tag:blogger.com,1999:blog-38420735732300054492024-03-23T03:15:14.338-07:00Erik W. Lentz's Personal Website Blog - on warp drive, dark matter, and moreA blog about axions, warp drive, and thoughts on science and technology by Erik W. Lentz. This blog is meant to contribute to promoting an interest in science.Unknownnoreply@blogger.comBlogger12125tag:blogger.com,1999:blog-3842073573230005449.post-14506004688243751272022-01-16T13:38:00.001-08:002022-01-21T21:47:06.333-08:00The Horizon Problem for Faster than Light Travel<p>Hello all,</p><p>As I have mentioned in previous posts, a major hurdle for autonomous FLT travel is the horizon problem. But what is a horizon? Is it some single well-defined concept or a catch-all term for many phenomena? In this post, I present some thoughts on horizons in general relativity theory and their implications for FTL travel using warp drives.</p><p>A horizon is a limit to the region that can be observed or communicated with in a space. On Earth, horizons can be used to refer to edge of what can be seen when looking out over a vast body of water like an ocean. The limit to what can be seen in this case is caused by the curvature of the Earth's surface and is a cute problem to calculate how distant the horizon appears given the height of an observer above the surface of a spherical Earth (ignoring effects of the Earth's atmosphere) that is straightforward to solve as it is embedded in a Euclidean geometry. </p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEg9DaywiyXW2bHZtzbuLQESkFwGYUuDRLMrRcJwVSzFloGZ714aOr8FNRpDKlvYgXaW513zQDT65ictkTIXfG3RGUFcImePYp__qLiR8N8mXleZ4BnG99D1o7KDVlWLd7RLGU2R0n0SHGZNNxxt2vvzPU875tlYcRJdQnz-1ZG_pTtrA3X5G0xvOlpn=s542" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="515" data-original-width="542" height="304" src="https://blogger.googleusercontent.com/img/a/AVvXsEg9DaywiyXW2bHZtzbuLQESkFwGYUuDRLMrRcJwVSzFloGZ714aOr8FNRpDKlvYgXaW513zQDT65ictkTIXfG3RGUFcImePYp__qLiR8N8mXleZ4BnG99D1o7KDVlWLd7RLGU2R0n0SHGZNNxxt2vvzPU875tlYcRJdQnz-1ZG_pTtrA3X5G0xvOlpn=s320" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;"><span style="font-size: x-small;">An observer looking out to the Earth's horizon from a height h above the surface.</span></div><p>In relativity theory, a horizon is also a boundary on the region which an observer can see. <b>Horizons pose an issue for superluminal warp drives as all known warp bubbles create horizons at superluminal speeds, implying that one cannot control the bubble to accelerate or decelerate it or pass communications between observers inside and outside the bubble once it is traveling faster than light. This is the essence of the horizon problem for FTL travel.</b> There are several different types of horizons that are commonly found in GR space-times. Here are several and some of their implications:</p><p><b>Light Cones:</b></p><p>This is not considered a true horizon, but I am mentioning it to better distinguish true horizons. A past light cone indicates the region of space-time an observer can see <b>at one moment</b>, as opposed to a more strict horizon which determines what all observers in a region of space-time can ever see. Light cones about a space-time event are defined as the points in space-time connected to that event by incoming light rays. The cone term comes from the shape of the region close to the point of observation when considering a flat 3 dimensional space-time (see figure). In 4 dimensional space-time, the cone takes the form of a sphere. Light ray trajectories are given by light-like, or null, vectors, the name given to non-zero vectors <b>X</b> that have zero (null) inner product via the space-time metric (g(<b>X</b>,<b>X</b>) = 0).</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjQip7ItpIBATUBdT7BO4qfTAlRMccCA7UnYblOQEIq-xviBOqJpGkf_1ndBsxEF0F5D_984zaU3mzb5Hd7DMwbnaX26QjYmx9zQGCEvFd6CvEGn9L0mibLwIi3iKBCdcevhuVIkSV2_JTMzYtYPTxTHOBHjhKIpirhBQPxdwS9o__m2inolb3AtILi=s477" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="464" data-original-width="477" height="311" src="https://blogger.googleusercontent.com/img/a/AVvXsEjQip7ItpIBATUBdT7BO4qfTAlRMccCA7UnYblOQEIq-xviBOqJpGkf_1ndBsxEF0F5D_984zaU3mzb5Hd7DMwbnaX26QjYmx9zQGCEvFd6CvEGn9L0mibLwIi3iKBCdcevhuVIkSV2_JTMzYtYPTxTHOBHjhKIpirhBQPxdwS9o__m2inolb3AtILi=s320" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;"><span style="text-align: left;"><span style="font-size: xx-small;"><a href="http://creativecommons.org/licenses/by-sa/3.0/" target="_blank">Stib at en.wikipedia, CC BY-SA 3.0</a> , via Wikimedia Commons</span></span></div><p>A light cone varies from moment to moment for a time-like observer such as a person. It is not meaningful on its own for differentiating what is contactable from what is not for observers moving with constant velocity. Case in point: a conversation using light beams can occur between any two time-like inertial observers in a flat space-time, it just might take a while.</p><p>The story is more complex for curved space-times, which is where we will find true horizons.</p><p><b>Killing Horizon:</b></p><p>Our first type of horizon is the Killing horizon. A Killing horizon (named after Wilhelm Killing) is defined as a null hypersurface (space of lesser dimension than the full space-time where all tangent vectors are light-like) given by the vanishing of the (metric) norm of a Killing vector field (also named after W. Killing). This definition appears similar to the light cone definition above extended for curved space-times, but is actually much more broadly useful as we will see. To understand this, we must learn what a Killing vector field (KVF) is.</p><p>A KVF is a smooth vector field, such as <b>V</b>, that preserves the space-time metric, meaning that the metric is unchanged when shifted in the direction of <b>V</b>. KVFs identify symmetries in a geometry, such as rotational symmetry and time invariance. </p><p>So, a Killing horizon is produced over regions where a KVF's norm vanishes (g(<b>V</b>,<b>V</b>) = 0). Killing horizons can identify when the vectors generating a symmetry of space-time change signature, transitioning from being space-like (g(<b>V</b>,<b>V</b>) > 0, using the "mostly plus" metric sign convention) to time-like (g(<b>V</b>,<b>V</b>) < 0) or vice versa.</p><p>For example, the space-time of a static rotating black hole has a symmetry that looks like time translation far away from the black hole and is generated by a KVF that changes from time-like to space-like in signature as it crosses a horizon called the ergosphere. Observers that cross this horizon will also change signature from time-like to space-like. I will talk more about the ergosphere Killing horizon and other black hole horizons below.</p><p><b>Black Hole Horizons:</b></p><p>The event horizon around spherical Schwarzschild Black Holes is likely the most popularly known horizon in GR. Less widely known is that Black Holes (BHs) can have up to four unique horizons when considering their spin and electric charge. For simplicity, we will limit ourselves to static BHs and their horizons. Dynamic horizons are of interest when considering the lifecycle of a BH, and a warp drive, and I may delve into them more in a future post. I touch on them briefly below in the context of cosmological horizons.</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEiPDzOYRNrdlRxY-_f2Loc5zGJG91fay6uJn1gq77KvVw_8ZX-uOpkvA1ogVAnnuWaaN1pwMyghdureuIG2RE59eGTUB2LgEMsCaNVmrVFq5m38bxCwjad5Vp94DcdfhW0MvbJasbxnfrGHc_DBjmqVbPNLdUpQ4CpR4ikInZ-DI846XZrIDfaUADQO=s692" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="692" data-original-width="595" height="320" src="https://blogger.googleusercontent.com/img/a/AVvXsEiPDzOYRNrdlRxY-_f2Loc5zGJG91fay6uJn1gq77KvVw_8ZX-uOpkvA1ogVAnnuWaaN1pwMyghdureuIG2RE59eGTUB2LgEMsCaNVmrVFq5m38bxCwjad5Vp94DcdfhW0MvbJasbxnfrGHc_DBjmqVbPNLdUpQ4CpR4ikInZ-DI846XZrIDfaUADQO=s320" width="275" /></a></div><div class="separator" style="clear: both; text-align: center;"><span style="font-size: x-small;">Cartoon of charges spinning BH structure, including some of the horizons (Original image found in <a href="http://dx.doi.org/10.3390/galaxies6020061" target="_blank">Tito and Pavlov 2018</a>)</span></div><div><br /></div><div>The horizons are determined where the radial or time components of the space-time metric (g<span style="font-size: x-small;">rr</span> and g<span style="font-size: x-small;">tt)</span> invert in sign (passing through zero). After a convenient choice of units, finding the zero points to the g<span style="font-size: x-small;">rr</span> and g<span style="font-size: x-small;">tt</span> metric components is equivalent to solving the respective equations:</div><div><br /></div><div style="text-align: center;">r<sup>2</sup> + e<sup>2</sup> + a<sup>2</sup> - 2 M*r = 0</div><div style="text-align: center;">r<sup>2</sup> + e<sup>2</sup> + a<sup>2</sup> Cos<sup>2</sup> θ - 2 M*r = 0</div><div style="text-align: center;"><br /></div><div>where M represents the total mass of the BH, e the total electric charge, and a is a parameter representing the spin of the BH. Solving for the radial coordinate r shows that horizon surfaces occur at</div><div><br /></div><div style="text-align: center;">r <sup><span style="text-align: left;">±</span></sup><sub>ergo</sub> = M <span style="text-align: left;">±</span> (M<sup>2</sup> - e<sup>2</sup> - a<sup>2</sup> Cos<sup>2</sup> θ)<sup>1/2</sup></div><div style="text-align: center;">r <sup><span style="text-align: left;">±</span></sup><sub>event</sub> = M <span style="text-align: left;">±</span> (M<sup>2</sup> - e<sup>2</sup> - a<sup>2</sup>)<sup>1/2</sup></div><div><br /></div><div>where the "±" indicates a choice in whether to use a plus or a minus. There are up to four unique solutions and horizons to this space-time, depending on the choice of spin and charge parameters. These four horizons also correspond to Killing horizons for KVFs which point in the time and radial directions far away from the BH, in the BH's rest frame of reference. The horizons at which the time-like KVF changes sign are known as the ergospheres, and the horizons at which the radial KVF changes sign are the event horizons. It is also worth noting that the singularity inside the BH is not point like in general, but shaped like a ring.</div><div><br /></div>The outermost horizon is called the (outer) ergosphere, also known as the stationary limit. The ergosphere shape is actually not spherical in general but oblate. Again, the outer ergosphere coincides with a time Killing horizon caused by the metric time component switching signs (negative to positive in our convention), and for a spinning BH is in part influenced by what is called "frame dragging", a geometric effect induced by the BH spin. For a spinning BH with zero electrical charge, the outer edge of the ergosphere signifies where the frame dragging effect exceeds the speed of light. Observers inside the ergosphere must be moving faster than light as measured by observers outside of the BH. Frame dragging is analogous to the effect used to create warp drives space-times. Stationary objects inside the ergosphere will fall further into the BH, but objects co-rotating quickly enough can maintain distance or even escape if they are imparted some thrust. As some objects can escape, the outer ergosphere does not qualify as an event horizon. An event horizon being defined as a boundary beyond which events cannot affect an observer.<div><p>The next horizon is the outer event horizon. This horizon coincides with the outer radial Killing horizon caused by the inversion of the metric's radial component, and implies that between the outer and inner horizons the metric has changed the signature of two of its four principle directions. Objects that are time-like far from the BH and fall past the event horizon have no known means of escape in classical physics. </p><p>The two innermost horizons sit immediately outside of the BH singularity. The inner horizons emulate the outer ergosphere and event horizon shape, but their behavior is opposite: inside the inner ergosphere the metric time component again becomes negative and the time KVF again becomes time-like; the radial metric component again becomes positive and the radial KVF again becomes space-like. The metric principle directions have regained the signature they had outside the BH.</p><p><b>Cosmological Horizons:</b></p><p>The last example here is from cosmology, and pulls together several of the principles we covered above. Models of big bang cosmology include multiple observational horizons set by the physics and history of our expanding universe. The context for this example with be FLRW (<span style="background-color: white; color: #202122;"><span style="font-family: inherit;">Friedmann–Lemaître–Robertson–Walker</span></span><span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;">)</span> model of cosmology where the stress-energy sources are homogeneous (take the same value everywhere at a given time) and isotropic (have no preferred direction), and for our purposes flat (the space portion of the space-time geometry is neither positively curved like a sphere nor negatively curved like a hyperbolic surface, but flat like Eulidean space). The only remaining geometric degree remaining is the time-dependent scale factor of the universe, a(t).</p><p>The near uniform expansion of our universe has served as inspiration for faster than light travel research due to the seemingly cumulative nature of separation speeds as objects become increasingly separated. As an analogy, think of points on a balloon separating as the balloon inflates: the points close together separate less rapidly than points with larger distance between them.</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEj4NoNvY_AWIFUgAP9OWtjmM9xwTw-kCAIG5yRuXGCqw9Js4xsytbYOPq1JPU6acY7RrbARVYYQJQqIAOLqQXajEj_d-yJVsxYyxGvsHDz2d57Q6Izo3F7PPRFmhmv5BpGpW3SVDeSzMkbDo140b9d-2S9ECCwqtu92LalTWSXhyG-QoCa_zNTNKNcc=s789" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="267" data-original-width="789" height="210" src="https://blogger.googleusercontent.com/img/a/AVvXsEj4NoNvY_AWIFUgAP9OWtjmM9xwTw-kCAIG5yRuXGCqw9Js4xsytbYOPq1JPU6acY7RrbARVYYQJQqIAOLqQXajEj_d-yJVsxYyxGvsHDz2d57Q6Izo3F7PPRFmhmv5BpGpW3SVDeSzMkbDo140b9d-2S9ECCwqtu92LalTWSXhyG-QoCa_zNTNKNcc=w622-h210" width="622" /></a></div><div class="separator" style="clear: both; text-align: center;">Boy blowing up balloon to demonstrate cosmological expansion. Image from <a href="https://www.worldcat.org/title/fun-with-astronomy/oclc/4839860&referer=brief_results" target="_blank">Fun with Astronomy</a></div><p>We are currently only able to see objects up to about 46.5 Gly (giga-light-years) away from the Earth, limited by a wall of primordial radiation called the cosmic microwave background (CMB) emitted from a cooling atomic plasma nearly 13.8 Gyr (giga-years) ago, when our universe was an estimated 380,000 years old. We cannot see further back using conventional telescopes as the opacity (optical scattering dense-ness) at times before the CMB is too large for light to have reached us undisturbed. This also known as the surface of last scattering for photons. There are also surfaces of last scattering for other types of radiation that reach beyond the CMB, such as neutrinos and gravitational waves. Though not true horizons, these surfaces are practical horizons as it is effectively impossible to perform observations beyond them using their radiation of choice.</p><p>Cosmologies also contain a general particle horizon, the farthest distance to emissions traveling at light speed could travel to the observer over the history of the universe. Its reach is limited by the lifetime of the universe, and the objects that can or cannot be seen by a particular observer are dependent on the location of the observer, so it is an observer-dependent horizon. Also, the edge of this observable region is not calculated simply by multiplying the age of the universe by the speed of light, otherwise we would only be able to observe to a distance of about 13.8 Gly, with the history of expansion making the particle horizon distance much larger.</p><p>Cosmologies can also contain event horizons. An event horizon to an observer is the distance beyond which we will never see light from. This horizon only exists if the cosmology expands eternally without slowing, as otherwise the horizon would shrink and one would eventually be able to see light emitted from objects at any distance. Note that like the particle horizon, the location of this horizon is still dependent on the location of the observer.</p><p>The last horizon type discussed here is the Hubble sphere (named after Edwin Hubble). The radius of the Hubble sphere is set by the distance at which co-moving objects are receding from us at light speed, meaning everything inside the sphere is co-moving away from us at less than the speed of light and everything outside the sphere is co-moving away faster than the speed of light.</p><p>Figures showing the relative locations of these horizons for an example cosmology <span style="text-align: center;">with 30% dark matter and 70% dark energy is below</span>.</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEidtVRowMim-Ki1crx4O2Lf_iN3s04EOFOHSL50g0z9ipjrW_kW_82enjSWldtvimy03iqpmpyfllrJaKaWh07Sjtsfz_V-E2NwZePKoqgX_B0I16elb0CZFSEU28WYcQzuzyHggPwXxs3BYAN4Vmd5qKQ_BZ_AwhsrOJUqyE-j2tiQianq1gOn9HNT=s1440" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1318" data-original-width="1440" height="516" src="https://blogger.googleusercontent.com/img/a/AVvXsEidtVRowMim-Ki1crx4O2Lf_iN3s04EOFOHSL50g0z9ipjrW_kW_82enjSWldtvimy03iqpmpyfllrJaKaWh07Sjtsfz_V-E2NwZePKoqgX_B0I16elb0CZFSEU28WYcQzuzyHggPwXxs3BYAN4Vmd5qKQ_BZ_AwhsrOJUqyE-j2tiQianq1gOn9HNT=w564-h516" width="564" /></a></div><div class="separator" style="clear: both; text-align: center;"><span style="font-size: x-small;"><br /></span></div><div class="separator" style="clear: both; text-align: center;"><span style="font-size: x-small;">Figures of evolving horizons for a cosmology with 30% dark matter and 70% dark energy. The original annotated version can be found on Dr. Tamara Davis's website <a href="https://people.smp.uq.edu.au/TamaraDavis/research.html" target="_blank">here</a>.</span></div><br /><p>You may have noticed that the event and Hubble horizons shown are no longer growing in proper distance/size (as measured by a fixed ruler) and are shrinking in co-moving size (as measured by rulers that scale with the cosmology's expansion factor a(t)). This is a feature of the cosmological expansion rate and that our universe appears to have entered a period of eternal accelerated expansion. The growing and shrinking of these horizons can be interrupted though if the expansion rate experiences a significant change driven by the stress-energy physics. Such is believed to have happened very early in our universe's history (too early to be visible on the plots), during and immediately following a period of rapid cosmological inflation. The early inflation quickly expanded nearby objects away from each other faster than the speed of light, creating a narrow event horizon and Hubble sphere if inflation had continued forever. But at the end of this early inflationary period, the expansion slowed considerably and the Hubble sphere and event horizon began to grow again, counterintuitively bringing regions that had fallen out of contact with each other back into observable contact again. This is an example of how a horizon can be reversed using the physics of stress-energy, though keep in mind the specialness of the unbounded cosmology.</p><p><b>Implications for Warp Drives:</b></p><p>We have seen that horizons come in several different varieties and have a range of behaviors. We have seen that some horizons can be turned on and off and even escaped, while others are much more strict. So what does this mean for warp drives? </p><p>While we cannot exactly match the horizons we see in the superluminal warp drives of my own paper <a href="https://doi.org/10.1088/1361-6382/abe692" target="_blank">Breaking the Warp Barrier</a> (also at <a href="https://arxiv.org/pdf/2006.07125.pdf">https://arxiv.org/pdf/2006.07125.pdf</a>) or others in the literature, we do see that of the horizons listed above, warp drive horizons most closely resemble the ergosphere of a rotating BH. This should not be too surprising as the warp drive geometries are formed from the same shift vector components of the metric that describe the frame dragging effect of the BH, and the ergosphere is also formed where frame dragging grows to the point where comoving observers exceed the speed of light.</p><p>That the horizon of one of these warp drives can support communication by being entered and escaped like an ergosphere, and can be turned on and off remains to be seen. Detailed calculations will tell, but I take these as promising hints that a future more carefully designed warp drive solution will be controllable at superluminal speeds. </p><div>Happy belated New Year,</div><div><br /></div><div>Erik</div></div><div><br /></div><div><div><b>Update 1/21/22:</b></div><div><b><br /></b></div><div>It was brought to my attention that some readers are having trouble finding a statement of the horizon problem. I have expanded on my non-technical statement of the horizon problem and typeset that text in boldface to make it more visible.</div></div>Erikhttp://www.blogger.com/profile/12492752467374888110noreply@blogger.com18tag:blogger.com,1999:blog-3842073573230005449.post-120685385104900132021-12-07T17:16:00.000-08:002021-12-07T17:16:12.677-08:00Warp Drive Soliton Inertia<p>Hello All,</p><p>In this post I log some thoughts on another proposed challenge to superluminal warp drives: acceleration of the soliton past the speed of light. In their paper published earlier this year, Bobrick and Martire (<a href="https://arxiv.org/abs/2102.06824v2" target="_blank">Introducing Physical Warp Drives</a>) state on page 17 of their pre-print that one cannot accelerate a subluminal warp drive to superluminal speeds for similar reasons as why one cannot accelerate masses moving through space-time to superluminal speeds:</p><blockquote><p><span style="background-color: #ffa400;">Warp drives can move superluminally only in the same sense as any ordinary
inertial mass, test mass, or any other object.</span> Namely, there is no known way of
accelerating regular material beyond the speed of light. However, one may postulate a
test particle which moves faster than light in relativity, in which case it may continue
moving inertially. <span style="background-color: #ffa400;">In the same way, <span>a</span></span><span style="background-color: #ffa400;">s warp drives are shells of material, there is
no known way of accelerating a warp drive beyond the speed of light.</span> However, one
may also postulate the warp drive shell to be in superluminal motion, just like the
hypothetical test particles, and the shell-like object will continue moving in the same
fashion. In this sense, superluminal warp drives are at least as hypothetically possible
as any other superluminal objects.</p></blockquote><p>The crux of the above statement is that a warp soliton acts like a point-like massive test body being moved through space-time, having constant non-vanishing inertia (resistance to acceleration) regardless of velocity. In relativity theory, constant inertia implies that a force acting on an object with set mass over a set period of proper time will produce the same change in spatial momentum. However, as a result of the invariance of the speed of light to non-accelerating reference frames, there is an asymptote in momentum and energy as a body approaches the speed of light relative to another time-like observer. </p><p>But does a warp drive act in this way? And if it does, what is its inertia?</p><p>Recall that the means we are using to create these solitons is intrinsically different as we are not accelerating objects through space-time, but manipulating spacetime itself to alter a ship's separation from an object at some arbitrary speed. As the concept of inertia is rooted in the former (the motion of objects through space-time), we must reconsider what inertia means in the context of warp drives. Since there is no well-defined acceleration mechanism in the literature for a warp drive, and I will not go to the trouble here of solving both the Einstein equation and the stress-energy-momentum conservation law in order to produce one, let us consider the difference between two warp drives at different velocities assuming that there is some acceleration-like process that can connect them.</p><p>Let us define inertia for a warp drive soliton as a normalized ratio of the infinitesimal change in momentum over infinitesimal change in energy of a soliton having experienced change in velocity along its original direction of motion</p><p style="text-align: center;">m = <b>r</b>.<span face="sans-serif" style="background-color: #f8f9fa; color: #202122; font-size: 12.32px; text-align: left;">δ</span><b>p</b>/<span face="sans-serif" style="background-color: #f8f9fa; color: #202122; font-size: 12.32px; text-align: left;">δ</span>E, </p><p>where <b>r</b> = <b>v</b> (c/|<b>v</b>|)<sup>2</sup> and <b>v</b> is in the velocity. Note that in the limit of a point-like particle moving through space-time, this quantity is the same as the particle mass.</p><p>Consider a warp bubble from my paper <a href="https://doi.org/10.1088/1361-6382/abe692" style="font-style: italic;" target="_blank">Breaking the Warp Barrier</a> of a given velocity <b>v</b>. The (Eulerian) energy of that bubble is given by E(|<b>v</b>|) and its (Eulerian) momentum is <b>p</b> = <b>0</b>, both calculated through the Einstein equation. The zero momentum of the soliton is something of a surprise considering the non-zero energy required to make the bubble. This is a result of the way this warp bubble is constructed, and indicates that the medium sourcing the soliton in net is <b>locally co-moving</b> with the bubble. It does not imply that nothing inside the bubble is moving. </p><p>Therefore, a change in soliton velocity would produce no change in momentum. The soliton has <b>zero inertia </b>according to the above definition, and may not be limited to subluminal speeds by that consideration alone. This is not to say that this warp bubble could easily be accelerated to superluminal speeds. There is still a challenge in the form of the dominant energy condition (DEC), which states that sources viewed from any future-pointing reference frame cannot locally move faster than light, which this soliton still violates as it nears the speed of light. I plan to discuss the DEC and the challenge it presents to FTL in more detail in a future post.</p><p>Have a good Tuesday,</p><p>Erik </p>Erikhttp://www.blogger.com/profile/12492752467374888110noreply@blogger.com17tag:blogger.com,1999:blog-3842073573230005449.post-8822772102679689422021-11-26T12:29:00.000-08:002021-11-26T12:29:07.275-08:00Reading Recommendations for Studying Warp Drives<p> Hello All,</p><p>I have received dozens of emails in the past months asking me for what essentially amounts to a warp drive course curriculum. This post contains a list of background reading I would recommend for those interested in studying and researching warp drives. Many technical topics have contributed to my research in this area so far, but I will narrow my selection to those that were the most useful in the research that led to <i><a href="https://iopscience.iop.org/article/10.1088/1361-6382/abe692">Breaking the Warp Barrier</a></i> (also at <a href="https://arxiv.org/abs/2006.07125v2">https://arxiv.org/abs/2006.07125v2</a>). These works are written at or above the level of upper undergraduate studies in physics or mathematics. Some of the text books are expensive, but may be available at your local library or through an inter-library loan (WorldCat links provided). Where available, links to freely available pre-prints are provided in addition to journal links for research articles.</p><p>I may expand on this list at a later date as my work in this area expands.</p><p><b>Text Books:</b></p><p></p><ul style="text-align: left;"><li><i><a href="https://www.worldcat.org/title/first-course-in-general-relativity/oclc/1273677107&referer=brief_results" target="_blank">A First Course in General Relativity</a>,</i> Bernard Schutz </li><li><a href="https://www.worldcat.org/title/introduction-to-differentiable-manifolds-and-riemannian-geometry/oclc/797239105&referer=brief_results" target="_blank">An Introduction to Differentiable Manifolds and Riemannian Geometry</a>, William Boothby </li><li><i><a href="https://www.worldcat.org/title/general-relativity/oclc/554287751&referer=brief_results" target="_blank">General Relativity</a></i>, Robert Wald</li><li><i><a href="https://www.worldcat.org/title/gravitation/oclc/1099299350&referer=brief_results" target="_blank">Gravitation</a></i>, Charles Misner, Kip Thorne, and John Wheeler</li><li><i><a href="https://www.worldcat.org/title/introduction-to-31-numerical-relativity/oclc/869682792&referer=brief_results" target="_blank">Introduction to 3+1 Numerical Relativity</a></i>, Miguel Alcubierre</li><li><i><a href="https://www.worldcat.org/title/classical-theory-of-fields/oclc/836899768&referer=brief_results" target="_blank">The Classical Theory of Fields, Course of Theoretical Physics Volume 2</a></i>, Lev Landau and Evgenii Lifshitz</li><li><i><a href="https://www.worldcat.org/title/methods-of-mathematical-physics-by-r-courant-and-d-hilbert/oclc/874264286" target="_blank">Methods of Mathematical Physics</a></i>, Richard Courant and David Hilbert</li></ul><p></p><p><b>Lecture Notes:</b></p><p></p><ul style="text-align: left;"><li><i><a href="https://arxiv.org/pdf/gr-qc/0703035.pdf" target="_blank">3+1 Formalism and Bases of Numerical Relativity</a></i>, Eric Gourgoulhon </li></ul><p></p><p><b>Research Papers:</b></p><p></p><ul style="text-align: left;"><li><i><a href="https://iopscience.iop.org/article/10.1088/0264-9381/11/5/001/pdf" target="_blank">The Warp Drive: Hyper-Fast Travel Within General Relativity</a></i>, Miguel Alcubierre (<a href="https://arxiv.org/abs/gr-qc/0009013v1">https://arxiv.org/abs/gr-qc/0009013v1</a>)</li><li><i><a href="https://arxiv.org/abs/gr-qc/0110086v3" target="_blank">Warp Drive with Zero Expansion</a></i>, Jose Natario (<a href="https://arxiv.org/abs/gr-qc/0110086">https://arxiv.org/abs/gr-qc/0110086</a>)</li><li><a href="https://journals.aps.org/prd/abstract/10.1103/PhysRevD.67.104013" target="_blank">Quantum inequalities do not forbid space-time shortcuts</a>, Sergey Krasnikov (<a href="https://arxiv.org/abs/gr-qc/0207057">https://arxiv.org/abs/gr-qc/0207057</a>)</li><li><a href="https://ui.adsabs.harvard.edu/abs/1983PhR....96..205B/abstract" target="_blank"><i>General Relativistic Axisymmetric Rotating Systems: Coordinates and Equations</i></a>, James Bardeen and Tsvi Piran</li></ul><p></p><p>Enjoy your Friday,</p><p>Erik</p>Erikhttp://www.blogger.com/profile/12492752467374888110noreply@blogger.com3tag:blogger.com,1999:blog-3842073573230005449.post-17459247431430750242021-06-13T09:23:00.001-07:002021-06-13T09:23:37.806-07:00The Judgement Call Podcast<p> Hello All,</p><p>I was on The Judgement Call Podcast with Torsten Jacobi recently. The recording can be found on the podcast website (<a href="https://judgmentcallpodcast.com/2021/06/94-erik-lentz-how-to-build-an-actual-warp-drive/">https://judgmentcallpodcast.com/2021/06/94-erik-lentz-how-to-build-an-actual-warp-drive/</a>) and YouTube (<a href="https://www.youtube.com/watch?v=ZYBD5z_LDSQ">https://www.youtube.com/watch?v=ZYBD5z_LDSQ</a>). We touched on a number of topics, including faster than light space travel, extra dimensions, the simulation hypothesis, and non-local phenomena. Torsten was an engaging host and, while I am not an expert on every topic we discussed, we had fun exploring possibilities.</p><p>Have a great Sunday,</p><p>Erik</p>Erikhttp://www.blogger.com/profile/12492752467374888110noreply@blogger.com13tag:blogger.com,1999:blog-3842073573230005449.post-15081390571869822992021-05-11T18:46:00.008-07:002021-11-02T19:54:49.549-07:00Response to new paper: "Generic warp drives violate the null energy condition"<p> Hello all,</p><p>I was pointed to a paper that appeared on the arXiv yesterday (<a href="https://arxiv.org/abs/2105.03079v1">https://arxiv.org/abs/2105.03079v1</a>). I am glad to see people taking a close look into my paper (<a href="https://doi.org/10.1088/1361-6382/abe692">https://doi.org/10.1088/1361-6382/abe692</a>, and the latest arXiv version <a href="https://arxiv.org/abs/2006.07125v2">https://arxiv.org/abs/2006.07125v2</a>) and the other recent papers on the topic of warp drives (<a href="https://doi.org/10.1088/1361-6382/abdf6e">https://doi.org/10.1088/1361-6382/abdf6e</a>, <a href="https://arxiv.org/abs/2104.06488">https://arxiv.org/abs/2104.06488</a>). I have also been contacted in the last months by several other researchers working to reproduce and expand upon my results. I wanted to make a running post on this paper and a series of posts on other appraisals of my work in this field and the resulting discussions.</p><p>After reading through their entire paper, I noticed that this new manuscript seems to have overlooked my complete discussion of the weak energy condition (WEC) contained in my published paper, instead referring to an early arXiv manuscript that tracked quantities of energy, momentum, etc. in only an Eulerian frame. This single frame of reference does not cover the breadth of the WEC, which requires an examination of the energy from the reference frames of all time-like observers. This is the new paper's stated central issue with my work, that it does not fully address the WEC. Fortunately, this limitation was addressed in the peer-review process and the final version of the paper published by <i>Classical and Quantum Gravity </i>contains a presentation of the WEC in full. </p><p>I contacted the authors of the new paper to point this out and they supplied a reply this evening as I was writing this post. I will make an update when I have had a chance to think through their new comments. At this point, I will say that the disagreement has yet to be resolved. </p><p><br /></p><p>Enjoy your Tuesday evening,</p><p>Erik</p><p><b>Update (6/25/21):</b></p><p>The conversation with the authors of `Generic Warp Drives' continued for a few days after the initial post. The reply to my initial message reiterated and elaborated on their positions in their manuscript. These new details that much of the confusion was regarding the origin of some components used in the WEC calculations. The authors assumed that the stress tensor principle values had been introduced <i>ad hoc </i>using an assumed form of the stress-energy from the plasma instead of directly from the soliton geometry and therefore did not demonstrate non-negative energy. In my response, I pointed out that every term used in the calculation was derived directly from the soliton's space-time metric, and therefore was a valid test of the WEC. There were other comments addressed in the correspondence, but this was the central point.</p><p>Also, the same group put out another paper this month (<a href="https://arxiv.org/abs/2106.05002">https://arxiv.org/abs/2106.05002</a>), as indicated in the comments. The new manuscript introduces engineered metrics in the form of beams of space-time curvature that can induce stresses such as pressure or tension forces or more general shearing stresses onto objects that pass into the beam path. This is an interesting paper exploring another science-fiction technology (tractor beams) within the scope of general relativity. I am still absorbing the paper's contents and may have more to say soon.</p><p><b>Update (7/25/21):</b></p><p>I recently gave a talk the (virtual) 16th Marcel Grossmann meeting (<a href="http://www.icra.it/mg/mg16/">http://www.icra.it/mg/mg16/</a>) where there was a mini-session warp drives. I mention this here as Matt Visser, one of the authors to the two papers above, was also invited to give a talk in that session. Dr. Visser mostly focused on the more recent tractor/pressor/stressor beam (<a href="https://arxiv.org/abs/2106.05002">https://arxiv.org/abs/2106.05002</a>), but also took the opportunity to review the shortcomings of the recent warp drive papers (Fell/Heisenberg, Bobrick/Martire, and my own). My own talk directly followed Dr. Visser's, allowing me to directly address his comments and demonstrate ow my soliton persists as an example of a positive energy warp drive. Discussion continued in the Q&A, where I think we were able to make progress.</p><p>You can watch the whole session on YouTube, which was recently released by the International Center for Relativistic Astrophysics Network (<a href="https://youtu.be/NqN1c-2fv8Y">https://youtu.be/NqN1c-2fv8Y</a>).</p><p><b>Update (11/2/21):</b></p><p>The 16th Marcel Grossmann meeting is putting out a proceedings, where speakers submit an article on the topic of their talk and the set is published as a collection (in this case by World Scientific). Matt Visser and his co-authors posted their contribution to the proceedings on the arXiv (<a href="https://arxiv.org/abs/2110.14926">https://arxiv.org/abs/2110.14926</a>). Unfortunately, not much from our conversation at the meeting made its way into the article, which essentially follows the contents of their tractor/pressure/stressor beam paper.</p>Erikhttp://www.blogger.com/profile/12492752467374888110noreply@blogger.com24tag:blogger.com,1999:blog-3842073573230005449.post-47243767435358802282021-04-10T17:50:00.002-07:002021-04-10T17:50:43.575-07:00Saturday Feature: Article in The Academic Times<p> Hello all,</p><p>An article about my warp drive paper came out recently, and I wanted to share it here. The article provides some further commentary made by other physicists about my paper, so it gives a better sense for where the topic stands among physicists. I was interviewed for this article and thought it was well done.</p><p>Here you go:</p><p><a href="https://academictimes.com/this-design-for-a-faster-than-light-warp-drive-is-making-waves-but-physicists-disagree-on-whether-its-possible/" target="_blank">https://academictimes.com/this-design-for-a-faster-than-light-warp-drive-is-making-waves-but-physicists-disagree-on-whether-its-possible/</a></p><p>Happy Saturday,</p><p>Erik</p>Erikhttp://www.blogger.com/profile/12492752467374888110noreply@blogger.com4tag:blogger.com,1999:blog-3842073573230005449.post-60156963617676677442021-04-07T18:41:00.000-07:002021-04-07T18:41:27.340-07:00Review and Response to Bobrick and Martire 2021Hello all,<br /><br />I have received quite a few requests by email, interview, reddit, social media, and even this blog for a response to the paper "Introducing Physical Warp Drives" (IPWD) by Alexey Bobrick and Gianni Martire (https://doi.org/10.1088/1361-6382/abdf6e). There has also some confusion about the difference between the papers. This paper came out very near to my own 'warp drive' paper (Hyper-Fast) and in the same journal, <i>Classical and Quantum Gravity</i>. This post addresses questions I have received about IPWD and its reference to an unfinished version of my own paper. Suffice it to say, I am somewhat disappointed in their assessment. For ease of access, I will be working from the most recent arXiv version of IPWD (https://arxiv.org/abs/2102.06824v2).<div><div><div><br /></div><div>For disclosure, I have been in contact with the authors since July 2020 when I gave a virtual seminar on my own paper to CENTRA (Center for Astrophysics and Gravitation) in Lisbon, Portugal (<a href="https://tecnico.ulisboa.pt/en/events/centra-seminar-erik-w-lentz/">https://tecnico.ulisboa.pt/en/events/centra-seminar-erik-w-lentz/</a>). We exchanged several messages and discussed our respective papers over Zoom during the following months.</div><h2 style="text-align: left;">Questions on IPWD have fallen into three topics:</h2><h3 style="text-align: left;"><b>What are the differences between IPWD and Hyper-Fast?</b></h3><div>The IPWD paper contains a review of the warp drive literature, defines a "general warp drive spacetime", studies a set of spherical (Schwarzschild) models of subluminal warp drives with positive energies as well as a set of axis-symmetric (Alcubierre) negative energy warp drives with some optimizations between shape and energy, and provides some comments concerning the acceleration of warp drives. </div><div><br /></div><div>My paper concentrates on deriving sufficient conditions for constructing a class of hyper-fast solitons (warp drives) that are capable of moving at an arbitrary speed (subluminal or superluminal) and are sourced by only positive energy densities. The process of constructing these new spacetimes is discussed rigorously in explicit mathematical detail. One example of a hyper-fast soliton is provided, and the properties of the sourcing energy and stresses are discussed in the context of a plasma.<span style="color: red;"> </span></div><h3 style="text-align: left;"><b>What is your take on the IPWD results?</b></h3><div>Overall I appreciated their review treatment of the published literature to date, though I would have liked to have seen more technical detail throughout the rest of the paper. I found myself particularly interested in the discussion of acceleration among warp drives and stress-energy-momentum conservation, but was ultimately let down by the depth of discussion, which could effectively be summarized as 'put a rocket on it'.</div><div><br /></div><div>The motivation for positive energy warp drives being limited to subluminal speeds seemed ad hoc, and the result seems to simply fall in line with previous work. I would expect a more thorough treatment of positive-energy soliton spacetimes to motivate their claim of a limit at light speed beyond the consideration of spherically symmetric (Schwarzschild) spacetimes.</div><div><br /></div><div>Additional questions I have:</div><div><ul style="text-align: left;"><li>What are the conditions on the spacetime geometry, particularly the metric's shift vector, due to stress-energy-momentum conservation?</li><li>How does the IPWD's so-called general warp drive model include my solution? The discussion in Section 2 is unclear on this point.</li></ul></div><h3 style="text-align: left;"><b>Do you have a response to IPWD's reference of Hyper-Fast?</b></h3><div><div>IPWD references my paper several times. References included statements that my findings of a positive energy soliton in general relativity capable of traveling superluminaly are done "without providing means to reproduce the study" and that the conclusions of IPWD "do not support the recent claim" of my paper. I am disappointed by this, considering the great detail I put into the manuscript laying out the conditions for creating such solutions.</div><div><br /></div><div><div>Much of the meat of Hyper-Fast is in the description of the underlying mathematics of the solution geometry, and is very audit-able. Further, I gathered feedback from colleagues and experts in general relativity before publication and added further detail to the manuscript through the peer-review process, refining the results. Mostly, I am disappointed that their paper made a rush to judgement by criticizing a draft version of my paper and not the full peer-reviewed published article. The authors did not contact me to check what updates may have been made to my manuscript during the months between our first contact and the posting date of IPWD by the journal or on the arXiv.</div></div><div><br /></div><div>Additional questions I have:</div><div><ul style="text-align: left;"><li>What further details would be helpful to understand and reasonably reproduce my positive energy soliton?</li><li>Is this satisfied by the published version of the paper?</li></ul></div></div><div>It is an exciting time for space travel! Moving forward, it is my hope that we can thoroughly examine all possibilities and help make warp drive a reality.</div><div><br /></div><div></div><div>Have a happy Wednesday,</div></div><div><br /></div><div>Erik</div></div>Unknownnoreply@blogger.com1tag:blogger.com,1999:blog-3842073573230005449.post-52095373021959603742021-03-25T17:58:00.003-07:002021-03-25T18:05:41.817-07:00Reddit Q&A<p> Hello all,</p><p>As mentioned in an earlier post, on March 9 there was some activity on /r/science on the press release for my warp drive paper (<a href="https://www.reddit.com/r/science/comments/m1gyyi/breaking_the_warp_barrier_for_fasterthanlight/)" style="background: rgb(255, 255, 255); color: #2196f3; font-family: Roboto, sans-serif; font-size: 15px; outline: 0px; text-decoration-line: none;" target="_blank">https://www.reddit.com/r/science/comments/m1gyyi/breaking_the_warp_barrier_for_fasterthanlight/</a>). Several questions came up that I thought required some follow-up. So instead of outing myself on Reddit, I will address here those questions that I can answer quickly. Others might appear in their own posts in the future.</p><p>1) Phonons as a source of negative mass:</p><div><iframe height="181" id="reddit-embed" sandbox="allow-scripts allow-same-origin allow-popups" scrolling="no" src="https://www.redditmedia.com/r/science/comments/m1gyyi/breaking_the_warp_barrier_for_fasterthanlight/gqeevvd/?depth=1&showmore=false&embed=true" width="640"></iframe></div><div>Theories of phonons (essentially density waves in a medium) can be described as an effective (quasi-)particle field with negative mass in the sense that they are repelled by gravitational fields of massive bodies such as the Earth. Examples of phonons include the sound waves in air, or most any other pressure/density waves over a massive, elastic, and nearly homogeneous medium. (<a href="https://en.wikipedia.org/wiki/Phonon">https://en.wikipedia.org/wiki/Phonon</a>). However, phonons are not likely to provide the "exotic matter" needed by the Alcubierre and other earlier warp drive models as the total mass density of the medium plus phonons will still be positive.</div><div><br /></div><div>The jury is still out on warp plasma in general.</div><div><br /></div><div>2) & 3) Creating the soliton and in-space collisions:</div><div><iframe height="270" id="reddit-embed" sandbox="allow-scripts allow-same-origin allow-popups" scrolling="no" src="https://www.redditmedia.com/r/science/comments/m1gyyi/breaking_the_warp_barrier_for_fasterthanlight/gqfnocc/?depth=1&showmore=false&embed=true" width="640"></iframe></div><div>Two great questions here. Not sure if I can ELI5, but I will do my best to be understandable.</div><div><br /></div><div>The paper does not cover how to create/accelerate one of these solitons/bubbles. They are 'constructed' mathematically to exist <i>ad infinitum</i>, meaning forever. Answering the question of how these solitons are created and accelerated will be crucial to forming experiments and addressing the horizon problem. </div><div>Note that the means of acceleration will still need to obey the relativity version of momentum conservation (stress-energy-momentum conservation).</div><div><br /></div><div>For the second question, gas, dust, and other objects impacted by the bubble has been discussed in the literature, at least for the Alcubierre drive. One study (<a href="https://arxiv.org/abs/1202.5708v1">https://arxiv.org/abs/1202.5708v1</a>) shows that objects are given a boost in speed in the direction of the soliton. For warp bubbles moving faster than light, these objects can become "time locked" in the shell of a constant velocity bubble where they are stuck in the leading edge. I may write another post on this once I have a better sense for how colliding objects will behave in this new class of warp bubble.</div><div><br /></div><div>4) Time dilation and trip duration:</div><div><iframe height="297" id="reddit-embed" sandbox="allow-scripts allow-same-origin allow-popups" scrolling="no" src="https://www.redditmedia.com/r/science/comments/m1gyyi/breaking_the_warp_barrier_for_fasterthanlight/gqeh7nh/?depth=1&showmore=false&embed=true" width="640"></iframe></div><div>Okay, so you kind of got me on a technicality here: the trip time as viewed from the ship traveling through space-time at near light speed can be shortened to a human lifetime by dilation effects. An earlier version of the press release actually accounted for this possibility, and indicated that it was more desirable for both the ship crew and their friends and family back on Earth to be alive for a reunion on the ship's return. This was altered by the editors to the third paragraph (<a href="https://www.uni-goettingen.de/en/3240.html?id=6192">https://www.uni-goettingen.de/en/3240.html?id=6192</a>). Personally I would rather be able to come home to my family and friends.</div><div><br /></div><div>In any case, even if you could find a rocket propellant with a high enough specific impulse to accelerate a rocket for years on end (<a href="https://en.wikipedia.org/wiki/Specific_impulse">https://en.wikipedia.org/wiki/Specific_impulse</a>), you would still be faced with the conundrum: Is it worth it to take the trip if the technology that would surpass you is developed while you are en route?<br /></div><div><br /></div><div>5) Warp bubble miniaturization:</div><div><iframe height="160" id="reddit-embed" sandbox="allow-scripts allow-same-origin allow-popups" scrolling="no" src="https://www.redditmedia.com/r/science/comments/m1gyyi/breaking_the_warp_barrier_for_fasterthanlight/gqfkcnp/?depth=1&showmore=false&embed=true" width="640"></iframe></div><div>This is actually the first step of several of the past energy reduction techniques ( see <a href="https://arxiv.org/abs/gr-qc/9905084v5">https://arxiv.org/abs/gr-qc/9905084v5</a> and <a href="https://arxiv.org/abs/gr-qc/0207057v3">https://arxiv.org/abs/gr-qc/0207057v3</a>). The equation for the total energy required scales as</div><div><br /></div><div><div class="separator" style="clear: both; text-align: center;">E_total ~ R<sup>2</sup>/w,</div><br />implying that by scaling down the radius (R) of the probe while keeping the bubble radius to shell width ratio (R/w) constant, that the energy scales linearly with bubble size. If we were to shrink the bubble from R=100m to R=1mm, the size of our miniature robotic ship. The energy would reduce by a factor of 100,000. This is still on the order of some large fraction of the Earth's mass. Plus, you would still have to find a way to put all that energy into such a confined space. Probably best to wait until the energy needs are reduced by some tens of orders of magnitude.</div><div><br /></div><div><br /></div><div>Have a good Thursday.</div><div><br /></div><div>Erik</div>
Erikhttp://www.blogger.com/profile/12492752467374888110noreply@blogger.com0tag:blogger.com,1999:blog-3842073573230005449.post-25790864588019403272021-03-14T10:59:00.000-07:002021-03-14T10:59:09.699-07:00Dr. John Cramer's "The Lentz Soliton FTL Drive" Article<p>Hello all, </p><p>I wanted to write a short post to acknowledge the first online article I found to talk about my hyper-fast soliton work. Back in November, long before the press release while my manuscript was still under peer review, Dr. John Cramer wrote a perspective on my paper in his long-standing science column "The Alternate View" in <i>Analog: Science Fiction and Fact Magazine</i>. John also happens to be professor emeritus at my own Ph.D. institution, the University of Washington. You can find the article in the archives of the Analog website<span style="color: #202124;"><span style="background-color: white; font-variant-ligatures: none; letter-spacing: 0.1px; white-space: pre-wrap;"> (</span><span><a href="https://www.analogsf.com" style="font-variant-ligatures: none; letter-spacing: 0.1px; white-space: pre-wrap;" target="_blank">https://www.analogsf.com</a><span style="font-variant-ligatures: none; letter-spacing: 0.1px; white-space: pre-wrap;">). </span></span></span>John also keeps a repository of his past articles at UW where the "The Lentz Soliton FTL Drive" can be found <span style="color: #202124;"><span><span style="font-variant-ligatures: none; letter-spacing: 0.1px; white-space: pre-wrap;">(</span><a href="https://www.npl.washington.edu/av/altvw209.html" style="font-variant-ligatures: none; letter-spacing: 0.1px; white-space: pre-wrap;" target="_blank">https://www.npl.washington.edu/av/altvw209.html</a><span style="font-variant-ligatures: none; letter-spacing: 0.1px; white-space: pre-wrap;">). </span></span></span>While there, you can also check out his other columns<span style="color: #202124;"><span><span style="font-variant-ligatures: none; letter-spacing: 0.1px; white-space: pre-wrap;"> (</span><a href="https://www.npl.washington.edu/av/av_index.html" style="font-variant-ligatures: none; letter-spacing: 0.1px; white-space: pre-wrap;" target="_blank">https://www.npl.washington.edu/av/av_index.html</a><span style="font-variant-ligatures: none; letter-spacing: 0.1px; white-space: pre-wrap;">).</span></span></span></p><p>Have a good Sunday.</p><p>Erik</p>Unknownnoreply@blogger.com1tag:blogger.com,1999:blog-3842073573230005449.post-22909451175744368392021-03-11T13:29:00.004-08:002021-03-11T17:09:28.104-08:00Announcement of talk<h2 style="text-align: left;">Hello all,</h2><p>As mentioned earlier, I was invited a few months ago by a colleague at <a href="https://prescott.erau.edu/about/planetarium/public-shows" target="_blank">Embry-Riddle Aeronautical University</a> in Prescott to give a live talk about my work. This is my alma mater, and I am happy to "be back", even if it is just virtually.</p><h4 style="text-align: left;">Here is a summary of my upcoming talk:</h4><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><p style="text-align: justify;">Solitons in space–time capable of transporting time-like observers at superluminal speeds have long been tied to violations of the weak, strong, and dominant energy conditions of general relativity. The negative-energy sources required for these solitons must be created through energy-intensive uncertainty principle processes as no such classical source is known in particle physics. This talk presents an approach for overcoming this barrier, explicitly constructing a class of soliton solutions that are capable of superluminal motion and sourced by purely positive energy densities.</p></blockquote><h4 style="text-align: left;"><span style="color: #0b5394;">Location:</span> Jim and Linda Lee Planetarium (virtually through YouTube)</h4><h4 style="text-align: left;"><span style="color: #0b5394;">Time: </span>12:00 PST (please check the link for your local time because daylight savings time begins in the USA on March 14, 2021)</h4><h4 style="text-align: left;"><span style="color: #0b5394;">Date: </span>March 18, 2021</h4><h4 style="text-align: left;"><span style="color: #0b5394;">Link: <a href="https://www.youtube.com/watch?v=6O8ji46VBK0">https://www.youtube.com/watch?v=6O8ji46VBK0</a></span></h4><div><br /></div><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhFIFwcmpxrBNqNzW32qxHB5EFaedeBKWWWM22uHoPPb-KUOV-l2VGRCrD1MkwY3ZMfgjaUBSuZS4OPvOSuSSx6jt5EZyybjOfjrvqTN7kyhjrdIJw49wyCRB3sVP1zHR8_pqDdxjgXmiI/s771/1614017128902.jpeg" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="575" data-original-width="771" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhFIFwcmpxrBNqNzW32qxHB5EFaedeBKWWWM22uHoPPb-KUOV-l2VGRCrD1MkwY3ZMfgjaUBSuZS4OPvOSuSSx6jt5EZyybjOfjrvqTN7kyhjrdIJw49wyCRB3sVP1zHR8_pqDdxjgXmiI/s320/1614017128902.jpeg" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">An image from a previous talk I gave (<b style="color: #212121; font-family: times; font-size: 14.6667px; font-variant-ligatures: none; text-align: left; white-space: pre-wrap;"><a href="https://eriklentzphd.blogspot.com/p/video-of-talks.html" target="_blank">"Breaking the Warp Barrier: Hyperfast Solitons in Einstein-Maxwell-Plasma Theory", CENTRA Seminar, Lisbon, Portugal, 23 July, 2020</a>)</b> illustrating a potential study of soliton morphology and spacecraft designs</td></tr></tbody></table><br /><p><br /></p><p>Feel free to leave me a comment here after the talk, if you have any questions and I can try to address them in a future blog post. </p><p>I'm back to a very busy week and hope you watch the talk <span face="Roboto, arial, sans-serif" style="background-color: white; color: #202124; font-size: 16px;">– </span>either live or afterwards!</p><p>Erik</p>Unknownnoreply@blogger.com4tag:blogger.com,1999:blog-3842073573230005449.post-21988748200098404552021-03-10T10:52:00.005-08:002021-03-11T17:11:16.528-08:00Reddit response and answers to some questions<h2 style="text-align: left;">Hey all,</h2><p>Exciting morning for me: I found out that my warp drive paper was the top subject on the /r/science subreddit at one point today (for the published version that requires a subscription to access go here: <a href="https://iopscience.iop.org/article/10.1088/1361-6382/abe692/meta" target="_blank">https://iopscience.iop.org/article/10.1088/1361-6382/abe692/meta</a> or for a draft version on the arXiv go here: <a href="https://arxiv.org/abs/2006.07125v2" target="_blank">https://arxiv.org/abs/2006.07125v2</a>) .</p><h2 style="text-align: left;">Reddit link (top one)</h2><p>Here is the main Reddit link: <a href="https://www.reddit.com/r/science/comments/m1gyyi/breaking_the_warp_barrier_for_fasterthanlight/)" target="_blank">https://www.reddit.com/r/science/comments/m1gyyi/breaking_the_warp_barrier_for_fasterthanlight/</a></p><p>I have gotten <b>a lot </b>of great comments and plan to respond to some of them soon-ish. It is a really exciting to see all this interest!</p><h2 style="text-align: left;">Response to some questions</h2><p>Also I have had quite a few questions about my response to another recent article (<a href="https://arxiv.org/pdf/2102.06824.pdf" target="_blank">https://arxiv.org/pdf/2102.06824.pdf</a>), which is critical of my work as of a pre-print I made available in the summer of 2020. I plan to dedicate an entire to post to this, since I have spoken to the group behind the paper personally about my research, but want to provide a thoughtful answer and so that is taking longer. For now, I will just say the following:</p><p>This other paper was accepted for publication before my final draft was available to read, so the authors did not have an opportunity to see the clarifications and additional work I added due to the peer review process. I was very detailed in my presentation of my approach, and consulted with a number of experts in general relativity (see the paper's acknowledgements). Additionally, I am working on making the supporting code available on GitHub after I find the time. </p><h2 style="text-align: left;">Next Steps</h2><p>My next steps will be (1) responding to some of the 2000+ comments at <a href="https://www.reddit.com/r/science/comments/m1gyyi/breaking_the_warp_barrier_for_fasterthanlight/)" target="_blank">https://www.reddit.com/r/science/comments/m1gyyi/breaking_the_warp_barrier_for_fasterthanlight/</a>, (2) working on some blog posts, and (3) working on a number of other projects. </p><p>Thanks for all of the comments and I hope that this new interest in faster-than-light travel continues and grows! Feel free to hop on the <a href="https://www.reddit.com/r/science/comments/m1gyyi/breaking_the_warp_barrier_for_fasterthanlight/," target="_blank">subreddit </a>and add your comment there or just add one below!</p><p>Have a good Wednesday,</p><p>Erik</p>Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-3842073573230005449.post-22323679304374875742021-03-03T17:35:00.007-08:002021-03-11T17:07:09.942-08:00Erik Lentz's first announcements<h2 style="text-align: left;"> Hello all, </h2><p>This first post is made on the date of this blog's creation. More posts are soon to follow, including</p><p></p><ul style="text-align: left;"><li>Upcoming online talk to be given on 18 March 2021 at 3pm Eastern Daylight Time (please check the YouTube page for your local time) for the <br />Science Speaker Series at the <a href="https://www.youtube.com/channel/UCVBTREXW9YnUpypzj2pMA4w">Jim and Linda Lee Planetarium</a>: <a data-saferedirecturl="https://www.google.com/url?q=https://youtu.be/6O8ji46VBK0&source=gmail&ust=1613973830367000&usg=AFQjCNGoh0XTqt_SHhw8PaOdMV1n3MQrZg" href="https://youtu.be/6O8ji46VBK0" rel="nofollow" style="background-color: white; border: 0px; box-shadow: none; box-sizing: border-box; color: #3a3a3a; font-family: Roboto, sans-serif; font-size: 15px; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: none; transition: all 0.2s linear 0s; vertical-align: baseline;" target="_blank">https://youtu.be/6O8ji46VBK0</a></li></ul><div class="separator" style="clear: both; text-align: center;"><a href="https://www.youtube.com/watch?v=6O8ji46VBK0&ab_channel=JimandLindaLeePlanetarium" style="margin-left: 1em; margin-right: 1em;" target="_blank"><img alt="Erik Lentz YouTube talk time" border="0" data-original-height="188" data-original-width="445" height="169" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_bASeTYHtT9QzEjxrkcYKi1bqXYIE1NfHQH_rri2RVCchNofwOTe8OLAKXFeW3vVGiLuD1UfUek0mBQ8FlB21CruCxIeMQDJvooUoXdFi9-ogREh3p6msyPYFtlz_K0maJz7iIZzsFSE/w400-h169/blog+post+1+image+1.png" width="400" /></a></div><br /><div><br /></div><ul style="text-align: left;"><li>Upcoming paper "Breaking the warp barrier: Hyper-fast solitons in Einstein-Maxwell-plasma theory" accepted to the journal <i><a href="https://iopscience.iop.org/journal/0264-9381" target="_blank">Classical and Quantum Gravity</a>: </i><a href="https://iopscience.iop.org/article/10.1088/1361-6382/abe692/meta" target="_blank">https://iopscience.iop.org/article/10.1088/1361-6382/abe692/meta</a></li></ul><div>For now, check me out on <a href="https://www.linkedin.com/in/erik-lentz-0a7a1539/">LinkedIn</a>.</div><div><br /></div><div>Back to my busy week!</div><div><br /></div><div>Erik </div><p></p>Unknownnoreply@blogger.com9