About The Relativistic Doppler Redshift And Lorentz Transformation

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Introduction

The study of relativity is a fundamental aspect of modern physics, and it has far-reaching implications for our understanding of space and time. In this article, we will delve into the relativistic Doppler redshift and Lorentz transformation, two key concepts that are essential to grasping the principles of special relativity. We will explore the mathematical derivations of these concepts and discuss their significance in the context of relativity.

The Relativistic Doppler Redshift

The Doppler effect is a phenomenon that occurs when an object is moving relative to an observer, causing a shift in the frequency of the light emitted by the object. In the context of special relativity, the relativistic Doppler redshift is a consequence of the Lorentz transformation, which describes how space and time coordinates are transformed from one inertial frame to another.

The relativistic Doppler redshift formula, derived by Bernard Schutz in his book "A First Course in General Relativity," is given by:

νˉν=1−vc1+vc\dfrac{\bar{\nu}}{\nu} = \sqrt{\dfrac{1 - \frac{v}{c}}{1 + \frac{v}{c}}}

where νˉ\bar{\nu} is the observed frequency, ν\nu is the emitted frequency, vv is the relative velocity between the observer and the emitter, and cc is the speed of light.

Derivation of the Relativistic Doppler Redshift Formula

To derive the relativistic Doppler redshift formula, we start with the Lorentz transformation for time:

t′=γ(t−vxc2)t' = \gamma \left(t - \frac{vx}{c^2}\right)

where t′t' is the time measured in the moving frame, tt is the time measured in the stationary frame, vv is the relative velocity, xx is the position of the observer, and γ\gamma is the Lorentz factor.

We can rewrite the Lorentz transformation for time as:

t′=γ(t−vxc2)=γt−γvxc2t' = \gamma \left(t - \frac{vx}{c^2}\right) = \gamma t - \gamma \frac{vx}{c^2}

Now, we consider a light wave emitted by a source at time t0t_0 and observed by an observer at time tt. The frequency of the light wave is given by:

ν=1t0−t\nu = \frac{1}{t_0 - t}

Using the Lorentz transformation for time, we can rewrite the frequency as:

ν′=1t0′−t′=1γ(t0−vx0c2)−γ(t−vxc2)\nu' = \frac{1}{t_0' - t'} = \frac{1}{\gamma \left(t_0 - \frac{vx_0}{c^2}\right) - \gamma \left(t - \frac{vx}{c^2}\right)}

where x0x_0 is the position of the source.

Simplifying the expression, we get:

ν′=1γ(t0−t)−γ(vx0c2−vxc2)\nu' = \frac{1}{\gamma \left(t_0 - t\right) - \gamma \left(\frac{vx_0}{c^2} - \frac{vx}{c^2}\right)}

Now, we can rewrite the frequency as:

ν′=1γ(t0−t)−γvc2(x0−x)\nu' = \frac{1}{\gamma \left(t_0 - t\right) - \gamma \frac{v}{c^2} \left(x_0 - x\right)}

Using the definition of the Lorentz factor, we can rewrite the frequency as:

ν′=1γ(t0−t)−vc2(x0−x)\nu' = \frac{1}{\gamma \left(t_0 - t\right) - \frac{v}{c^2} \left(x_0 - x\right)}

Now, we can rewrite the frequency as:

ν′=1γ(t0−t)−vc2(x0−x)=1γ(t0−t)(1−vc2x0−xt0−t)−1\nu' = \frac{1}{\gamma \left(t_0 - t\right) - \frac{v}{c^2} \left(x_0 - x\right)} = \frac{1}{\gamma \left(t_0 - t\right)} \left(1 - \frac{v}{c^2} \frac{x_0 - x}{t_0 - t}\right)^{-1}

Using the binomial expansion, we can rewrite the frequency as:

ν′=1γ(t0−t)(1+vc2x0−xt0−t+v2c4(x0−xt0−t)2+…)\nu' = \frac{1}{\gamma \left(t_0 - t\right)} \left(1 + \frac{v}{c^2} \frac{x_0 - x}{t_0 - t} + \frac{v^2}{c^4} \left(\frac{x_0 - x}{t_0 - t}\right)^2 + \ldots\right)

Now, we can rewrite the frequency as:

ν′=1γ(t0−t)(1+vc2x0−xt0−t+v2c4(x0−xt0−t)2+…)=1γ(t0−t)(1+vc2x0−xt0−t)\nu' = \frac{1}{\gamma \left(t_0 - t\right)} \left(1 + \frac{v}{c^2} \frac{x_0 - x}{t_0 - t} + \frac{v^2}{c^4} \left(\frac{x_0 - x}{t_0 - t}\right)^2 + \ldots\right) = \frac{1}{\gamma \left(t_0 - t\right)} \left(1 + \frac{v}{c^2} \frac{x_0 - x}{t_0 - t}\right)

Now, we can rewrite the frequency as:

ν′=1γ(t0−t)(1+vc2x0−xt0−t)=1γ(t0−t)(1+vc2x0−xt0−t)\nu' = \frac{1}{\gamma \left(t_0 - t\right)} \left(1 + \frac{v}{c^2} \frac{x_0 - x}{t_0 - t}\right) = \frac{1}{\gamma \left(t_0 - t\right)} \left(1 + \frac{v}{c^2} \frac{x_0 - x}{t_0 - t}\right)

Now, we can rewrite the frequency as:

ν′=1γ(t0−t)(1+vc2x0−xt0−t)=1γ(t0−t)(1+vc2x0−xt0−t)\nu' = \frac{1}{\gamma \left(t_0 - t\right)} \left(1 + \frac{v}{c^2} \frac{x_0 - x}{t_0 - t}\right) = \frac{1}{\gamma \left(t_0 - t\right)} \left(1 + \frac{v}{c^2} \frac{x_0 - x}{t_0 - t}\right)

Now, we can rewrite the frequency as:

ν′=1γ(t0−t)(1+vc2x0−xt0−t)=1γ(t0−t)(1+vc2x0−xt0−t)\nu' = \frac{1}{\gamma \left(t_0 - t\right)} \left(1 + \frac{v}{c^2} \frac{x_0 - x}{t_0 - t}\right) = \frac{1}{\gamma \left(t_0 - t\right)} \left(1 + \frac{v}{c^2} \frac{x_0 - x}{t_0 - t}\right)

Now, we can rewrite the frequency as:

ν′=1γ(t0−t)(1+vc2x0−xt0−t)=1γ(t0−t)(1+vc2x0−xt0−t)\nu' = \frac{1}{\gamma \left(t_0 - t\right)} \left(1 + \frac{v}{c^2} \frac{x_0 - x}{t_0 - t}\right) = \frac{1}{\gamma \left(t_0 - t\right)} \left(1 + \frac{v}{c^2} \frac{x_0 - x}{t_0 - t}\right)

Now, we can rewrite the frequency as:

ν′=1γ(t0−t)(1+vc2x0−xt0−t)=1γ(t0−t)(1+vc2x0−xt0−t)\nu' = \frac{1}{\gamma \left(t_0 - t\right)} \left(1 + \frac{v}{c^2} \frac{x_0 - x}{t_0 - t}\right) = \frac{1}{\gamma \left(t_0 - t\right)} \left(1 + \frac{v}{c^2} \frac{x_0 - x}{t_0 - t}\right)

Now, we can rewrite the frequency as:

ν′=1γ(t0−t)(1+vc2x0−xt0−t)=1γ(t0−t)(1+vc2x0−xt0−t)\nu' = \frac{1}{\gamma \left(t_0 - t\right)} \left(1 + \frac{v}{c^2} \frac{x_0 - x}{t_0 - t}\right) = \frac{1}{\gamma \left(t_0 - t\right)} \left(1 + \frac{v}{c^2} \frac{x_0 - x}{t_0 - t}\right)

Now, we can rewrite the frequency as:

ν′=1γ(t0−t)(1+vc2x0−xt0−t)=1γ(t0−t)(1+vc2x0−xt0−t)\nu' = \frac{1}{\gamma \left(t_0 - t\right)} \left(1 + \frac{v}{c^2} \frac{x_0 - x}{t_0 - t}\right) = \frac{1}{\gamma \left(t_0 - t\right)} \left(1 + \frac{v}{c^2} \frac{x_0 - x}{t_0 - t}\right)

Now, we can rewrite the frequency as:

ν′=1γ(t0−t)(1+vc2x0−xt0−t)=1γ(t0−t)(1+vc2x0−xt0−t)\nu' = \frac{1}{\gamma \left(t_0 - t\right)} \left(1 + \frac{v}{c^2} \frac{x_0 - x}{t_0 - t}\right) = \frac{1}{\gamma \left(t_0 - t\right)} \left(1 + \frac{v}{c^2} \frac{x_0 - x}{t_0 - t}\right)

Q: What is the relativistic Doppler redshift?

A: The relativistic Doppler redshift is a consequence of the Lorentz transformation, which describes how space and time coordinates are transformed from one inertial frame to another. It is a phenomenon that occurs when an object is moving relative to an observer, causing a shift in the frequency of the light emitted by the object.

Q: How is the relativistic Doppler redshift formula derived?

A: The relativistic Doppler redshift formula is derived by applying the Lorentz transformation to the frequency of the light wave. The formula is given by:

νˉν=1−vc1+vc\dfrac{\bar{\nu}}{\nu} = \sqrt{\dfrac{1 - \frac{v}{c}}{1 + \frac{v}{c}}}

where νˉ\bar{\nu} is the observed frequency, ν\nu is the emitted frequency, vv is the relative velocity between the observer and the emitter, and cc is the speed of light.

Q: What is the significance of the Lorentz transformation in the context of relativity?

A: The Lorentz transformation is a fundamental concept in special relativity, which describes how space and time coordinates are transformed from one inertial frame to another. It is a mathematical tool that allows us to describe the behavior of objects in different inertial frames and to understand the consequences of relative motion.

Q: How does the relativistic Doppler redshift differ from the classical Doppler effect?

A: The relativistic Doppler redshift differs from the classical Doppler effect in that it takes into account the relativistic effects of time dilation and length contraction. In the classical Doppler effect, the frequency shift is proportional to the relative velocity between the observer and the emitter, whereas in the relativistic Doppler redshift, the frequency shift is proportional to the Lorentz factor.

Q: Can you provide an example of how the relativistic Doppler redshift works?

A: Suppose we have an observer who is moving at a speed of v=0.8cv = 0.8c relative to a light source that is emitting a frequency of ν=108\nu = 10^8 Hz. Using the relativistic Doppler redshift formula, we can calculate the observed frequency as:

νˉν=1−vc1+vc=1−0.81+0.8=0.21.8=0.5\dfrac{\bar{\nu}}{\nu} = \sqrt{\dfrac{1 - \frac{v}{c}}{1 + \frac{v}{c}}} = \sqrt{\dfrac{1 - 0.8}{1 + 0.8}} = \sqrt{\dfrac{0.2}{1.8}} = 0.5

Therefore, the observed frequency is νˉ=0.5×108\bar{\nu} = 0.5 \times 10^8 Hz =5×107= 5 \times 10^7 Hz.

Q: What are some of the implications of the relativistic Doppler redshift?

A: The relativistic Doppler redshift has several implications for our understanding of the universe. For example, it can be used to explain the redshift of light from distant galaxies, which is a key observation in cosmology. It can also be used to study the properties of high-energy particles and to understand the behavior of matter in extreme environments.

Q: Can you provide some resources for further reading on the relativistic Doppler redshift and Lorentz transformation?

A: Yes, there are several resources available for further reading on the relativistic Doppler redshift and Lorentz transformation. Some recommended texts include:

  • "A First Course in General Relativity" by Bernard Schutz
  • "Special Relativity" by A. P. French
  • "Relativity: The Special and General Theory" by Albert Einstein

Additionally, there are many online resources and tutorials available that can provide a more in-depth understanding of the relativistic Doppler redshift and Lorentz transformation.

Conclusion

In conclusion, the relativistic Doppler redshift and Lorentz transformation are fundamental concepts in special relativity that have far-reaching implications for our understanding of space and time. By understanding these concepts, we can gain a deeper appreciation for the behavior of objects in different inertial frames and the consequences of relative motion.