Historic question regarding SR and c*t

[Trysse]

TL;DR Summary

Was Einstein the first to multiply time with the speed of light to give it the same unit of measurement as spatial distance?

This is a historical question regarding SR. Has any physicist before EInstein multiplied the time coordinate with the speed of light? If yes who was it? What is a good book or paper that discusses why and how Einstein (or others before him) have come up with the idea to multiply c and t?

When Einstein introduced c*t as a unit for the time coordinate in 1905, did he do so only because it allowed him to treat spatial and temporal separation as somehow mathematically equivalent? Or did he already have in mind, to treat time as a "quasi spatial" dimension?

 

[DrClaude]

Has any physicist before EInstein multiplied the time coordinate with the speed of light?

Oh yes, and long before!

You can start by reading https://en.wikipedia.org/wiki/History_of_Lorentz_transformations

 

[Ibix]

Also worth noting that Einstein introduced $ct$ because he wanted to know how far light traveled in the time interval $t$. That this turns out to have a deeper significance ($c$ as a natural conversion factor between units of time and distance) only became clear later, as far as I understand.

 

[vanhees71]

I think the first hint at a relativistic spacetime model was with Woldemar Voigt in the late 1880ies, when he found a transformation involving space and time which left the (free?) Maxwell equations invariant, which was pretty close to the Lorentz transformation, but he took this more as a mathematical curiosity than a physically significant discovery. Also Heaviside pretty early thought in this direction. He (as well as Sommerfeld) derived also the radiation of a faster-than-light electron. As we know today that makes no sense for particles moving in a vacuum, but the math can be used for particles moving in a medium with a refractive index $n>1$, moving faster than the phase velocity of light in this medium, leading to Cherenkov radiation.

 

[robphy]

---

When Einstein introduced c*t as a unit for the time coordinate in 1905, did he do so only because it allowed him to treat spatial and temporal separation as somehow mathematically equivalent? Or did he already have in mind, to treat time as a "quasi spatial" dimension?

Do you have the specific 1905 reference for this claim?

• https://en.wikisource.org/wiki/On_the_Electrodynamics_of_Moving_Bodies_(1920_edition)
  
  In § 3. Theory of Co-ordinate and Time-Transformation from a stationary system to a system which moves relatively to this with uniform velocity.

To every value of (x, y, z, t) which fully determines the position and time of an event in the stationary system,

I don't see (x, y, z, ct) or (x/c, y/c, z/c, t).​

Certainly in an equation, one may see a combination like x-ct or t-x/c,
as found in (say) the wave equation, associated with folks like d'Alembert (1747) and Gauss (1766).
Use of such combinations in an equation doesn't mean Einstein introduced "ct" as a "time coordinate".

• In Minkowski's 1907 "Space and Time",

Through the world postulate an identical treatment of the four identifying quantities x, y, z, t becomes possible.
...
The cone $$c^2t^2-x^2-y^2-z^2=0$$...

---

"ct" might be a product (pun intended) of Sommerfeld.

• In an edition of
  The Principle of Relativity: A Collection of Original Memoirs on the Special and General Theory of Relativity
  by Lorentz, Einstein, Minkowski, and Weyl
  with Notes by Sommerfeld
  and translated by Perrett and Jeffery (Dover, 1923),
  a footnote on a page 92 says

* Sommerfeld seems to take ct as a co-ordinate in the graph in place of t as used by Minkowski.-TRANS.

---

While Einstein should be rightfully credited with developing Special Relativity,
together with its physical interpretation,
other developments like the notion of "spacetime", its geometric interpretation, and geometrized units
were developments due to
others like Minkowski, Sommerfeld, Robb, etc... (that is, it wasn't all due to Einstein).

 

[robphy]

In this paper by Minkowski
https://de.wikisource.org/wiki/Das_Relativitätsprinzip_(Minkowski)
it appears that Minkowski sets (quoting from Minkowski) "$c=1$",
then introduces the "imaginary time"-related coordinate $x_4=it$.

In context,
copy-pasting and cleaning up some LaTeX... (I didn't edit any words or characters)

Die Zeiteinheit mag so gewählt werden, daß $c=1$ wird, d. h. also $1/3.10^{10}$ Sek. bei der Längeneinheit 1 cm. Es soll nun ${\displaystyle x_{1},\ x_{2},\ x_{3}}$ statt x, y, z geschrieben werden, und ferner soll ${\displaystyle x_{4}}$ für it gesetzt werden. Es ist dann natürlich ${\displaystyle x_{4}}$ I am folgenden immer eine rein imaginäre Größe. Jener quadratische Ausdruck geht in die Form

$${\displaystyle x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}}$$

and Google translate gives...
copy-pasting and cleaning up some LaTeX... (I didn't edit any words or characters)

The time unit may be chosen so that ${\displaystyle c=1}$ will, ie so ${\displaystyle 1/3.10^{10}}$ sec. in the unit of length 1 cm. It should now ${\displaystyle x_{1},\ x_{2},\ x_{3}}$ instead of x, y, z should be written, and furthermore should ${\displaystyle x_{4}}$ be set for it . It's natural then ${\displaystyle x_{4}}$ in the following always a purely imaginary quantity. That square expression goes into shape

$${\displaystyle x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}}$$

 

[Histspec]

Let's not forget Poincaré (july 1905, published 1906):
https://en.wikisource.org/wiki/Translation:On_the_Dynamics_of_the_Electron_(July)
page 168:

We regard
$$\begin{array}{ccccccc} x, & & y, & & z, & & t\sqrt{-1},\\ \\\delta x, & & \delta y, & & \delta z, & & \delta t\sqrt{-1},\\ \\\delta_{1}x, & & \delta_{1}y, & & \delta_{1}z, & & \delta_{1}t\sqrt{-1},\end{array}$$

as the coordinates of three points P, P', P" in a 4-dimensional space. We see that the Lorentz transformation is a rotation of that space around the origin, regarded as fixed.

(Note that he set c=1)

 

[Trysse]

Do you have the specific 1905 reference for this claim?

It wasn't meant as claim. But I take the point: It was a sloppy formulation that had an (implicit and) unjustified reference to a source.

So, could I say/write:

In physics, events are often referenced by a four-dimensional tuple (t,x,y,z). The first coordinate describes when an event takes place expressed in the time elapsed since some other event. The last three coordinates describe the location of the event in a Cartesian coordinate system expressed in spatial distances. In the current interpretation of the Theory of Special Relativity, the time coordinate is treated as a quasi-space-like distance. This is achieved by multiplying time by the speed of light, expressing a temporal separation between two events as if it was a spatial displacement.

Although Albert Einstein’s paper “On the Electrodynamics of Moving Bodies” is considered to be the foundation of modern Special Relativity, he didn’t treat spatial and temporal separation as equivalent in his paper. In his paper, temporal separation is still what can be measured with clocks and spatial separation is what can be measured with rulers.

The feat of unifying time and space can be attributed to Herman Minkowski. He most prominently made this connection in his 1907 presentation “Space and time”.

 

[martinbn]

...
Gauss (1766)
...

Sorry for the off topic, but even Gauss wasn't that good to study the wave equation 11 years before he was born.

 

[robphy]

Sorry for the off topic, but even Gauss wasn't that good to study the wave equation 11 years before he was born.

Oops… I think I meant Euler.

 

[topsquark]

Sorry for the off topic, but even Gauss wasn't that good to study the wave equation 11 years before he was born.

With Gauss nothing surprises me.

-Dan

 

[Ibix]

In the current interpretation of the Theory of Special Relativity, the time coordinate is treated as a quasi-space-like distance.

No. Coordinates and distances are different things.

This is achieved by multiplying time by the speed of light, expressing a temporal separation between two events as if it was a spatial displacement.

No. It just enables us to use the same units for time intervals and spacelike intervals. There is still a clear distinction between spacelike and timelike separations imposed by the metric.

In his paper, temporal separation is still what can be measured with clocks and spatial separation is what can be measured with rulers.

This is still the case in any interpretation of relativity, including Minkowski's. You can't make a ruler measure time.

 

[DrGreg]

In the current interpretation of the Theory of Special Relativity, the time coordinate is treated as a quasi-space-like distance. This is achieved by multiplying time by the speed of light, expressing a temporal separation between two events as if it was a spatial displacement.

That is indeed the way that 4D spacetime is usually introduced. The coordinates are $(ct,\, x,\, y,\, z)$ and the metric components are
$$ g_{\mu\nu} =
\begin{pmatrix}
-1 & 0 & 0 &0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}
$$But it doesn't have to be that way. You could use coordinates $(t,\, x,\, y,\, z)$ with metric components
$$ g_{\mu\nu} =
\begin{pmatrix}
-c^2 & 0 & 0 &0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}
$$But nobody seems to do it that way. And once you get beyond introductory relativity, most people adopt the convention of units in which $c=1$ and so the difference between the two approaches vanishes.

I guess the logic is that beginners will feel more comfortable with all four dimensions having the same units, though I think it may promote a misconception that space (not spacetime) has a hidden fourth dimension through which we move as time (which they incorrectly perceive as a 5th independent variable) progresses.