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Measuring Distances in Cosmology

I have re c ently written an article for my local amateur astronomical society about distance measurements in cosmology. There is a lot of confusion about what we mean by “distance” on the vast such scales and the article was written to help clear up of these misunderstandings.

It is an expanded version of a shorter blog post I wrote on this topic in early 2021. As I thought it might be of interest to many of my readers, I have published it her e.

——————————————————————————————

I recently read an article from a popular astronomy website called Universe Today.

It stated that:

“…the CMB [cosmic microwave background radiation] is visible at a distance of 13.8 billion light years in all directions from Earth, leading scientists to determine that this is the true age of the Universe. “

This statement isn’t quite correct. A light year is the distance light travels in a year and so is a unit of distance not time. So, the age of anything in the Universe cannot be measured in light years and, as I’ll explain later, the radiation we detect as the CMB actually lies at a distance of 46 billion light years.

—————————————————————————————————————–

Definition: a light year is a unit of distance . It is the distance which light travels in a year and is equal to  approximately 9 461 000 000 000 km

 Reading the article prompted me to write about what we actually mean by “distance” when we are dealing on the vast scales which occur in cosmology. This is an area around which there is a lot of confusion. For example, the furthest galaxy from us GN-z11 is often quoted as lying at a “distance of 32 billion light years”. This frequently causes puzzlement because the Universe is generally believed to be only 13.8 billion years old, and nothing can travel faster than light. So, surely nothing can be further away than 13.8 billion light years?  The situation arises because of the way the main distance measure  used by cosmologists, called the proper distance, is defined.

In fact, the proper distance is only one of the many different ways to define distance used in cosmology and in this article I’ll talk about these definitions and how they differ.

The proper distance and the light travel distance

When we look at a distant galaxy then the finite speed of light means that its light will have taken millions (or even billions) of years to have reached us.  if we could construct a cosmological-sized ruler between the galaxy and the Earth then we would measure a quantity known as the proper distanceto the galaxy . The proper distance is the most widely used measure in cosmology and when cosmologists used the term “distance” without further qualification they usually mean the proper distance.

Definition: the proper distance is the separation between two a distant objects at a given time which would be measured by an imaginary cosmological sized ruler.

The proper distance isn’t of course directly measurable! It is estimated by using other techniques and making assumptions about the way the Universe is expanding. Because the Universe is expanding, the proper distance between two distant objects (which aren’t held together by gravity) will increase over time. As an illustration:

  • Suppose that a photon of light is received today from a galaxy, which 300 million years ago was at a proper distance of 297 million light years from Earth.
  • The Earth has been moving away from this photon all the time it has been travelling towards us, so the photon will have to travel further than 297 million light years to reach us.
  • In fact, the photon will have travelled 300 million light years by the time it reaches Earth and will have taken 300 million years to do so.
  • When the photon reaches us today, the galaxy is now at a proper distance of 303 million light years from us .

light travel distance vs proper distance

In the example above the distance which the light has actually travelled, known as the light travel distance, is 300 million light years. This is three million light years lower than the current proper distance.

Definition: the light travel distance  between two objects at a given time is the distance travelled by photons emitted by one object (in the example above the distant galaxy) and detected on the other object (in this case the Earth)

When we are dealing with extremely distant objects there are large differences between the light travel distance and the proper distance. If we take the example of the furthest known galaxy GN-z11, then its light we see today was emitted 13.4 billion light years ago, when the Universe was only 3% of its current age. When this light was emitted, the Earth, and the Solar System did not exist and the Milky Way was still in the Early stages of formation.

light travel distance vs proper distance

GN-z11 – Image credit Wikimedi a Commons

In the case of GN-z11.

  • The light we see today was emitted 13.4 billion years ago when GN-z11 was at a proper distance of 2.67 billion light years from the Milky Way.
  • This light has been travelling for 13.4 billion light years to reach us. So, the light travel distance to GN-z11 is 13.4 billion light years.

If we could construct an imaginary ruler between the Milky Way and GN-z11 it would be 32 billion light years long. So, its current proper distance is  32 billion light years

light travel distance vs proper distance

The Comoving distance

The comoving distance is another distance measure which is sometimes used by cosmologists. It is a variation on the proper distance which factors out the expansion of the Universe, giving a distance that does not change in time due to the expansion of space.  The comoving distance between two objects will change if they have any additional relative motion on top of that due to the expansion of the Universe.

Another way to visualise the comoving distance between two distant objects is the distance measured by an imaginary cosmological sized ruler, which is stretching at exactly the same rate the Universe is expanding.

Definition: the comoving distance between two distant objects at any given time in the past or future is defined to be the proper distance at the present time .

So, if we consider two objects which are moving apart solely due to the expansion of the Universe (and for no other reason), the proper distance between them increases over time but the comoving distance does not change.

The comoving distance is discussed in more detail in the notes at the end of this article.

Other distance measures

If we measure the velocity that galaxies are moving away from us due to the expansion of the Universe and plot it as a function of their distance, then we get a graph like that shown below.

light travel distance vs proper distance

Finally, to complete the list, two other distance measures which are sometimes used by astronomers are the luminosity distance and the angular size distance.

Definition: the luminosity distance is how far away a distant object of known actual brightness would be to have the apparent brightness we observe.

Definition: the angular size distance is how far away a distant object of known actual size would be to have the apparent size we observe.

How far away is the Cosmic Microwave background?

Going back to the topic at the start of this article, the early Universe was too hot for atoms to have existed. It consisted of a plasma of positively charged hydrogen and helium ions and negatively charged electrons. Electromagnetic radiation, of which light is an example, cannot pass through the plasma. When the Universe was about 400 000 years old (an era which astronomers call the recombination time*) it had cooled to around 2700 degrees C and all the ions and electrons had combined to make atoms. The Universe became transparent to radiation. Photons could pass unhindered through the hydrogen and helium gases.

* Although widely used, the term recombination time is misleading. It implies (wrongly) that atoms previously existed in the very early Universe, were ionised later and, at the recombination time, the ions and electron were recombined back into atoms. This is not the case it was far too hot for atoms to have existed in the very early Universe.

The weak radiation we observe today is a relic from the recombination time. The photons we detect were last scattered 13.8 billion years ago by the hot plasma. But, due to the expansion of the Universe, the region they were scattered from is a spherical shell of points lying at a proper distance of 46 billion light years from Earth.

light travel distance vs proper distance

In the diagram below a slice has been taken through this spherical shell . The locations inside the sphere (shaded pale yellow) lie closer than 46 billion light years.  The CMB photons from these regions will have taken less than 13.8 billion years to reach us and will have arrived in the past. The locations outside the sphere (shaded pale blue) lie further away than 46 billion light years. The photons from these regions will take longer than 13.8 billion years to reach us and will arrive in the future.

light travel distance vs proper distance

Appendix A Further detail on the Comoving distance

If we consider any two objects which are far enough away from each other so that they are not bound together by gravity then the cosmic scale factor is the ratio of their proper distance at time t (D(t)) to their current proper distance (D o ).

It is given the symbol a(t) and is defined as:

 a(t) = D(t) / D o

The cosmic scale factor is.

  • equal to zero at t=0, the instant of the Big Bang
  • equal to one at the current age of the Universe

Clearly, as the Universe expands, and objects move further apart the cosmic scale factor increases.

light travel distance vs proper distance

So the comoving distance (D CM ) can be defined as the proper distance divided by the scale factor.

D CM = D(t)/a(t)

(In many astronomy textbooks the comoving distance is denoted by the Greek letter χ (lowercase chi))

For objects which only get further apart (i.e., their proper distance increases) as a result of the expansion of the Universe the comoving distance between them will not change over time.

light travel distance vs proper distance

As the Universe expands, the proper distance to the galaxy increases but the comoving distance does not change. This is also shown in the graph below.

light travel distance vs proper distance

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Published by Steve Hurley

Hi I am Steve Hurley. I work in the IT industry. I studied for a PhD in astronomy in the 1980s. Outside work my real passion is explaining scientific concepts to a non-scientific audience. My blog (explainingscience.org) covers various scientific topics, but primarily astronomy. It is written in a style that it is easily understandable to the non scientist. Publications and videos For links to my books and videos please visit www.explainingscience.org View all posts by Steve Hurley

26 thoughts on “Measuring Distances in Cosmology”

Surely it’s 32-2.67=29.33/2=14.67+2.67=17.34 ly, in 13.4 y. How is that possible!? Is it because H0 is going down or the expanding space carries light to a degree along with it?

Ye’s but what distance, in ly, did the light from GN-z11 actually cover? For surely that is what determines its Intensity. Which is what really interests me.

Of what use is the comoving distance, and when do U start it?

You raise an interesting point, the comoving distance between two objects is the proper distance at the current time. It does not increase as the Universe expands. Although its use it not immediately obvious it is a distance measure often used by cosmologists in scientific papers.

Is the cosmological redshift/z equal to the recession/al velocity when the light started or when it arrived, or something inbetween? And can U equate it with the distance, directly, without resorting to V=H0D?

Thank you for your comment.

When we talk about redshift we mean the redshift of radiation received by the observer . For example if we observe two spectral lines from a distant galaxy at wavelengths of 505 nm and 606 nm which are emitted at 500 nm and 600 nm when the source is at rest, then the spectral lines have been redshifted by Δλ/ λ = 0.01. This is usually interpreted as the doppler shift due to the galaxy moving away from us. But a better explanation is that space itself is expanding in the sense that the distance that the distance between two objects (in this case us and the distant galaxy) not held together by another force such a gravity increases over time.

Hubble’s law shows that the redshift (usually given the symbol z) of an object is proportional to its distance (D) (measured by the various methods discussed in this post https://explainingscience.org/2020/12/10/dark-energy-an-unexpected-finding/ .

If we measure the redshift of an object it can be used to gives us a quantity known as the redshift distance cz/Ho

Thank U, for Ur reply. . OK, so what do they mean by “The distant type Ia SNS are fainter, and therefore further, away, than expected”? Relative to what? Their z and D=v/H0=cz/H0? OK, so why not just say that H0 is decreasing, rather than that the expansion is accelerating? And, they do say that H0 IS decreasing!? . So what do they actually mean by the accelerating expansion? That the rate of expansion is increasing? Meaning what, exactly? That distances are increasing? And yet they say that velocities are decreasing!? I don’t know. “Third base! AH'”!

I coulda/shoulda said: “Are they ‘avin’ a laugh!? Are they ‘avin’ a laugh”!? . I know that SP, AR and BS are really smart/sharp, guys, but/t I have read a few articles in the SA by SP and AR, and watched a few 1 hr long videos in (the) YT by AR and BS, and AR is a great talker and BS is very entertaining, if not informative, but they never explain the problem! They complain that the theorists won’t do their jub and explain it, away, to them, but why don’t they do their jub and explain the problem, to us, first!? . Now, I don’t actually have a problem with the AE and DE. My model predicts them. It says that the Universe has to be either expanding or contracting, unless it’s on a cusp, and the expansion has to be either accelerating or decelerating… but I still don’t understand how they came to the conclusion that it is accelerating!?

Sorry, . H0 IS the current rate of expansion, and they say that it is decreasing, and yet the expansion is accelerating!? See what I mean. WTF!? What they man is that the velocity of these distant SNs is increasing. Well it would have to be, coz H0 is positive? . Now I know what it would mean if H0 was 0. The Universe would stop, expanding. So what sort of expansion would cause these velocities to stay constant, with increasing distance/s? D is increasing but v is not, so H?0 would have to be decreasing, by a lot? I am gonna have to write a program to simulate/model this. It’s the only way, I’ll ever understand it. Und I vill, if it’s the last thing I do!

Why is D in v=H0D in uppercase? Isn’t distance normally in lowercase?

Sir, I don´t understand your second example about galaxy GN-z11. If the the light travel distance to GN-z11 is 13.4 billion light years, the Earth has moved 13.4-2.67 = 10.73 billion light years. Therfore, would not be the proper distance 10.73 + 2.67 + 10.73 = 24.13 billion light years, instead 32? I´m sorry if I´m wrong.

You raise a good point, but your suggestion an oversimplification. The proper distance between GN-z11 depends on how the Universe has expanded since the photons were emitted. I will update the post to reflect this.

Will you be kind to share the new article on “measuring distances in cosmology”? I would love to see what you have put down.

Hi, This post is pretty much as the the article I wrote for my local astronomical society

Will be kind to share the new article on measuring distances in astronomy? I will be delighted to see what you have come up with.

[…] Source link […]

This is so very good, thank you! Whenever I’ve heard that a galaxy is for example, 5 billion years “away” and that the light we see was emitted 5 billion years ago, I’ve always wondered, “okay but where is that galaxy now? It’s been moving away for 5 billion years.” That seems rarely touched on in popular articles so I really like this post for touching on it and showing that the present distance can be calculated with some reliability. Also, it was especially cool to see calculations for how far the galaxy was in the past when the light on our eyes was emitted. I think this is the first time I’ve had a proper (!) appreciation of that side of the question.

Fun and mind-stretching stuff!

Thanks Kevin,

Distances in cosmology is certainly something around which there’s a lot of confusion/misunderstanding. Your might want to check out my YouTube channel as well. Some of the videos on the General Astronomy playlist may be of particular interest.

https://www.youtube.com/c/ExplainingScience/playlists

I came here trying to find an answer to the question of the definition of year in regards to the age of the universe. Is the usage here the same as an earth year? 13 billion earth years? If so, isn’t that arbitrary since earth years are simply a result of the orbit around the sun which is further constrained in definition by the Distance relative to the sun. An earth year, Saturn year, all measured by orbital speed but distance of the object seems indispensable in defining “year”. Or is the age of the universe version of year a light year? So again, distance. If it is a light year, what is the relative “location” ,which I know is a horrible word in this context- but what is the relative guage for determining what a light year would be? Light travels at a certain speed, regardless of if the observer is on earth or Saturn. Yet each of these are limited by themselves essentially it seems. So, what is the proper meaning of year in the age of universe. If the answer is that the definition isn’t really about time, then what is meant? I’m barely even an amateur on any of this so go easy if some of or all of what I said is incorrect. All I’m looking for is an answer to the “year” question applied to the age of the universe.

Like Liked by 1 person

the question you ask is very interesting and perfectly valid. What we mean when we talk about a “year” is often glossed over.

In astronomy, when we talk about years, we normally mean Julian years. A Julian year is a measurement of time and is defined as exactly 365.25 days. Where a “day” is defined as having exactly 86400 SI seconds.

Because the Earth’s rotation speed varies a little and is slowing down gradually. an SI second is defined in terms of a transitions of a caesium atom rather than being a fraction of the mean solar day (the natural day due the apparent motion of the Sun across the sky).

I hope this helps…

Kind regards,

[…] By Steve Hurley, Posted on December 1, 2021 […]

Isn’t a cosmological sized ruler an imaginary cosmic sized ruler?

Very nice reading. I was expecting unfortunately an article about the Cepheid variables… maybe next time.

It is down on my ever growing list of “topics to write about”. Although I touched very briefly on this topic in a previous post

Dark energy an unexpected finding

[…] This post has now been superseded by a more detailed post Measuring Distances in Cosmology […]

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Cosmic Distances

The trinary star Alpha Centauri, hangs above the horizon of Saturn

The space beyond Earth is so incredibly vast that units of measure which are convenient for us in our everyday lives can become GIGANTIC. Distances between the planets, and especially between the stars, can become so big when expressed in miles and kilometers that they're unwieldy. So for cosmic distances, we switch to whole other types of units: astronomical units, light years and parsecs.

Astronomical units, abbreviated AU, are a useful unit of measure within our solar system. One AU is the distance from the Sun to Earth's orbit, which is about 93 million miles (150 million kilometers). When measured in astronomical units, the 886,000,000-mile (1,400,000,000-kilometer) distance from the Sun to Saturn's orbit, is a much more manageable 9.5 AU. So astronomical units are a great way to compress truly astronomical numbers to a more manageable size.

Astronomical units also make it easy to think about distances between solar system objects. They make it easy to see that Jupiter orbits five times farther from the Sun than Earth, and that Saturn is twice as far from the Sun as Jupiter. (This is because, technically, you're expressing every distance as a ratio of the distance from Earth to the Sun. Convenient!)

For much greater distances — interstellar distances — astronomers use light years. A light year is the distance a photon of light travels in one year, which is about 6 trillion miles (9 trillion kilometers, or 63,000 AU). Put another way, a light year is how far you'd travel in a year if you could travel at the speed of light, which is 186,000 miles (300,000 kilometers) per second. (By the way, you can't travel at the speed of light, as far as we know, but that's a whole other story...) Like AU, light years make astronomical distances more manageable. For example, the nearest star system to ours is the triple star system of Alpha Centauri , at about 4.3 light years away. That's a more manageable number than 25 trillion miles, 40 trillion kilometers or 272,000 AU.

Light years also provide some helpful perspective on solar system distances: the Sun is about 8 light minutes from Earth. (And yes, there are also light seconds !) And because light from objects travels at light speed , when you see the Sun, or Jupiter or a distant star, you're seeing it as it was when the light left it, be that 8 minutes, tens of minutes or 4.3 years ago. And this is fundamental to the idea that when we're looking farther out into space, we're seeing farther back in time. (Think about it: you're seeing all the stars in the sky at different times in history — some a few years ago, others hundreds of years ago — all at the same time!)

Finally, parsecs. This is the unit used when the number of light years between objects climbs into the high thousands or millions. One parsec is 3.26 light years. The origin of this unit of measure is a little more complicated, but it's related to how astronomers measure widths in the sky. Astronomers use "megaparsecs" — a megaparsec is 1 million parsecs — for intergalactic distances, or the scale of distances between the galaxies.

And at the point when distances between galaxies become so epic that even megaparsecs get unwieldy, astronomers talk about distances in terms of how much a galaxy's light has been shifted toward longer, redder wavelengths by the expansion of the universe — a measure known as "redshift." Now that's astronomical.

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Speed of Light Calculator

Table of contents

With this speed of light calculator, we aim to help you calculate the distance light can travel in a fixed time . As the speed of light is the fastest speed in the universe, it would be fascinating to know just how far it can travel in a short amount of time.

We have written this article to help you understand what the speed of light is , how fast the speed of light is , and how to calculate the speed of light . We will also demonstrate some examples to help you understand the computation of the speed of light.

What is the speed of light? How fast is the speed of light?

The speed of light is scientifically proven to be the universe's maximum speed. This means no matter how hard you try, you can never exceed this speed in this universe. Hence, there are also some theories on getting into another universe by breaking this limit. You can understand this more using our speed calculator and distance calculator .

So, how fast is the speed of light? The speed of light is 299,792,458 m/s in a vacuum. The speed of light in mph is 670,616,629 mph . With this speed, one can go around the globe more than 400,000 times in a minute!

One thing to note is that the speed of light slows down when it goes through different mediums. Light travels faster in air than in water, for instance. This phenomenon causes the refraction of light.

Now, let's look at how to calculate the speed of light.

How to calculate the speed of light?

As the speed of light is constant, calculating the speed of light usually falls on calculating the distance that light can travel in a certain time period. Hence, let's have a look at the following example:

  • Source: Light
  • Speed of light: 299,792,458 m/s
  • Time traveled: 100 seconds

You can perform the calculation in three steps:

Determine the speed of light.

As mentioned, the speed of light is the fastest speed in the universe, and it is always a constant in a vacuum. Hence, the speed of light is 299,792,458 m/s .

Determine the time that the light has traveled.

The next step is to know how much time the light has traveled. Unlike looking at the speed of a sports car or a train, the speed of light is extremely fast, so the time interval that we look at is usually measured in seconds instead of minutes and hours. You can use our time lapse calculator to help you with this calculation.

For this example, the time that the light has traveled is 100 seconds .

Calculate the distance that the light has traveled.

The final step is to calculate the total distance that the light has traveled within the time . You can calculate this answer using the speed of light formula:

distance = speed of light × time

Thus, the distance that the light can travel in 100 seconds is 299,792,458 m/s × 100 seconds = 29,979,245,800 m

What is the speed of light in mph when it is in a vacuum?

The speed of light in a vacuum is 670,616,629 mph . This is equivalent to 299,792,458 m/s or 1,079,252,849 km/h. This is the fastest speed in the universe.

Is the speed of light always constant?

Yes , the speed of light is always constant for a given medium. The speed of light changes when going through different mediums. For example, light travels slower in water than in air.

How can I calculate the speed of light?

You can calculate the speed of light in three steps:

Determine the distance the light has traveled.

Apply the speed of light formula :

speed of light = distance / time

How far can the speed of light travel in 1 minute?

Light can travel 17,987,547,480 m in 1 minute . This means that light can travel around the earth more than 448 times in a minute.

Harris-Benedict calculator uses one of the three most popular BMR formulas. Knowing your BMR (basal metabolic weight) may help you make important decisions about your diet and lifestyle.

Everyone knows biking is fantastic, but only this Car vs. Bike Calculator turns biking hours into trees! 🌳

The photoelectric effect calculator computes the kinetic energy of electrons ejected from material by incident light.

Speed of light

The speed of light in the medium. In a vacuum, the speed of light is 299,792,458 m/s.

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  • 5.4 Length Contraction
  • Introduction
  • 1.1 The Propagation of Light
  • 1.2 The Law of Reflection
  • 1.3 Refraction
  • 1.4 Total Internal Reflection
  • 1.5 Dispersion
  • 1.6 Huygens’s Principle
  • 1.7 Polarization
  • Key Equations
  • Conceptual Questions
  • Additional Problems
  • Challenge Problems
  • 2.1 Images Formed by Plane Mirrors
  • 2.2 Spherical Mirrors
  • 2.3 Images Formed by Refraction
  • 2.4 Thin Lenses
  • 2.5 The Eye
  • 2.6 The Camera
  • 2.7 The Simple Magnifier
  • 2.8 Microscopes and Telescopes
  • 3.1 Young's Double-Slit Interference
  • 3.2 Mathematics of Interference
  • 3.3 Multiple-Slit Interference
  • 3.4 Interference in Thin Films
  • 3.5 The Michelson Interferometer
  • 4.1 Single-Slit Diffraction
  • 4.2 Intensity in Single-Slit Diffraction
  • 4.3 Double-Slit Diffraction
  • 4.4 Diffraction Gratings
  • 4.5 Circular Apertures and Resolution
  • 4.6 X-Ray Diffraction
  • 4.7 Holography
  • 5.1 Invariance of Physical Laws
  • 5.2 Relativity of Simultaneity
  • 5.3 Time Dilation
  • 5.5 The Lorentz Transformation
  • 5.6 Relativistic Velocity Transformation
  • 5.7 Doppler Effect for Light
  • 5.8 Relativistic Momentum
  • 5.9 Relativistic Energy
  • 6.1 Blackbody Radiation
  • 6.2 Photoelectric Effect
  • 6.3 The Compton Effect
  • 6.4 Bohr’s Model of the Hydrogen Atom
  • 6.5 De Broglie’s Matter Waves
  • 6.6 Wave-Particle Duality
  • 7.1 Wave Functions
  • 7.2 The Heisenberg Uncertainty Principle
  • 7.3 The Schrӧdinger Equation
  • 7.4 The Quantum Particle in a Box
  • 7.5 The Quantum Harmonic Oscillator
  • 7.6 The Quantum Tunneling of Particles through Potential Barriers
  • 8.1 The Hydrogen Atom
  • 8.2 Orbital Magnetic Dipole Moment of the Electron
  • 8.3 Electron Spin
  • 8.4 The Exclusion Principle and the Periodic Table
  • 8.5 Atomic Spectra and X-rays
  • 9.1 Types of Molecular Bonds
  • 9.2 Molecular Spectra
  • 9.3 Bonding in Crystalline Solids
  • 9.4 Free Electron Model of Metals
  • 9.5 Band Theory of Solids
  • 9.6 Semiconductors and Doping
  • 9.7 Semiconductor Devices
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  • 10.1 Properties of Nuclei
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  • 11.1 Introduction to Particle Physics
  • 11.2 Particle Conservation Laws
  • 11.3 Quarks
  • 11.4 Particle Accelerators and Detectors
  • 11.5 The Standard Model
  • 11.6 The Big Bang
  • 11.7 Evolution of the Early Universe
  • B | Conversion Factors
  • C | Fundamental Constants
  • D | Astronomical Data
  • E | Mathematical Formulas
  • F | Chemistry
  • G | The Greek Alphabet

Learning Objectives

By the end of this section, you will be able to:

  • Explain how simultaneity and length contraction are related.
  • Describe the relation between length contraction and time dilation and use it to derive the length-contraction equation.

The length of the train car in Figure 5.8 is the same for all the passengers. All of them would agree on the simultaneous location of the two ends of the car and obtain the same result for the distance between them. But simultaneous events in one inertial frame need not be simultaneous in another. If the train could travel at relativistic speeds, an observer on the ground would see the simultaneous locations of the two endpoints of the car at a different distance apart than observers inside the car. Measured distances need not be the same for different observers when relativistic speeds are involved.

Proper Length

Two observers passing each other always see the same value of their relative speed. Even though time dilation implies that the train passenger and the observer standing alongside the tracks measure different times for the train to pass, they still agree that relative speed, which is distance divided by elapsed time, is the same. If an observer on the ground and one on the train measure a different time for the length of the train to pass the ground observer, agreeing on their relative speed means they must also see different distances traveled.

The muon discussed in Example 5.3 illustrates this concept ( Figure 5.9 ). To an observer on Earth, the muon travels at 0.950 c for 7.05 μs from the time it is produced until it decays. Therefore, it travels a distance relative to Earth of:

In the muon frame, the lifetime of the muon is 2.20 μs. In this frame of reference, the Earth, air, and ground have only enough time to travel:

The distance between the same two events (production and decay of a muon) depends on who measures it and how they are moving relative to it.

Proper length L 0 L 0 is the distance between two points measured by an observer who is at rest relative to both of the points.

The earthbound observer measures the proper length L 0 L 0 because the points at which the muon is produced and decays are stationary relative to Earth. To the muon, Earth, air, and clouds are moving, so the distance L it sees is not the proper length.

Length Contraction

To relate distances measured by different observers, note that the velocity relative to the earthbound observer in our muon example is given by

The time relative to the earthbound observer is Δ t , Δ t , because the object being timed is moving relative to this observer. The velocity relative to the moving observer is given by

The moving observer travels with the muon and therefore observes the proper time Δ τ . Δ τ . The two velocities are identical; thus,

We know that Δ t = γ Δ τ . Δ t = γ Δ τ . Substituting this equation into the relationship above gives

Substituting for γ γ gives an equation relating the distances measured by different observers.

Length contraction is the decrease in the measured length of an object from its proper length when measured in a reference frame that is moving with respect to the object:

where L 0 L 0 is the length of the object in its rest frame, and L is the length in the frame moving with velocity v .

If we measure the length of anything moving relative to our frame, we find its length L to be smaller than the proper length L 0 L 0 that would be measured if the object were stationary. For example, in the muon’s rest frame, the distance Earth moves between where the muon was produced and where it decayed is shorter than the distance traveled as seen from the Earth’s frame. Those points are fixed relative to Earth but are moving relative to the muon. Clouds and other objects are also contracted along the direction of motion as seen from muon’s rest frame.

Thus, two observers measure different distances along their direction of relative motion, depending on which one is measuring distances between objects at rest.

But what about distances measured in a direction perpendicular to the relative motion? Imagine two observers moving along their x -axes and passing each other while holding meter sticks vertically in the y -direction. Figure 5.10 shows two meter sticks M and M ′ M ′ that are at rest in the reference frames of two boys S and S ′ , S ′ , respectively. A small paintbrush is attached to the top (the 100-cm mark) of stick M ′ . M ′ . Suppose that S ′ S ′ is moving to the right at a very high speed v relative to S, and the sticks are oriented so that they are perpendicular, or transverse, to their relative velocity vector. The sticks are held so that as they pass each other, their lower ends (the 0-cm marks) coincide. Assume that when S looks at his stick M afterwards, he finds a line painted on it, just below the top of the stick. Because the brush is attached to the top of the other boy’s stick M ′ , M ′ , S can only conclude that stick M ′ M ′ is less than 1.0 m long.

Now when the boys approach each other, S ′ , S ′ , like S, sees a meter stick moving toward him with speed v . Because their situations are symmetric, each boy must make the same measurement of the stick in the other frame. So, if S measures stick M ′ M ′ to be less than 1.0 m long, S ′ S ′ must measure stick M to be also less than 1.0 m long, and S ′ S ′ must see his paintbrush pass over the top of stick M and not paint a line on it. In other words, after the same event, one boy sees a painted line on a stick, while the other does not see such a line on that same stick!

Einstein’s first postulate requires that the laws of physics (as, for example, applied to painting) predict that S and S ′ , S ′ , who are both in inertial frames, make the same observations; that is, S and S ′ S ′ must either both see a line painted on stick M, or both not see that line. We are therefore forced to conclude our original assumption that S saw a line painted below the top of his stick was wrong! Instead, S finds the line painted right at the 100-cm mark on M. Then both boys will agree that a line is painted on M, and they will also agree that both sticks are exactly 1 m long. We conclude then that measurements of a transverse length must be the same in different inertial frames .

Example 5.5

Calculating length contraction, solution for (a).

  • Identify the knowns: L 0 = 4.300 ly; γ = 30.00 . L 0 = 4.300 ly; γ = 30.00 .
  • Identify the unknown: L .
  • Express the answer as an equation: L = L 0 γ . L = L 0 γ .
  • Do the calculation: L = L 0 γ = 4.300 ly 30.00 = 0.1433 ly. L = L 0 γ = 4.300 ly 30.00 = 0.1433 ly.

Solution for (b)

  • Identify the known: γ = 30.00 . γ = 30.00 .
  • Identify the unknown: v in terms of c .
  • Express the answer as an equation. Start with: γ = 1 1 − v 2 c 2 . γ = 1 1 − v 2 c 2 . Then solve for the unknown v/c by first squaring both sides and then rearranging: γ 2 = 1 1 − v 2 c 2 v 2 c 2 = 1 − 1 γ 2 v c = 1 − 1 γ 2 . γ 2 = 1 1 − v 2 c 2 v 2 c 2 = 1 − 1 γ 2 v c = 1 − 1 γ 2 .
  • Do the calculation: v c = 1 − 1 γ 2 = 1 − 1 ( 30.00 ) 2 = 0.99944 v c = 1 − 1 γ 2 = 1 − 1 ( 30.00 ) 2 = 0.99944 or v = 0.9994 c . v = 0.9994 c .

Significance

People traveling at extremely high velocities could cover very large distances (thousands or even millions of light years) and age only a few years on the way. However, like emigrants in past centuries who left their home, these people would leave the Earth they know forever. Even if they returned, thousands to millions of years would have passed on Earth, obliterating most of what now exists. There is also a more serious practical obstacle to traveling at such velocities; immensely greater energies would be needed to achieve such high velocities than classical physics predicts can be attained. This will be discussed later in the chapter.

Why don’t we notice length contraction in everyday life? The distance to the grocery store does not seem to depend on whether we are moving or not. Examining the equation L = L 0 1 − v 2 c 2 , L = L 0 1 − v 2 c 2 , we see that at low velocities ( v < < c ) , ( v < < c ) , the lengths are nearly equal, which is the classical expectation. But length contraction is real, if not commonly experienced. For example, a charged particle such as an electron traveling at relativistic velocity has electric field lines that are compressed along the direction of motion as seen by a stationary observer ( Figure 5.12 ). As the electron passes a detector, such as a coil of wire, its field interacts much more briefly, an effect observed at particle accelerators such as the 3-km-long Stanford Linear Accelerator (SLAC). In fact, to an electron traveling down the beam pipe at SLAC, the accelerator and Earth are all moving by and are length contracted. The relativistic effect is so great that the accelerator is only 0.5 m long to the electron. It is actually easier to get the electron beam down the pipe, because the beam does not have to be as precisely aimed to get down a short pipe as it would to get down a pipe 3 km long. This, again, is an experimental verification of the special theory of relativity.

Check Your Understanding 5.4

A particle is traveling through Earth’s atmosphere at a speed of 0.750 c . To an earthbound observer, the distance it travels is 2.50 km. How far does the particle travel as viewed from the particle’s reference frame?

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Access for free at https://openstax.org/books/university-physics-volume-3/pages/1-introduction
  • Authors: Samuel J. Ling, Jeff Sanny, William Moebs
  • Publisher/website: OpenStax
  • Book title: University Physics Volume 3
  • Publication date: Sep 29, 2016
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  • Book URL: https://openstax.org/books/university-physics-volume-3/pages/1-introduction
  • Section URL: https://openstax.org/books/university-physics-volume-3/pages/5-4-length-contraction

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  • Distance and Lookback Time

The expansion of the universe makes measuring the distance to distant objects complicated. Rather than using distance units, astronomers measure the redshift (z) of distant objects such as galaxies. z corresponds to the number of years the light from an object has traveled to reach us. This is not the distance to the object in light years, however, because the universe has been expanding as the light traveled and the object is now much farther away. The more distant an object, the more it will be redshifted . Some very distant objects may emit energy in the ultraviolet or even higher energy wavelengths. As the light travels great distances and is redshifted, its wavelength may be shifted by a factor of 10. So light that starts out as ultraviolet may be become infrared by the time it gets to us! The highest known redshifts are from galaxies producing gamma ray bursts . The highest confirmed redshift is for a galaxy called UDFy-38135539 with a z value of 8.6, which corresponds to a light travel time of about 13.1 billion years. This means the light we see now left the galaxy about 600 million years after the Big Bang! The galaxy is now 30.384 billion light years away from us due to the expansion of the universe during the time the light from the galaxy traveled to us. As the universe expands, the space between galaxies is expanding. The more distance between us and a galaxy, the more quickly the galaxy will appear to be moving away from us.

It is important to remember that although such distant galaxies can appear to be moving away from us at near the speed of light, the galaxy itself is not traveling so fast. Its motion away from us is due to the expansion of the space between us.

light travel distance vs proper distance

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5.5: Length Contraction

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Learning Objectives

By the end of this section, you will be able to:

  • Explain how simultaneity and length contraction are related.
  • Describe the relation between length contraction and time dilation and use it to derive the length-contraction equation.

The length of the train car in Figure \(\PageIndex{1}\) is the same for all the passengers. All of them would agree on the simultaneous location of the two ends of the car and obtain the same result for the distance between them. But simultaneous events in one inertial frame need not be simultaneous in another. If the train could travel at relativistic speeds, an observer on the ground would see the simultaneous locations of the two endpoints of the car at a different distance apart than observers inside the car. Measured distances need not be the same for different observers when relativistic speeds are involved.

A photo of a TGV high speed train

Proper Length

Two observers passing each other always see the same value of their relative speed. Even though time dilation implies that the train passenger and the observer standing alongside the tracks measure different times for the train to pass, they still agree that relative speed, which is distance divided by elapsed time, is the same. If an observer on the ground and one on the train measure a different time for the length of the train to pass the ground observer, agreeing on their relative speed means they must also see different distances traveled.

The muon discussed previously illustrates this concept (Figure \(\PageIndex{2}\)). To an observer on Earth, the muon travels at 0.950 c for 7.05 μs from the time it is produced until it decays. Therefore, it travels a distance relative to Earth of:

\[ \begin{align*} L_0 &= v\Delta t \\[4pt] &= (0.950)(3.00 \times 10^8 \, m/s)(7.05 \times 10^{-6}s) \\[4pt] &= 2.01 \, km. \end{align*} \nonumber \]

In the muon frame, the lifetime of the muon is 2.20 μs. In this frame of reference, the Earth, air, and ground have only enough time to travel:

\[ \begin{align*} L &= v\Delta r \\[4pt] &= (0.950)(3.00 \times 10^8 \, m/s)(2.20 \times 10^{-6}s) \\[4pt] &= 0.627 \, km. \end{align*} \nonumber \]

The distance between the same two events (production and decay of a muon) depends on who measures it and how they are moving relative to it.

Definition: Proper Length

Proper length \(L_0\) is the distance between two points measured by an observer who is at rest relative to both of the points.

The earthbound observer measures the proper length \(L_0\) because the points at which the muon is produced and decays are stationary relative to Earth. To the muon, Earth, air, and clouds are moving, so the distance L it sees is not the proper length.

CNX_UPhysics_38_04_CloudMuon.jpg

Length Contraction

To relate distances measured by different observers, note that the velocity relative to the earthbound observer in our muon example is given by

\[v = \dfrac{L_0}{\Delta t}. \nonumber \]

The time relative to the earthbound observer is \(Δt\), because the object being timed is moving relative to this observer. The velocity relative to the moving observer is given by

\[v = \dfrac{L}{\Delta \tau}. \nonumber \]

The moving observer travels with the muon and therefore observes the proper time \(\Delta \tau\). The two velocities are identical; thus,

\[\dfrac{L_0}{\Delta t} = \dfrac{L}{\Delta \tau}. \label{eq10} \]

We know that \(\Delta t = \gamma \Delta \tau\) and substituting this into Equation \ref{eq10} gives

\[L = \dfrac{L_0}{\gamma}. \nonumber \]

Substituting for \(γ\) gives an equation relating the distances measured by different observers.

Definition: Lenght Contraction

Length contraction is the decrease in the measured length of an object from its proper length when measured in a reference frame that is moving with respect to the object:

\[L = L_0\sqrt{1 - \dfrac{v^2}{c^2}} \label{contraction} \]

where \(L_0\) is the length of the object in its rest frame, and \(L\) is the length in the frame moving with velocity \(v\).

If we measure the length of anything moving relative to our frame, we find its length L to be smaller than the proper length \(L_0\) that would be measured if the object were stationary. For example, in the muon’s rest frame, the distance Earth moves between where the muon was produced and where it decayed is shorter than the distance traveled as seen from the Earth’s frame. Those points are fixed relative to Earth but are moving relative to the muon. Clouds and other objects are also contracted along the direction of motion as seen from muon’s rest frame.

Thus, two observers measure different distances along their direction of relative motion, depending on which one is measuring distances between objects at rest.

But what about distances measured in a direction perpendicular to the relative motion? Imagine two observers moving along their x -axes and passing each other while holding meter sticks vertically in the y -direction. Figure \(\PageIndex{3}\) shows two meter sticks M and M' that are at rest in the reference frames of two boys S and S', respectively. A small paintbrush is attached to the top (the 100-cm mark) of stick M'. Suppose that S' is moving to the right at a very high speed v relative to S, and the sticks are oriented so that they are perpendicular, or transverse, to their relative velocity vector. The sticks are held so that as they pass each other, their lower ends (the 0-cm marks) coincide. Assume that when S looks at his stick M afterwards, he finds a line painted on it, just below the top of the stick. Because the brush is attached to the top of the other boy’s stick M', S can only conclude that stick M' is less than 1.0 m long.

A skateboarder moving to the right with velocity v is holding a ruler vertically. The bottom of the ruler is labeled as zero, and its top as 100 cm. A paintbrush is attached to the upper end of the ruler. The skateboarder is labeled S prime and his ruler is labeled M prime. To the skateboarder’s right stands a boy holding a vertical 100 cm ruler at the same height as the skateboarder’s ruler. The stationary boy is labeled S and his ruler is labeled M.

Now when the boys approach each other, S', like S, sees a meter stick moving toward him with speed v . Because their situations are symmetric, each boy must make the same measurement of the stick in the other frame. So, if S measures stick M' to be less than 1.0 m long, S' must measure stick M to be also less than 1.0 m long, and S' must see his paintbrush pass over the top of stick M and not paint a line on it. In other words, after the same event, one boy sees a painted line on a stick, while the other does not see such a line on that same stick!

Einstein’s first postulate requires that the laws of physics (as, for example, applied to painting) predict that S and S', who are both in inertial frames, make the same observations; that is, S and S' must either both see a line painted on stick M, or both not see that line. We are therefore forced to conclude our original assumption that S saw a line painted below the top of his stick was wrong! Instead, S finds the line painted right at the 100-cm mark on M. Then both boys will agree that a line is painted on M, and they will also agree that both sticks are exactly 1 m long. We conclude then that measurements of a transverse length must be the same in different inertial frames .

Example \(\PageIndex{4}\): Calculating Length Contraction

Suppose an astronaut, such as the twin in the twin paradox discussion, travels so fast that \(\gamma = 30.00\). (a) The astronaut travels from Earth to the nearest star system, Alpha Centauri, 4.300 light years (ly) away as measured by an earthbound observer. How far apart are Earth and Alpha Centauri as measured by the astronaut? (b) In terms of c , what is the astronaut’s velocity relative to Earth? You may neglect the motion of Earth relative to the sun (Figure \(\PageIndex{4}\)).

imageedit_9_2293138876.png

First, note that a light year (ly) is a convenient unit of distance on an astronomical scale—it is the distance light travels in a year. For part (a), the 4.300-ly distance between Alpha Centauri and Earth is the proper distance \(L_0\), because it is measured by an earthbound observer to whom both stars are (approximately) stationary. To the astronaut, Earth and Alpha Centauri are moving past at the same velocity, so the distance between them is the contracted length L . In part (b), we are given \(\gamma\), so we can find \(v\) by rearranging the definition of \(\gamma\) to express \(v\) in terms of \(c\).

Solution for (a)

For part (a):

  • Identify the knowns: \(L_0 = 4.300 \, ly\);\(\gamma = 30.00.\)
  • Identify the unknown: L .
  • Express the answer as an equation: \(L = \dfrac{L_0}{\gamma}\).

\[\begin{align*} L &= \dfrac{L_0}{\gamma} \\[4pt] &= \dfrac{4.300 \, ly}{30.00} \\[4pt] &= 0.1433 \, ly. \end{align*} \nonumber \]

Solution for (b)

For part (b):

  • Identify the known: \(\gamma = 30.00\).
  • Identify the unknown: v in terms of c .

\[\gamma = \dfrac{1}{\sqrt{1 - \dfrac{v^2}{c^2}}}. \nonumber \]

\[\begin{align*} \gamma^2 &= \dfrac{1}{1 - \dfrac{v^2}{c^2}} \\[4pt] \dfrac{v^2}{c^2} &= 1 - \dfrac{1}{\gamma^2} \\[4pt] \dfrac{v}{c} &= \sqrt{1 - \dfrac{1}{\gamma^2}}. \end{align*} \nonumber \]

\[\begin{align*} \dfrac{v}{c} &= \sqrt{1 - \dfrac{1}{\gamma^2}} \\[4pt] &= \sqrt{1 - \dfrac{1}{(30.00)^2}} \\[4pt] &= 0.99944\end{align*} \nonumber \]

\[v = 0.9994 \, c. \nonumber \]

Significance: Remember not to round off calculations until the final answer, or you could get erroneous results. This is especially true for special relativity calculations, where the differences might only be revealed after several decimal places. The relativistic effect is large here (\(\gamma = 30.00\)), and we see that v is approaching (not equaling) the speed of light. Because the distance as measured by the astronaut is so much smaller, the astronaut can travel it in much less time in her frame.

People traveling at extremely high velocities could cover very large distances (thousands or even millions of light years) and age only a few years on the way. However, like emigrants in past centuries who left their home, these people would leave the Earth they know forever. Even if they returned, thousands to millions of years would have passed on Earth, obliterating most of what now exists. There is also a more serious practical obstacle to traveling at such velocities; immensely greater energies would be needed to achieve such high velocities than classical physics predicts can be attained. This will be discussed later in the chapter.

Why don’t we notice length contraction in everyday life? The distance to the grocery store does not seem to depend on whether we are moving or not. Examining Equation \ref{contraction}, we see that at low velocities ( \(v \ll c\)), the lengths are nearly equal, which is the classical expectation. However, length contraction is real, if not commonly experienced. For example, a charged particle such as an electron traveling at relativistic velocity has electric field lines that are compressed along the direction of motion as seen by a stationary observer (Figure \(\PageIndex{5}\)). As the electron passes a detector, such as a coil of wire, its field interacts much more briefly, an effect observed at particle accelerators such as the 3-km-long Stanford Linear Accelerator (SLAC). In fact, to an electron traveling down the beam pipe at SLAC, the accelerator and Earth are all moving by and are length contracted. The relativistic effect is so great that the accelerator is only 0.5 m long to the electron. It is actually easier to get the electron beam down the pipe, because the beam does not have to be as precisely aimed to get down a short pipe as it would to get down a pipe 3 km long. This, again, is an experimental verification of the special theory of relativity.

An electron is shown traveling with a horizontal velocity v in a tube. The electric field lines point toward the electron, but are compressed into a cone above and below the electron.

Exercise \(\PageIndex{1}\)

A particle is traveling through Earth’s atmosphere at a speed of \(0.750c\). To an earthbound observer, the distance it travels is 2.50 km. How far does the particle travel as viewed from the particle’s reference frame?

\[L = L_0\sqrt{1 - \dfrac{v^2}{c^2}} = (2.50 \, km)\sqrt{1 - \dfrac{(0.750c)^2}{c^2}} = 1.65 \, km \nonumber \]

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  1. Comoving and proper distances

    Comoving distance and proper distance. Comoving distance is the distance between two points measured along a path defined at the present cosmological time. For objects moving with the Hubble flow, it is deemed to remain constant in time. The comoving distance from an observer to a distant object (e.g. galaxy) can be computed by the following ...

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  3. Measuring Distances in Cosmology

    This light has been travelling for 13.4 billion light years to reach us. So, the light travel distance to GN-z11 is 13.4 billion light years. If we could construct an imaginary ruler between the Milky Way and GN-z11 it would be 32 billion light years long. So, its current proper distance is 32 billion light years. The Comoving distance

  4. The difference between comoving and proper distances in defining the

    The 1st graph depicts proper distance as a function of time, and the 2nd depicts comoving distance as a function of time. The 3rd graph is essentially the same as the second, but the time axis is rescaled in such a way that light rays are moving on straight lines at $45^\circ$ angles.

  5. PDF Distances in Cosmology

    In a flat universe, the proper distance to an object is just its coordinate distance, s(t) = a(t) · r. Because sin−1(x) > x and sinh−1(x) < x, in a closed universe (k > 0) the proper distance to an object is greater than its coordinate distance, while in an open universe (k < 0) the proper distance to an object is less than its

  6. PDF Distance measures in cosmology

    The comoving distance happens to be equivalent to the proper motion distance (hence the name DM), de ned as the ratio of the actual transverse velocity (in distance over time) of an object to its proper motion (in radians per unit time) (Weinberg, 1972, pp 423{424). The proper motion distance is plotted in Figure 1.

  7. Distance measure

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  8. Cosmic Distances

    A light year is the distance a photon of light travels in one year, which is about 6 trillion miles (9 trillion kilometers, or 63,000 AU). Put another way, a light year is how far you'd travel in a year if you could travel at the speed of light, which is 186,000 miles (300,000 kilometers) per second.

  9. 6.2: Relation Between Events- Timelike, Spacelike, or Lightlike

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    A fourth distance is based on the light travel time: D ltt = c*(t o-t em). People who say that the greatest distance we can see is c*t o are using this distance. But D ltt = c*(t o-t em) is not a very useful distance because it is very hard to determine t em, the age of the Universe at the

  11. Light Travel Time Distance

    In fact the Hubble law with time variable Hubble parameter is satisfied exactly at all times by the metric radial distance D (t) which is the spatial separation at the common time t, so. "velocity" = dD/dt = H (t)D (t). This is not true for the light travel time distance. The rate of change of the light travel time distance with observation ...

  12. 28.3: Length Contraction

    First note that a light year (ly) is a convenient unit of distance on an astronomical scale—it is the distance light travels in a year. For part (a), note that the 4.300 ly distance between the Alpha Centauri and the Earth is the proper distance \(l_0\), because it is measured by an Earth-bound observer to whom both stars are (approximately ...

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    In principle, if we measure distances and redshifts for objects at a variety of distances we could then infer a(t) and k. The general relationship between redshift and luminosity distance is contained in these equations: c∫1 ae da a2H = ∫d 0 dr √1 − kr2. and. dlum = d(1 + z) with 1 + z = 1 / ae.

  14. 28.3 Length Contraction

    First note that a light year (ly) is a convenient unit of distance on an astronomical scale—it is the distance light travels in a year. For part (a), note that the 4.300 ly distance between the Alpha Centauri and the Earth is the proper distance L 0 L 0 size 12{L rSub { size 8{0} } } {} , because it is measured by an Earth-bound observer to ...

  15. Speed of Light Calculator

    The final step is to calculate the total distance that the light has traveled within the time. You can calculate this answer using the speed of light formula: distance = speed of light × time. Thus, the distance that the light can travel in 100 seconds is 299,792,458 m/s × 100 seconds = 29,979,245,800 m. FAQs.

  16. Distance Measures in Cosmology

    4. COMOVING DISTANCE (LINE-OF-SIGHT) A small comoving distance D C between two nearby objects in the Universe is the distance between them which remains constant with epoch if the two objects are moving with the Hubble flow. In other words, it is the distance between them which would be measured with rulers at the time they are being observed (the proper distance) divided by the ratio of the ...

  17. Distances in Cosmology and Theory of Relativity

    The proper distance (or cosmological proper distance, physical distance, ordinary distance) is a distance between two nearby events in the frame in which they occur at the same time. It is the distance measured by a ruler at the time of observation. ... The light travel distance (or LTD) is a distance from us to a distant object, ...

  18. How do you calculate comoving distance and light's travel distance

    Yes, you multiply those integrals by the Hubble distance. It's like a cosmological base distance. You generally can't calculate those integrals by algebra, you have to use a numerical method, like Simpson's rule.

  19. 5.4 Length Contraction

    First, note that a light year (ly) is a convenient unit of distance on an astronomical scale—it is the distance light travels in a year. For part (a), the 4.300-ly distance between Alpha Centauri and Earth is the proper distance L 0 , L 0 , because it is measured by an earthbound observer to whom both stars are (approximately) stationary.

  20. Distance and Lookback Time

    Rather than using distance units, astronomers measure the redshift (z) of distant objects such as galaxies. z corresponds to the number of years the light from an object has traveled to reach us. ... The highest confirmed redshift is for a galaxy called UDFy-38135539 with a z value of 8.6, which corresponds to a light travel time of about 13.1 ...

  21. 5.5: Length Contraction

    First, note that a light year (ly) is a convenient unit of distance on an astronomical scale—it is the distance light travels in a year. For part (a), the 4.300-ly distance between Alpha Centauri and Earth is the proper distance \(L_0\), because it is measured by an earthbound observer to whom both stars are (approximately) stationary.

  22. Ned Wright's Light Travel Time Converter

    The light travel time was 9.283 Gyr. The comoving radial distance, which goes into Hubble's law, is 4343.2 Mpc or 14.166 Gly. The comoving volume within redshift z is 343.173 Gpc 3 . The angular size distance D A is 1771.5 Mpc or 5.7778 Gly. This gives a scale of 8.588 kpc/". The luminosity distance D L is 10648.1 Mpc or 34.730 Gly. 1 Gyr ...

  23. Distance measures (cosmology)

    Cosmological proper distance. The distance between two points measured along a path defined at a constant cosmological time. The cosmological proper distance should not be confused with the more general proper length or proper distance. Light travel time or lookback time. This is how long ago light left an object of given redshift.