hubble constant and $D$ is the proper distance. This relationship emerges naturally from the expansion of the universe. Gravitational Redshift Gravitational redshift occurs when light escapes from a gravitational potential well—the stronger the gravity, the larger the redshift. This effect arises from time dilation near massive objects: clocks run slower in strong gravitational fields, which stretches the wavelength of light. For a non-rotating spherical mass $M$, the gravitational redshift observed at distance $r$ is: $$z = \left(1 - \frac{2GM}{rc^2}\right)^{-1/2} - 1$$ This can also be expressed in terms of the escape velocity $v{\text{esc}} = \sqrt{2GM/r}$: $$1 + z = \frac{1}{\sqrt{1 - (v{\text{esc}}/c)^2}}$$ Extreme case: Near the event horizon of a black hole (where $r = 2GM/c^2$), the gravitational redshift diverges to infinity. This means light emitted from the event horizon would be infinitely redshifted by the time it reaches a distant observer—in fact, light cannot escape from inside the event horizon at all. Summary: Key Formulas For quick reference, here are the most important redshift formulas you need to know: Doppler redshift (non-relativistic): $$z \approx \frac{v{\parallel}}{c}$$ Relativistic Doppler (radial motion): $$1 + z = \gamma\left(1 + \frac{v{\parallel}}{c}\right)$$ Transverse (perpendicular) redshift: $$z \approx \frac{1}{2}\left(\frac{v}{c}\right)^2$$ Cosmological redshift: $$1 + z = \frac{a0}{a{\text{emission}}}$$ Hubble's law (low redshift approximation): $$v \approx H0 D$$ Schwarzschild gravitational redshift: $$z = \left(1 - \frac{2GM}{rc^2}\right)^{-1/2} - 1$$ <extrainfo> Historical Context Understanding how redshift was discovered and interpreted provides important context for the field. Early Observations and Hubble's Discovery Vesto Slipher made the first extragalactic redshift measurements in 1912, observing that the Andromeda Galaxy showed a blueshift, indicating it was moving toward Earth. Over the following years, Slipher measured redshifts for many other nebulae and found that most were moving away from us. Edwin Hubble revolutionized our understanding by combining Slipher's redshift measurements with distance estimates derived from Cepheid variable stars in 1929. This was groundbreaking because it established that these nebulae were actually separate galaxies far outside the Milky Way, and that there was a simple relationship between their distance and redshift. This discovery led to the formulation of Hubble's law, which became foundational evidence for an expanding universe. Theoretical Framework Before Hubble's observational discoveries, Alexander Friedmann had already derived dynamic cosmological solutions in 1922 (now called the Friedmann equations), which describe how an expanding universe evolves. Similarly, Georges Lemaître independently derived similar equations in 1927, providing crucial theoretical support for the idea of cosmic expansion. Discovery of Quasars In the 1960s, astronomers discovered extremely luminous point sources called quasars (quasi-stellar objects). Initially, their nature was mysterious, and some wondered if they might be nearby stellar objects. However, redshift measurements revealed that quasars were actually extremely distant galaxies—some with redshifts of $z > 6$—making them among the most distant and luminous objects in the universe. This discovery demonstrated that redshift could be used to probe the very early universe. </extrainfo>