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Light clocks with multidimensional time

A previous post on this subject is here. One reference for this post is V. A. Ugarov’s Special Theory of Relativity (Mir, 1979).

A light clock is a device with an emission-reflection-reception cycle of light that registers the current time and stance in units of cycle length and duration. Consider two identical light clocks, at first in their reference frames at rest, K, K´ (left). Then, as the light clock in K´ moves relative to K with uniform motion at velocity v (right), from K observes the following:

light clocks, at rest & in motion

The left illustration shows one cycle length of the light path (i.e., wavelength), L, and one cycle duration (i.e., period), T, at rest in reference frames K, K´ (left). For the reference frame K´, in motion relative to reference frame K, call the arc length of one cycle of the light path x<. Call the distance between the beginning and ending points of one cycle x. For the reference frame K´ relative to reference frame K, call the arc time of one cycle of the light path t<. Call the distime between the beginning and ending instants of one cycle t.

Following Ugarov: Observing clock time rates in the two frames K and K´ moving relative to each other, one can only compare the reading of one clock time from one frame with readings of several clock times from another frame, because two clock times from different reference frames occur at the same point in space only once. In one of the frames there must be at least two clock times which are supposed to be synchronized. For the sake of definiteness we shall be comparing one clock time, t<, from the frame K´ with two clock times from the frame K, at the point in the beginning and end of a cycle.

Let a clock and a light source be located at the origin O´ of the frame K´. A mirror is set perpendicular to the L axis at the distance L/2 from the light source (and the clock). A light signal is transmitted from the source to the mirror from which it is reflected back and returns to the origin point O´ with the period TL/c. Both the light source and the mirror are at rest in the frame K´ and the signal travels there and back along the same straight line.

Now let us consider the propagation of the same signal in the frame K relative to which the source and the mirror move to the right together with the frame K´ at the velocity v. Although the signal was sent from the two coincident origins, O and O´, the reflection from the mirror will occur at another point x/2 of the frame K and the reception of the reflected signal at the point x of the axis. In this way the path of the signal in the frame K traces out two sides of an equilateral triangle.

As the path travelled by light in the frame K is greater than that in the frame K´, one can expect that the period T between the sending and reception of the signal, when measured in the frame K, will be greater than t. Indeed, the observer from the frame K will certify that the two events, i.e. the emitting of light from the origin O´ and its return to the origin O´, occur at the two different points of space. The period T between these two events in the frame K will be measured in this case by the two clocks removed from each other by the distance vt along the motion direction. The velocity of light is equal to c in all reference frames. Therefore, we obtain:

(x</2)² = (ct</2)² = (vt/2)² + (L/2)².

Given t< = t and collecting t< from this equation, we get

t<²(1 − v²/c²) = (L/c)²,

  t<= (L/c)/√(1 − v²/c²) = γ (L/c)

where γ = 1/√(1 − v²/c²).

Considering that L/c = T, then

t< = γ T.

Since both events occurred at the same point in the frame K´, they were registered by means of the same clock. A time interval between events registered by means of the same clock (which implies that the events occurred at the same point of space) is referred to as a proper-time interval between these events. Of course, a time interval of which the initial and the final moments are registered at different points of the reference frame and, consequently, by means of different clocks will not be a proper-time interval between events.


Following Ugarov but with Euclidean time: Observing clock stance rates in the two frames K and K´ moving relative to each other, one can only compare the reading of one clock stance from one frame with readings of several clock stances from another frame, because two clock stances from different reference frames occur at the same instant in time only once. In one of the frames there must be at least two clock stances which are supposed to be synstancized. For the sake of definiteness we shall be comparing one clock stance x from the frame K´ with two clock stances from the frame K, at the instant of the first and last points of a cycle.

Let a clock and a light source be chronated at the origin instant O´ of the frame K´. A mirroring event occurs parallel to the t⊥ axis at the distime T/2 from the light source (and the clock) perpendicular to the relative motion. A light signal is transmitted from the source to the mirror from which it is reflected back and returns to the origin instant O´ with the wavelength L = cT. Both the light source and the mirror are at rest in the frame K´ and the signal travels there and back along the same straight line.

Now let us consider the propagation of the same signal in the frame K relative to which the source and mirror move to the right together with the frame K´ at the velocity v. Although the signal was sent from the two coincident origin instants, O and O´, the reflection from the mirror will occur at another instant t/2 of the frame K and the reception of the reflected signal at the instant t. In this way the path of the signal in the frame K traces out two sides of an equilateral triangle.

As the time path travelled by light in the frame K is greater than that in the frame K´, one can expect that the wavelength L between the sending and reception of the signal, when measured in the frame K, will be greater than x<. Indeed, the observer from the frame K will certify that the two events, i.e. the emitting of light from the origin O´ and its return to the origin O´, occur at the two different instants of time. The wavelength L between these two events in the frame K will be measured in this case by the two clock stances removed from each other by the distime x/v along the motion direction. The velocity of light is equal to c in all reference frames. Therefore, we obtain:

(t</2)² = (x</2v)² = (x/2c)² + (T/2)².

Given x< = x and collecting x< from this equation, we get

x<²(1 − v²/c²) = (cT)²,

x<= (cT)²/√(1 − v²/c²) = γ (cT),

where γ = 1/√(1 − v²/c²).

Considering that cT = L, then

x< = L/γ.

Since both events occurred at the same instant in the frame K´, they were registered by means of the same clock stance. A length interval between events registered by means of the same clock stance (which implies that the events occurred at the same instant of time) is referred to as a proper-length interval between these events. Of course, a length interval of which the initial and the final moments are registered at different instants of the reference frame and, consequently, by means of different clock stances will not be a proper-length interval between events.

~

The moving light clock has x< = vt. and x< = ct<. Note: if c = ∞, then t|| = 0; x< = x; and t< = t. If v = 0, then t = 0; x< = x||; and t< = t||.

From the Euclidean metric for space we have: (x</2)² = (x/2)² + (x||/2)² . Combine this with the above to get:

(ct</2)² = (vt/2)² + (ct||/2)².

Divide by c² to get:

(t<)² = (βt)² + (t||)².

If x< = x above, then

(x<)² = (x</β)² + x||², or

(x<)² (1 − 1/β²) = x||².

From the Euclidean metric for time we have: (t2)² = (t)² + (t||)². Combine this with the above to get:

(x</c)² = (x/v)² + (x||/c)².

Multiply by c² to get:

(x<)² = (x/β)² + x||²,

which is a weighted metric.

Can we infer x|| = ct||?

Basic Gospel

Basic Gospel Message

Vic Scaravilli is a Catholic who put the following on his website here. I’m reposting it (with permission) and note that Evangelicals would agree that this is the gospel, with some nuances about baptism.

The Basic Gospel Message

By Vic Scaravilli

God loves each one of us. He loves me and He loves you with an unconditional love. You are precious in His eyes.

There is something that has kept us separated from God, something that has kept us from experiencing His love in our lives. That something is called sin. The result of sin is spiritual death. We have all sinned and never can be perfect.

Does that mean we can never know and experience God’s love? No, because God loved us so much He sent His only Son to die for each one of us. Jesus is the only bridge that takes us from our sin to the love of God. By His death and resurrection, Jesus opened the gates of heaven for everyone.

This is called salvation. It is the free gift of eternal life that is completely given to us by God’s grace. Salvation is the life in Jesus that begins now and will be for all eternity.

The free gift of salvation must be accepted in order for it to be our own. We must experience an internal conversion experience that changes our hearts. Once we accept Jesus into our hearts and allow Him to come into our lives, we begin to experience His love.

Read more →

Synopsis of the Gospel

A previous post here gave a summary of the Gospel. The following comes from Rev. David Harper’s blog entry, The power of story:

Here’s a synopsis.

1. God created humankind in His image for fellowship and partnership, entrusting to us stewardship of His earth. (Gen. 1:28)  

2. Because of sin, in which we all participate, our fellowship with God and one another has fractured (Gen. 3:1-19, 4:8; Rom. 3:23).

3. God sent His Son, Jesus, as the promised Messianic King and Son of God, come to earth in human form to become one with us. (Rom. 1:3-4; Phil. 2:4ff.).

4. By his death and resurrection, Jesus atoned for our sin, and secured our justification by grace, (1 Cor. 15:3ff.). He has broken the dominion of sin and evil over us (Col. 2:13-15), restored us to right relationship with the Father, and made us the firstfruits of His new creation. (James 1:18)

 5. He has given us His Holy Spirit to empower us to do the works that Jesus did, enlisting us in His plan and purpose to make the whole creation new. (John 14:12ff, Acts 2:1ff, Eph. 1:9-10, 3:8-12

6. At his return, Jesus will complete what he began by the renewal of the entire material creation, and the resurrection of our bodies (Rom 8: 18ff.).)

For more from Rev. Harper, see his website Things New & Old. One thing he is known for is expressed in his talk on Three Streams, One River.

Basic Gospel Message

Vic Scaravilli is a Catholic who put the following on his website here. I’m reposting it (with permission) and note that Evangelicals would agree that this is the gospel, with some nuances about baptism.

The Basic Gospel Message

By Vic Scaravilli

God loves each one of us. He loves me and He loves you with an unconditional love. You are precious in His eyes.

There is something that has kept us separated from God, something that has kept us from experiencing His love in our lives. That something is called sin. The result of sin is spiritual death. We have all sinned and never can be perfect.

Does that mean we can never know and experience God’s love? No, because God loved us so much He sent His only Son to die for each one of us. Jesus is the only bridge that takes us from our sin to the love of God. By His death and resurrection, Jesus opened the gates of heaven for everyone.

This is called salvation. It is the free gift of eternal life that is completely given to us by God’s grace. Salvation is the life in Jesus that begins now and will be for all eternity.

The free gift of salvation must be accepted in order for it to be our own. We must experience an internal conversion experience that changes our hearts. Once we accept Jesus into our hearts and allow Him to come into our lives, we begin to experience His love.

In order to become children of God, we must be born again. The sacrament of Baptism forgives your sins, gives the gift of the Holy Spirit, restores the grace lost by Adam, and makes you a member of His family.

This is the gift of salvation. We have the opportunity of eternal life because of what Jesus did for you and me. All we have to do is personally accept it by faith and be obedient to Him. Baptism and conversion are required to begin this process.

Jesus paid a debt He did not owe because we owed a debt we couldn’t pay.