Doppler Effect in Light: The Redshift and the Expanding Universe
How a simple frequency shift in starlight revealed one of the greatest discoveries in human history - the expanding universe.
The Cosmic Speedometer
The Doppler Effect in light is not just another physics formula - it's the tool that allowed astronomers to measure the motion of galaxies and discover that our universe is expanding. This single concept connects classroom physics to the grandest scales of cosmology.
🌌 The Big Picture
When Edwin Hubble observed that light from distant galaxies was redshifted, he realized they were moving away from us. This led to the revolutionary conclusion: The universe is expanding.
🚀 Quick Navigation
1. Doppler Effect in Light: The Basic Idea
From Sound to Light
You're familiar with the Doppler effect for sound - the changing pitch of a passing ambulance siren. For light, the principle is similar but with crucial differences:
Sound Waves
- Needs medium to propagate
- Observer velocity matters
- Source velocity matters
- Separate formulas for different cases
Light Waves
- No medium required
- Only relative velocity matters
- Governed by relativity
- Single relativistic formula
The Color Shift
Source moving toward observer
Wavelength decreases
Frequency increases
Source moving away from observer
Wavelength increases
Frequency decreases
Real-world Example: Police Radar
Police radar guns use the Doppler effect in microwaves (a form of light) to measure your car's speed. The frequency shift tells them how fast you're moving!
2. The Relativistic Doppler Formula
The Key Formula for JEE
For light waves, we use the relativistic Doppler formula since light speed is constant for all observers:
When source and observer are moving apart:
$$ f' = f \sqrt{\frac{1 - \frac{v}{c}}{1 + \frac{v}{c}}} $$
Where $f'$ = observed frequency, $f$ = source frequency, $v$ = relative speed, $c$ = speed of light
For Wavelength (More Commonly Used)
Since $c = f\lambda$, we can write the formula in terms of wavelength:
$$ \lambda' = \lambda \sqrt{\frac{1 + \frac{v}{c}}{1 - \frac{v}{c}}} $$
This shows why receding objects appear redshifted ($\lambda' > \lambda$).
Approximation for Small Velocities
For $v << c$ (most astronomical cases except very distant galaxies), we can use the simplified formula:
$$ \frac{\Delta \lambda}{\lambda} \approx \frac{v}{c} $$
Where $\Delta \lambda = \lambda' - \lambda$ is the wavelength shift
JEE Problem Example
Problem: A galaxy shows a redshift of 0.01. Calculate its recession velocity.
Solution: Using $\frac{\Delta \lambda}{\lambda} \approx \frac{v}{c}$
$v = 0.01c = 0.01 \times 3 \times 10^8 = 3 \times 10^6$ m/s
The galaxy is moving away at 3,000 km/s!
3. Redshift: The Cosmic Evidence
Measuring Galactic Motions
Astronomers use spectral lines as cosmic fingerprints. Each element emits/absorbs light at specific wavelengths. When these lines shift, we can calculate velocities:
The Redshift Parameter (z)
Astronomers define redshift using the parameter $z$:
$$ z = \frac{\Delta \lambda}{\lambda} = \frac{\lambda_{observed} - \lambda_{rest}}{\lambda_{rest}} $$
For small velocities: $z \approx \frac{v}{c}$
For relativistic velocities: $z = \sqrt{\frac{1 + \frac{v}{c}}{1 - \frac{v}{c}}} - 1$
Nearby Galaxy Example
Andromeda Galaxy: Blue shifted
z = -0.001 → Moving toward us at 300 km/s
Will collide with Milky Way in ~4 billion years!
Distant Galaxy Example
GN-z11 (most distant known): Red shifted
z = 11.1 → Moving away at incredible speed
We see it as it was 13.4 billion years ago!
Types of Redshift
1. Doppler Redshift
Caused by actual motion through space. This is what we've been discussing.
2. Cosmological Redshift
Caused by the expansion of space itself. As light travels through expanding space, its wavelength stretches.
Key insight: This is the primary cause of redshift for very distant galaxies!
3. Gravitational Redshift
Caused by light climbing out of gravitational wells. Predicted by General Relativity.
4. Hubble's Law: The Expanding Universe
The Discovery That Changed Everything
In 1929, Edwin Hubble made a revolutionary discovery by plotting galaxy distances against their redshifts:
Hubble's Law
$$ v = H_0 \times d $$
Where:
$v$ = recession velocity
$H_0$ = Hubble constant
$d$ = distance to galaxy
The Hubble Constant
The current best value: $H_0 \approx 70$ km/s/Mpc
This means: For every megaparsec (3.26 million light-years) of distance, a galaxy moves away 70 km/s faster.
Implications of Hubble's Law
Not just galaxies moving through space, but space itself is expanding
If we reverse the expansion, everything was together ~13.8 billion years ago
Redshift becomes a tool to measure vast cosmic distances
JEE Calculation Example
Problem: A galaxy has redshift z = 0.1. Estimate its distance.
Solution: For small z: $v \approx zc = 0.1 \times 3 \times 10^5$ km/s = 30,000 km/s
Using Hubble's Law: $d = \frac{v}{H_0} = \frac{30,000}{70} \approx 429$ Mpc
That's about 1.4 billion light-years away!
🎯 JEE Examination Focus
Must-Know Formulas
- Relativistic Doppler: $f' = f\sqrt{\frac{1-v/c}{1+v/c}}$
- Redshift: $z = \frac{\Delta\lambda}{\lambda}$
- Small velocity: $z \approx \frac{v}{c}$
- Hubble's Law: $v = H_0 d$
Common Question Types
- Calculate redshift from velocity
- Find velocity from spectral shift
- Hubble's Law distance calculations
- Conceptual questions about universe expansion
Quick Problem-Solving Approach
- Identify if it's source moving or observer moving
- Use relativistic formula for light
- For small v/c, use approximation
- Relate to Hubble's Law for cosmic distances
📝 Practice Problems
1. A star moving away from Earth at 0.1c. Calculate the redshift parameter z.
2. The hydrogen alpha line (656.3 nm) is observed at 700 nm from a galaxy. Find its recession velocity.
3. A galaxy has recession velocity 15,000 km/s. Estimate its distance using Hubble's Law.
🔭 Modern Connections
Unexpected acceleration of universe expansion found through supernova redshifts (1998 Nobel Prize)
Cosmic Microwave Background is redshifted from original visible/UV light to microwave wavelengths
GPS satellites must account for both special and general relativistic Doppler effects for accuracy
From Classroom to Cosmos
The Doppler effect in light connects fundamental physics to the grandest scales of our universe. What begins as a formula in your JEE preparation reveals the expanding cosmos itself.