Logout Open In App
Open In App
Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

Chapter 18: Problem 30

Consider two radual light s?gnals (null geodesics) received at the spatial origin of coordinates at times \(t_{0}\) and \(t_{0}+\Delta t_{0}\), emitted from \(x=x_{1}\) (or \(r=r_{1}\) in the case of the flat models) at time \(t=t_{1}

Short Answer

Expert verified
After a series of derivations from the properties of null geodesics and the scale factor for an expanding universe, we find that the observer experiences a redshift, i.e. 1 + z = \( \frac{a(t_{0})}{a(t_{1})}\) in the case of an expanding universe.

Step by step solution

01

Identify given relations

In this case, the radial light signals received are null geodesics, which means that their interval is zero. From this assumption and the definition of a geodesic, we can set up the following relation: \(ds^2=0=-dt^2+d\rho^2\), where \(\rho=a(t)x\). \[ds^2 = c^2dt^2 - a(t)^2dr^2 = 0\] Hence, we can derive from this that: \[dt= a(t)dr\] when considered for the two times, \(t_0\) and \(t_1\).
02

Derive relation between times and radial light signal

Knowing that \(dt=a(t)dr\), we can integrate both sides over the time interval \(\Delta t\), resulting in \[\int_{t_1}^{t_0} dt = \int_{r_1}^0 a(t) dr\] By a change of variables as \(a(t)dr=dr'\), we get \(t_0-t_1=\Delta t_0= \int_{r_1}^0 dr' . \) Or similarly, integrating the differential dt from \(t1\) to \(t0\) will yield \(\Delta t_0\). This also corresponds to the travel time of the light signal.
03

Derive Redshift

Redshift \(z\) is defined as the change in wavelength or time interval observed as a result of the expansion of the universe. Therefore, the redshift is related to the time intervals as \(1+z = \frac {\Delta t_0} {\Delta t_1}\). We can understand the terms \(\Delta t_0\) and \(\Delta t_1\) as the time interval between the reception of two light signals at the observer and the time interval between the emission of these two light signals at the source, respectively. As a result of the universe's expansion, the light signal is stretched causing the observer to see the signal 'redshifted'.
04

Conclude Result

The scale factor \(a(t)\) characterizes the size of the universe and it increases with time in an expanding universe. As \(a(t_0) > a(t_1)\) because \(t_0>t_1\), it means the Universe expands as time goes from \(t_1\) to \(t_0\), and from the result at step 3, we know that the redshift \(z\) is greater than 0 as \(\Delta t_0 > \Delta t_1\). Combining these results, we can conclude that the observer experiences a redshift in the case of an expanding universe and is given by \[1+z=\frac{a(t_{0})}{a(t_{1})}\].

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Null Geodesics
In the context of General Relativity, null geodesics describe the paths taken by light through spacetime. These paths are called "null" because the interval along them is zero. To put it simply, in the spacetime diagram, light travels the shortest distance possible and exactly along paths that maintain this interval as zero.

Understanding this concept is crucial when analyzing light signals in expanding universes. Light travels along these paths unaffected by the passage of time experienced by observers.

Since the interval is zero, this equation holds: \[ds^2 = 0 = c^2 dt^2 - a(t)^2 dr^2\]. The variable \(a(t)\) represents the scale factor in cosmology, and \(dr\) is a radial component in these equations. By equating this, we derive the relationship and understand that the timing of light signals depends heavily on the evolution of the universe's expansion.
Scale Factor in Cosmology
The scale factor, represented as \(a(t)\), is an essential concept in cosmology to describe how the universe expands or contracts over time. In an expanding universe, the scale factor increases as time progresses.

The reason this is important is that it allows conversion between co-moving and proper distances in calculations, which simplifies computations in cosmology. The change in \(a(t)\) conveys the universe's dynamics, directly influencing quantities like redshift.

When we observe distant objects, the increase in the scale factor over time explains why wavelengths of light expand (or shift to longer wavelengths), resulting in the observed redshift. More intuitively, as \(a(t_0) > a(t_1)\), this increase can be seen as stretching of all forms of waves, including light waves, leading to the redshift as explained in the original derivation.
Integration in Physics
Integration is a mathematical process essential in physics, especially in deriving relations from differential equations that describe physical laws and phenomena.

In the context of our problem, integration helps us find relationships between time intervals and space covered as light travels through the expanding universe.
  • By integrating both sides of the relation \(dt = a(t)dr\), we retrieve the time interval \(\Delta t\) that tells us about the light's journey.
  • This process also allows us to compare the time intervals of emission and reception of light signals in different stages of the universe's expansion.
  • Such analysis ultimately leads to insights like the derivation of redshift, where the proportion of change reflects how the universe uses scales and signals over time.
Using integration provides a framework for understanding complex systems and is the keystone of many breakthroughs in cosmology and general relativity.
Proper Time in General Relativity
Proper time is a fundamental concept in general relativity, often compared to the "personal time" experienced by an observer. Unlike coordinate time, which remains the same for all observers regardless of their state of motion, proper time varies and depends on the observer's path through spacetime.

Proper time can be thought of as the time ticked away on a clock moving along with an observer, considering all paths and deviations.

In cosmological contexts, proper time helps us assess the actual experience of time as the universe evolves.
  • In our problem, proper time is crucial because comparing it between events allows observers to quantify variations like redshift directly.

  • Such comparisons show real discrepancies in time intervals \((\Delta t_0 and \Delta t_1)\) experienced by light signals as seen from varying cosmic scales.

  • Thus, proper time anchors our understanding of how spreading space affects time perception across the cosmos.
Assessing proper time enables meaningful conclusions about cosmic events and supports the broader understanding of relativity's impact on observing the universe.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Problem \(18.12\) (a) Show that in a pscudo-Riemannian space the action principle $$ \delta \int_{t_{1}}^{b_{2}} L, \mathrm{~d} t=0 $$ where \(L=g_{\mu \nu} \dot{x}^{\mu} \dot{x}^{\nu}\) gives rise to geodesic equations with affine parameter \(t\). (b) For the sphere of radius \(a\) in polar coordinates, $$ \mathrm{ds}^{2}=a^{2}\left(\mathrm{~d} \theta^{2}+\sin ^{2} \theta d \phi^{2}\right) $$ use this variation principle to write out the equations of geodesics, and read off from them the Christoffel symbols \(\Gamma_{v \rho^{*}}^{\mu}\) where \(L=g_{k v} \dot{x}^{\mu} \dot{x}^{v}\) gives rise to geodesic equations with affine parameter \(t\). (b) For the sphere of radius \(a\) in polar coordinates, $$ \mathrm{ds}^{2}=a^{2}\left(\mathrm{~d} \theta^{2}+\sin ^{2} \theta \mathrm{d} \phi^{2}\right) $$ use this variation principle to write out the equations of geodesics, and read off from them the Christoffel symbols \(\Gamma_{i \rho^{*}}^{\mu}\) (c) Verify by direct substitution in the geodcsic equations, that \(L=\theta^{2}+\sin ^{2} \theta \dot{\phi}^{2}\) is a constant along the geodesics and use this to show that the general solution of the geodesic equations is grven by $$ b \cot \theta=-\cos \left(\phi-\phi_{0}\right) \text { where } b, \phi_{0}=\text { const. } $$ (d) Show that these curves are great circles on the sphere

Compute the Euler-Lagrange equations and energy-stress tensor for a scalar ficld Lagrangian in general relatwity given by $$ L_{5}=-\psi_{-\mu} \psi_{1} g^{n}-m^{2} \psi^{2} $$ Verify \(T^{n v},=0\)

In the Schwarzschild solution show the only possible closed photon path is a circular orbit at \(r=3 m\), and show that it is unstable.

(a) A particle falls radially inwards from rest at in finity in a Schwarzschild solution. Show that it will arrive at \(r=2 m\) m a finite proper time after crossing some fixed reference position \(r_{0}\), but that coordinate time \(t \rightarrow \infty\) as \(r \rightarrow 2 m\). (b) On an infalling extended body compute the tidal force in a radual direction, by parallel propagating a tetrad (only the radial spacelike unit vector need he considered) and calculating \(R_{1414}\). (c) Estimate the total tidal force on a person of height \(1.8 \mathrm{~m}\), weighing \(70 \mathrm{~kg}\), fallang head-first into a solar mass black hole \(\left(M_{3}=2 \times 10^{10} \mathrm{~kg}\right)\), as he crosses \(r=2 \mathrm{~m}\).

(a) For a perfect flud in general relat?uty, $$ T_{\mu v}=\left(\rho c^{2}+P\right) U_{\mu} U_{v}+P g_{\beta v} \quad\left(U^{\mu} U_{y}=-1\right) $$ show that the conservation identities \(T^{\mu v}, v=0\) imply \(\rho_{v} U^{v}+\left(\rho c^{2}+P\right) U_{v^{2}}^{*}\) \(\left(\rho c^{2}+P\right) U_{,}^{\mu} U^{v}+P_{v}\left(g^{\mu \prime}+U^{\mu} U^{\nu}\right)\) (c) In the Newtonian approximatuon where $$ U_{\mu}=\left(\frac{1}{c},-1\right)+O\left(\beta^{2}\right), \quad P=O\left(\beta^{2}\right) \rho c^{2}, \quad\left(\beta=\frac{v}{c}\right) $$ where \(|\beta| \ll 1\) and \(g_{\mu \nu}=\eta_{h^{x}}+\epsilon h_{\mu \nu}\) with \(\epsilon \ll 1\), show that $$ h_{+\mu} \approx-\frac{2 \phi}{c^{2}}, \quad h_{i j} \approx-\frac{2 \phi}{c^{2}} \delta_{j} \quad \text { where } \quad \nabla^{2} \phi=4 \pi G \rho $$ and \(h_{t 4}=O(\beta) h_{44}\) Show in this approxumation that the equations \(T^{\mu t},=0\) approvimate to $$ \frac{\partial \rho}{\partial t}+\nabla \cdot(\rho v)=0, \quad \rho \frac{d v}{d t}=-\nabla P-\rho \nabla \phi $$
See all solutions

Recommended explanations on Combined Science Textbooks

Synergy

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free